Basile Herlemont
email: herlemont.basile@hotmail.fr

Oleg Ogievetsky

Dif ferential Calculus on h-Deformed Spaces

Keywords: differential operators, Yang-Baxter equation, reduction algebras, universal enveloping algebra, representation theory, Poincaré-Birkhoff-Witt property, rings of fractions 2010 Mathematics Subject Classification: 16S30, 16S32, 16T25, 13B30, 17B10, 39A14

We construct the rings of generalized differential operators on the h-deformed vector space of gl-type. In contrast to the q-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators Diff h,σ (n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings Diff h,σ (n).

Introduction

As the coordinate rings of q-deformed vector spaces, the coordinate rings of h-deformed vector spaces are defined with the help of a solution of the dynamical Yang-Baxter equation. The coordinate rings of h-deformed vector spaces appeared in several contexts. In [START_REF] Bytsko | q-analog of model space and the Clebsch-Gordan coefficients generating matrices[END_REF] it was observed that such coordinate rings generate the Clebsch-Gordan coefficients for GL [START_REF] Bergman | The diamond lemma for ring theory[END_REF]. These coordinate rings appear in the study of the cotangent bundle to a quantum group [START_REF] Alekseev | a toy model for conformal field theory[END_REF] and in the study of zero-modes in the WZNW model [START_REF] Alekseev | a toy model for conformal field theory[END_REF][START_REF] Furlan | Quantum matrix algebra for the SU(n) WZNW model[END_REF][START_REF] Hadjiivanov | Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebras[END_REF].

The coordinate rings of h-deformed vector spaces appear naturally in the theory of reduction algebras. The reduction algebras [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF][START_REF] Mickelsson | Step algebras of semi-simple subalgebras of Lie algebras[END_REF][START_REF] Tolstoy | Fortieth anniversary of extremal projector method for Lie symmetries[END_REF][START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF] are designed to study the decompositions of representations of an associative algebra B with respect to its subalgebra B . Let B be the universal enveloping algebra of a reductive Lie algebra g. Let M be a g-module and B the universal enveloping algebra of the semi-direct product of g with the abelian Lie algebra formed by N copies of M . Then the corresponding reduction algebra is precisely the coordinate ring of N copies of h-deformed vector spaces.

We restrict our attention to the case g = gl(n). Let V be the tautological gl(n)-module and V * its dual. We denote by V (n, N ) the reduction algebra related to N copies of V and by V * (n, N ) the reduction algebra related to N copies of V * .

In this article we develop the differential calculus on the h-deformed vector spaces of gl-type as it is done in [START_REF] Wess | Covariant differential calculus on the quantum hyperplane[END_REF] for the q-deformed spaces. Formulated differently, we study the consistent, in the sense, explained in Section 3.2.1, pairings between the rings V (n, N ) and V * (n, N ). A consistent pairing allows to construct a flat deformation of the reduction algebra, related to N copies of V and N copies of V * . We show that for N > 1 or N > 1 the pairing is essentially unique. However it turns out that for N = N = 1 the result is surprisingly different from that for q-deformed vector spaces. The consistency leads to an over-determined system of finitedifference equations for a certain rational function σ, which we call "potential", in n variables. The solution space W can be described as follows. Let K be the ground ring of characteristic 0 and K[t] the space of univariate polynomials over K. Then W is isomorphic to K[t] n modulo the (n -1)-dimensional subspace spanned by n-tuples (t j , . . . , t j ) for j = 0, 1, . . . , n -2. Thus for each σ ∈ W we have a ring Diff h,σ (n) of generalized h-deformed differential operators. The polynomial solutions σ are linear combinations of complete symmetric polynomials; they correspond to the diagonal of K[t] n . The ring Diff h,σ (n) admits the action of the so-called Zhelobenko automorphisms if and only if the potential σ is polynomial.

In Section 2 we give the definition of the coordinate rings of h-deformed vector spaces of gl-type.

Section 3 starts with the description of two different known pairings between h-deformed vector spaces, that is, two different flat deformations of the reduction algebra related to V ⊕ V * . The first deformation is the ring Diff h (n) which is the reduction algebra, with respect to gl n , of the classical ring of polynomial differential operators. The second ring is related to the reduction algebra, with respect to gl n , of the algebra U(gl n+1 ). These two examples motivate our study. Then, in Section 3, we formulate the main question and results. We present the system of the finite-difference equations resulting from the Poincaré-Birkhoff-Witt property of the ring of generalized h-deformed differential operators. We obtain the general solution of the system and establish the existence of the potential. We give a characterization of polynomial potentials. We describe the centers of the rings Diff h,σ (n) and construct an isomorphism between a certain ring of fractions of the ring Diff h,σ (n) and a certain ring of fractions of the Weyl algebra. We describe a family of the lowest weight representations and calculate the values of central elements on them. We establish the uniqueness of the deformation in the situation when we have several copies of V or V * .

Section 4 contains the proofs of the statements from Section 3. Notation. We denote by S n the symmetric group on n letters. The symbol s i stands for the transposition (i, i + 1).

Let h(n) be the abelian Lie algebra with generators hi , i = 1, . . . , n, and U(n) its universal enveloping algebra. Set hij = hihj ∈ h(n). We define Ū(n) to be the ring of fractions of the commutative ring U(n) with respect to the multiplicative set of denominators, generated by the elements hij + a -1 , a ∈ Z, i, j = 1, . . . , n, i = j. Let

ψ i := k : k>i hik , ψ i := k : k<i hik and χ i := ψ i ψ i , i = 1, . . . , n. (1.1)
Let ε j , j = 1, . . . , n, be the elementary translations of the generators of U(n), ε j : hi → hi + δ j i . For an element p ∈ Ū(n) we denote ε j (p) by p[ε j ]. We shall use the finite-difference operators ∆ j defined by

∆ j f := f -f [-ε j ].
We denote by e L , L = 0, . . . , n, the elementary symmetric polynomials in the variables h1 , . . . , hn , and by e(t) the generating function of the polynomials e L ,

e L = i 1 <•••<i L hi 1 • • • hi L , e(t) = n L=0 e L t L = n i=1 1 + hi t . We denote by R ∈ EndŪ (n) Ū(n) n ⊗Ū (n) Ū(n) n the standard solution of the dynamical Yang-Baxter equation a,b,u R ij ab R bk ur [-ε a ] R au mn = a,b,u R jk ab [-ε i ] R ia mu R ub nr [-ε m ]
of type A. The nonzero components of the operator R are

R ij ij = 1 hij , i = j, and R ij ji =      h2 ij -1 h2 ij , i < j, 1, i ≥ j. (1.2)
We shall need the following properties of R:

R ij kl [ε i + ε j ] = R ij kl , i, j, k, l = 1, . . . , n, (1.3)
R ij kl = 0 if (i, j) = (k, l) or (l, k), (1.4) R 2 = Id. (1.5) We denote by Ψ ∈ EndŪ (n) Ū(n) n ⊗Ū (n) Ū(n) n the dynamical version of the skew inverse of the operator R, defined by k,l Ψ ik jl R ml nk [ε m ] = δ i n δ m j . (1.6)
The nonzero components of the operator Ψ are, see [START_REF] Khoroshkin | Diagonal reduction algebra and the reflection equation[END_REF],

Ψ ij ij = Q + i Q - j 1 hij + 1 , Ψ ij ji =      1, i < j, hij -1 2 hij hij -2 , i > j, (1.7)
where

Q ± i = χ i [±ε i ] χ i .

Coordinate rings of h-deformed vector spaces

Let F(n, N ) be the ring with the generators x iα , i = 1, . . . , n, α = 1, . . . , N , and hi , i = 1, . . . , n, with the defining relations

hi hj = hj hi , i, j = 1, . . . , n, (2.1) hi x jα = x jα hi + δ j i , i, j = 1, . . . , n, α = 1, . . . , N. (2.2)
We shall say that an element

f ∈ F(n, N ) has an h(n)-weight ω ∈ h(n) * if hi f = f hi + ω hi , i = 1, . . . , n. (2.
3)

The ring U(n) is naturally the subring of F(n, N ). Let F(n, N ) := Ū(n) ⊗ U(n) F(n, N ). The coordinate ring V(n, N ) of N copies of the h-deformed vector space is the factor-ring of F(n, N ) by the relations

x iα x jβ = k,l R ij kl x kβ x lα , i, j = 1, . . . , n, α, β = 1, . . . , N. (2.4)
The ring V(n, N ) is the reduction algebra, with respect to gl n , of the semi-direct product of gl n and the abelian Lie algebra V ⊕V ⊕• • •⊕V (N times) where V is the (tautological) n-dimensional gl n -module. According to the general theory of reduction algebras [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF][START_REF] Khoroshkin | Rings of fractions of reduction algebras[END_REF][START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF], V(n, N ) is a free left (or right) Ū(n)-module; the ring V(n, N ) has the following Poincaré-Birkhoff-Witt property:

given an arbitrary order on the set x iα , i = 1, . . . , n, α = 1, . . . , N , the set of all ordered monomials in x iα is a basis of the left Ū(n)-module V(n, N ).

(2.5)

Moreover, if {R kl ij } n i,j,k,l=1 is an arbitrary array of functions in hi , i = 1, . . . , n, then the Poincaré-Birkhoff-Witt property of the algebra defined by the relations (2.4), together with the weight prescriptions (2.2), implies that R satisfies the dynamical Yang-Baxter equation when N ≥ 3.

Similarly, let F * (n, N ) be the ring with the generators ∂iα , i = 1, . . . , n, α = 1, . . . , N , and hi , i = 1, . . . , n, with the defining relations (2.1) and hi ∂jα = ∂jα hi -δ j i , i, j = 1, . . . , n, α = 1, . . . , N.

(2.6)

Let F * (n, N ) := Ū(n) ⊗ U(n) F * (n, N ).
The h(n)-weights are defined by the same equation (2.3). The coordinate ring V * (n, N ) of N copies of the "dual" h-deformed vector space is the factor-ring of F * (n, N ) by the relations

∂lα ∂kβ = i,j ∂jβ ∂iα R ij kl , k, l = 1, . . . , n, α, β = 1, . . . , N. (2.7)
Again, the ring V * (n, N ) is the reduction algebra, with respect to gl n , of the semi-direct product of gl n and the abelian Lie algebra

V * ⊕ V * ⊕ • • • ⊕ V * (N times) where V * is the gl n -module, dual to V . The ring V * (n, N ) is a free left (or right) Ū(n)-module; it has a similar to V(n, N ) Poincaré-Birkhoff-Witt property:
given an arbitrary order on the set ∂iα , i = 1, . . . , n, α = 1, . . . , N , the set of all ordered monomials in ∂iα is a basis of the left Ū(n)-module V * (n, N ).

(2.8)

Again, the Poincaré-Birkhoff-Witt property of the algebra defined by the relations (2.7), together with the weight prescriptions (2.6), implies that R satisfies the dynamical Yang-Baxter equation when N ≥ 3.

For N = 1 we shall write V(n) and V * (n) instead of V(n, 1) and V * (n, 1).

3 Generalized rings of h-deformed dif ferential operators

Two examples

Before presenting the main question we consider two examples.

1. We denote by W n the algebra of polynomial differential operators in n variables. It is the algebra with the generators X j , D j , j = 1, . . . , n, and the defining relations

X i X j = X j X i , D i D j = D j D i , D i X j = δ j i + X j D i , i, j = 1, . . . , n.
The map, defined on the set {e ij } n i,j=1 of the standard generators of gl n by

e ij → X i D j ,
extends to a homomorphism U(gl n ) → W n . The reduction algebra of W n ⊗ U(gl n ) with respect to the diagonal embedding of U(gl n ) was denoted by Diff h (n) in [START_REF] Khoroshkin | Diagonal reduction algebra and the reflection equation[END_REF]. It is generated, over Ū(n), by the images x i and ∂ i , i = 1, . . . , n, of the generators X i and D i . Let

∂i := ∂ i ψ i ψ i [-ε i ] ,
where the elements ψ i are defined in (1.1). Then

x i ∂j = k,l ∂k R ki lj x l -δ i j σ (Diff) i , (3.1)
where σ (Diff) i = 1, i = 1, . . . , n. The h(n)-weights of the generators are given by (2.2) and (2.6). Moreover, the set of the defining relations, over Ū(n), for the generators x i and ∂i , i = 1, . . . , n, consists of (2.4), (2.7) (with N = 1) and (3.1) (see [START_REF] Khoroshkin | Diagonal reduction algebra and the reflection equation[END_REF]Proposition 3.3]).

The algebra Diff h (n, N ), formed by N copies of the algebra Diff h (n), was used in [START_REF] Khoroshkin | Mickelsson algebras and representations of Yangians[END_REF] for the study of the representation theory of Yangians, and in [START_REF] Khoroshkin | Diagonal reduction algebra and the reflection equation[END_REF] for the R-matrix description of the diagonal reduction algebra of gl n (we refer to [START_REF] Khoroshkin | Diagonal reduction algebras of gl type[END_REF][START_REF] Khoroshkin | Structure constants of diagonal reduction algebras of gl type[END_REF] for generalities on the diagonal reduction algebras of gl type).

2. Identifying each n × n matrix a with the larger matrix ( a 0 0 0 ) gives an embedding of gl n into gl n+1 . The resulting reduction algebra R

U(gl n+1 ) gl n , or simply R gl n+1 gl n
, was denoted by AZ n in [START_REF] Zhelobenko | Representations of reductive Lie algebras[END_REF]. It is generated, over Ū(n), by the elements x i , y i , i = 1, . . . , n, and hn+1 = z -(n + 1), where x i and y i are the images of the standard generators e i,n+1 and e n+1,i of U(gl n+1 ) and z is the image of the standard generator e n+1,n+1 . Let

∂i := y i ψ i ψ i [-ε i ] ,
where the elements ψ i are defined in (1.1) (they depend on h1 , . . . , hn only). The h(n)-weights of the generators are given by (2.2) and (2.6) while

hn+1 x i = x i hn+1 -1 , hn+1 ∂i = ∂i hn+1 + 1 , i = 1, . . . , n.
The set of the remaining defining relations consists of (2.4), (2.7) (with N = 1) and

x i ∂j = k,l ∂k R ki lj x l -δ i j σ (AZ) i , (3.2)
where

σ (AZ) i = -hi + hn+1 + 1, i = 1, . . . , n.
The algebra AZ n was used in [START_REF] Van Den Hombergh | Harish-Chandra modules and representations of step algebra[END_REF] for the study of Harish-Chandra modules and in [START_REF] Zhelobenko | Classical groups. Spectral analysis of finite-dimensional representations[END_REF] for the construction of the Gelfand-Tsetlin bases [START_REF] Gel | Finite-dimensional representations of the group of unimodular matrices[END_REF].

The algebra AZ n has a central element

h1 + • • • + hn + hn+1 . (3.3)
In the factor-algebra AZ n of AZ n by the ideal, generated by the element (3.3), the relation (3.2) is replaced by

x i ∂j = k,l ∂k R ki lj x l -δ i j σ (AZ) i , (3.4)
with

σ (AZ) i = -hi - n k=1
hk + 1, i = 1, . . . , n.

Main question and results

Main question

Both rings, Diff h (n) and AZ n satisfy the Poincaré-Birkhoff-Witt property. The only difference between these rings is in the form of the zero-order terms σ (Diff) i and σ

(AZ) i in the crosscommutation relations (3.1) and (3.4) (compare to the ring of q-differential operators [START_REF] Wess | Covariant differential calculus on the quantum hyperplane[END_REF] where the zero-order term is essentially -up to redefinitions -unique). It is therefore natural to investigate possible generalizations of the rings Diff h (n) and AZ n . More precisely, given n elements σ 1 , . . . , σ n of Ū(n), we let Diff h (σ 1 , . . . , σ n ) be the ring, over Ū(n), with the generators x i and ∂i , i = 1, . . . , n, subject to the defining relations (2.4), (2.7) (with N = 1) and the oscillator-like relations

x i ∂j = k,l ∂k R ki lj x l -δ i j σ i . (3.5)
The weight prescriptions for the generators are given by (2.2) and (2.6). The diagonal form of the zero-order term (the Kronecker symbol δ i j in the right hand side of (3.5)) is dictated by the h(n)-weight considerations.

We shall study conditions under which the ring Diff h (σ 1 , . . . , σ n ) satisfies the Poincaré-Birkhoff-Witt property. More specifically, since the rings V(n) and V * (n) both satisfy the Poincaré-Birkhoff-Witt property, our aim is to study conditions under which Diff

h (σ 1 , . . . , σ n ) is isomorphic, as a Ū(n)-module, to V * (n) ⊗Ū (n) V(n).
The assignment

deg x i = deg ∂i = 1, i = 1, . . . , n, (3.6)
defines the structure of a filtered algebra on Diff h (σ 1 , . . . , σ n ). The associated graded algebra is the homogeneous algebra Diff h (0, . . . , 0). This homogeneous algebra has the desired Poincaré-Birkhoff-Witt property because it is the reduction algebra, with respect to gl n , of the semi-direct product of gl n and the abelian Lie algebra V ⊕ V * . The standard argument shows that the ring Diff h (σ 1 , . . . , σ n ) can be viewed as a deformation of the homogeneous ring Diff h (0, . . . , 0): for the generating set x i , ∂i , where x i = x i , all defining relations are the same except (3.5) in which σ i gets replaced by σ i ; one can consider as the deformation parameter. Thus our aim is to study the conditions under which this deformation is flat.

Poincaré-Birkhof f-Witt property

It turns out that the Poincaré-Birkhoff-Witt property is equivalent to the system of finitedifference equations for the elements σ 1 , . . . , σ n ∈ Ū(n).

Proposition 3.1. The ring Diff h (σ 1 , . . . , σ n ) satisf ies the Poincaré-Birkhoff-Witt property if and only if the elements σ 1 , . . . , σ n ∈ Ū(n) satisfy the following linear system of finite-difference equations

hij ∆ j σ i = σ i -σ j , i, j = 1, . . . , n. (3.7)
We postpone the proof to Section 4.1.

∆-system

The system (3.7) is closely related to the following linear system of finite-difference equations for one element σ ∈ Ū(n):

∆ i ∆ j hij σ = 0, i, j = 1, . . . , n. (3.8)
We shall call it the "∆-system". The ∆-system can be written in the form hij ∆ j ∆ i σ = ∆ i σ -∆ j σ, i, j = 1, . . . , n.

We describe the most general solution of the system (3.8).

Definition 3.2. Let W j , j = 1, . . . , n, be the vector space of the elements of Ū(n) of the form π hj χ j where π hj is a univariate polynomial in hj , and χ j is defined in (1.1). Let W be the sum of the vector spaces W j , j = 1, . . . , n.

Theorem 3.3. An element σ ∈ Ū(n) satisfies the system (3.8) if and only if σ ∈ W.
The proof is in Section 4.2.

The sum W j is not direct.

Definition 3.4. Let H be the K-vector space formed by linear combinations of the complete symmetric polynomials H L , L = 0, 1, 2, . . . , in the variables h1 , . . . , hn ,

H L = i 1 ≤•••≤i L hi 1 • • • hi L . Lemma 3.5. (i) Let L ∈ Z ≥0 . We have n j=1 hL j χ j = 0, L = 0, 1, . . . , n -2, H L-n+1 , L ≥ n -1.
(3.9)

(ii) The space H is a subspace of W. Moreover, an element σ ∈ U(n) satisfies the system (3.8) if and only if σ ∈ H, that is,

H = W ∩ U(n). (3.10)
The symmetric group S n acts on the ring Ū(n) and on the space W by permutations of the variables h1 , . . . , hn . We have

H = W Sn , (3.11)
where W Sn denotes the subspace of S n -invariants in W.

(iii) Select j ∈ {1, . . . , n}. Then we have a direct sum decomposition

W = k : k =j W k ⊕ H. (3.12)
The proof is in Section 4.2.

Let t be an auxiliary indeterminate. We have a linear map of vector spaces

K[t] n → W defined by (π 1 , . . . , π n ) → n j=1 π j hj χ j .
It follows from Lemma 3.5 that this map is surjective and its kernel is the vector subspace of K[t] n spanned by n-tuples (t j , . . . , t j ) for j = 0, 1, . . . , n -2. The image of the diagonal in K[t] n , formed by n-tuples (π, . . . , π), is the space H.

Potential

We shall give a general solution of the system (3.7).

Proposition 3.6. Assume that the elements σ 1 , . . . , σ n ∈ Ū(n) satisfy the system (3.7). Then there exists an element σ ∈ Ū(n) such that

σ i = ∆ i σ, i = 1, . . . , n.
We shall call the element σ the "potential" and write Diff h,σ (n

) instead of Diff h (σ 1 , . . . , σ n ) if σ i = ∆ i σ, i = 1, . . . , n.
According to Proposition 3.1, the ring Diff h,σ (n) satisfies the Poincaré-Birkhoff-Witt property iff the potential σ satisfies the ∆-system (3.8).

In Section 4.4 we give two proofs of Proposition 3.6. In the first proof we directly describe the space of solutions of the system (3.7). As a by-product of this description we find that the potential exists and moreover belongs to the space W.

The second proof uses a partial information contained in the system (3.7) and establishes only the existence of a potential and does not immediately produce the general solution of the system (3.7). Given the existence of a potential, the general solution is then obtained by Theorem 3.3.

Let H be the K-vector space formed by linear combinations of the complete symmetric polynomials H L , L = 1, 2, . . . , and let

W = k : k =1 W k ⊕ H . (3.13)
The potential σ is defined up to an additive constant, and it will be sometimes useful to uniquely define σ by requiring that σ ∈ W .

A characterization of polynomial potentials

The polynomial potentials σ ∈ W can be characterized in different terms. The rings Diff h (n) and AZ n admit the action of Zhelobenko automorphisms q1 , . . . , qn-1 [START_REF] Khoroshkin | Mickelsson algebras and Zhelobenko operators[END_REF][START_REF] Zhelobenko | Extremal cocycles on Weyl groups[END_REF]. Their action on the generators x i and ∂i , i = 1, . . . , n, is given by (see [START_REF] Khoroshkin | Diagonal reduction algebra and the reflection equation[END_REF]) The proof is in Section 4.5.

qi x i = -x i+1 hi,i+1 hi,i+1 -1 , qi x i+1 = x i , qi x j = x j , j = i, i + 1, qi ( ∂i ) = - hi,i+1 -1 hi,i+1 ∂i+1 , qi ∂i+1 = ∂i , qi ∂j = ∂j , j = i, i + 1, qi hj = hs i (j) . ( 3
In the examples discussed in Section 3.1, the ring Diff h (n) corresponds to σ = H 1 and the ring AZ

n corresponds to σ = -H 2 = - i,j : i≤j hi hj , ∆ i H 2 = hi + n k=1 hk -1.
The question of constructing an associative algebra which contains U(gl n ) and whose reduction with respect to gl n is Diff h,σ (n) for σ = H k , k > 2, will be discussed elsewhere.

Center

In [START_REF] Ogievetsky | Rings of h-deformed differential operators[END_REF] we have described the center of the ring Diff h (n). The center of the ring Diff h,σ (n), σ ∈ W, admits a similar description. Let Γ i := ∂i x i for i = 1, . . . , n.

Let

c(t) = i e(t) 1 + hi t Γ i -ρ(t) = n k=1 c k t k-1 , (3.15)
where t is an auxiliary variable and ρ(t) a polynomial of degree n -1 in t with coefficients in Ū(n).

Proposition 3.8.

(i) Let σ ∈ W and σ j = ∆ j σ, j = 1, . . . , n. The elements c 1 , . . . , c n are central in the ring Diff h,σ (n) if and only if the polynomial ρ satisfies the system of finite-difference equations

∆ j ρ(t) = e(t) 1 + hj t σ j . (3.16)
(ii) For an arbitrary σ ∈ W the system (3.16) admits a solution. Since the system (3.16) is linear, it is sufficient to present a solution for an element σ ∈ W k for each k = 1, . . . , n, that is, for

σ = A hk χ k ,
where A is a univariate polynomial.

(3.17)

The solution of the system (3.16) for the element σ of the form (3.17) is, up to an additive constant from K,

ρ(t) = e(t) 1 + hk t σ.
(iii) The center of the ring Diff h,σ (n) is isomorphic to the polynomial ring K[t 1 , . . . , t n ]; the isomorphism is given by t j → c j , j = 1, . . . , n.

The proof is in Section 4.6.

Rings of fractions

In [START_REF] Ogievetsky | Rings of h-deformed differential operators[END_REF] we have established an isomorphism between certain rings of fractions of the ring Diff h (n) and the Weyl algebra W n . It turns out that when we pass to the analogous ring of fractions of the ring Diff h,σ (n), we loose the information about the potential σ. Thus we obtain the isomorphism with the same, as for the ring Diff h (n), ring of fractions of the Weyl algebra W n . We denote, as for the ring Diff h (n), by S -1

x Diff h,σ (n) the localization of the ring Diff h,σ (n) with respect to the multiplicative set S x generated by x j , j = 1, . . . , n. Lemma 3.9. Let σ and σ be two elements of the space W , see (3.13).

(i) The rings S -1

x Diff h,σ (n) and S -1

x Diff h,σ (n) are isomorphic. (ii) However, the rings Diff h,σ (n) and Diff h,σ (n) are isomorphic, as filtered rings over Ū(n) (where the filtration is defined by (3.6)), if and only if

σ = γσ for some γ ∈ K * .
The proof is in Section 4.7.

Lowest weight representations

The ring Diff h,σ (n) has an n-parametric family of lowest weight representations, similar to the lowest weight representations of the ring Diff h (n), see [START_REF] Ogievetsky | Rings of h-deformed differential operators[END_REF]. We recall the definition. Let D n be an Ū(n)-subring of Diff h,σ (n) generated by { ∂i } n i=1 . Let λ := {λ 1 , . . . , λ n } be a sequence, of length n, of complex numbers such that λ i -λ j / ∈ Z for all i, j = 1, . . . , n, i = j. Denote by M λ the one-dimensional K-vector space with the basis vector | . The formulas hi : where

| → λ i | , ∂i : | → 0, i = 1, . . . , n, (3.18
ε = ε 1 + • • • + ε n . (3.19)
The proof is in Section 4.8.

Several copies

The coexistence of several copies imposes much more severe restrictions on the flatness of the deformation. Namely, let L be the ring with the generators x iα , i = 1, . . . , n, α = 1, . . . , N , and ∂jβ , j = 1, . . . , n, β = 1, . . . , N subject to the following defining relations. The h(n)weights of the generators are given by (2.2) and (2.6). The generators x iα satisfy the relations (2.4). The generators ∂jβ satisfy the relations (2.7). We impose the general oscillatorlike cross-commutation relations, compatible with the h(n)-weights, between the generators x iα and ∂jβ :

x iα ∂jβ = k,l ∂kβ R ki lj x lα -δ i j σ iαβ , i, j = 1, . . . , n, α = 1, . . . , N , β = 1, . . . , N,
with some σ iαβ ∈ Ū(n).

Lemma 3.11. Assume that at least one of the numbers N and N is bigger than 1. Then the ring L has the Poincaré-Birkhoff-Witt property if and only if

σ iαβ = σ αβ for some σ αβ ∈ K. (3.20)
The proof is in Section 4.9.

Making the redefinitions of the generators, x iα A α α x iα and ∂iβ B β β ∂iβ with some A ∈ GL(N , K) and B ∈ GL(N, K) we can transform the matrix σ αβ to the diagonal form, with the diagonal (1, . . . , 1, 0, . . . , 0). Therefore, the ring L is formed by several copies of the rings Diff h (n), V(n) and V * (n).

4 Proofs of statements in Section 3. The explicit form of the defining relations for the ring Diff h (σ 1 , . . . , σ n ) is

x i x j = hij + 1 hij x j x i , 1 ≤ i < j ≤ n, (4.1
)

∂i ∂j = hij -1 hij ∂j ∂i , 1 ≤ i < j ≤ n, (4.2)
x i ∂j =      ∂j x i , 1 ≤ i < j ≤ n, hij hij -2 hij -1 2 ∂j x i , n ≥ i > j ≥ 1, (4.3)
x i ∂i = j 1 1 -hij ∂j x j -σ i , i = 1, . . . , n. (4.4)
Proof of Proposition 3.1. We can consider (4.1), (4.2) and (4.3) as the set of ordering relations and use the diamond lemma [START_REF] Bergman | The diamond lemma for ring theory[END_REF][START_REF] Bokut | Embeddings into simple associative algebras[END_REF] for the investigation of the Poincaré-Birkhoff-Witt property. The relations (4.1), (4.2) and (4.3) are compatible with the h(n)-weights of the generators x i and ∂i , i = 1, . . . , n, so we have to check the possible ambiguities involving the generators x i and ∂i , i = 1, . . . , n, only. The properties (2.5)

= hi , ∂i = ϕ i x i , x i = ∂i ϕ -1 i , (4.6)
where

ϕ i := ψ i ψ i [-ε i ] = k : k>i hik hik -1 , i = 1, . . . , n.
By using the anti-automorphism we reduce the check of the ambiguity x j x k ∂i to the check of the ambiguity x i ∂j ∂k .

Since the associated graded algebra with respect to the filtration (3.6) has the Poincaré-Birkhoff-Witt property, we have, in the check of the ambiguity x i ∂j ∂k , to track only those ordered terms whose degree is smaller than 3. We use the symbol u l.d.t. to denote the part of the ordered expression for u containing these lower degree terms. Check of the ambiguity x i ∂j ∂k . We calculate, for i, j, k = 1, . . . , n,

x i ∂j ∂k l.d.t. = u,v R ui vj [ε u ] ∂u x v -δ i j σ i ∂k l.d.t. = - u R ui kj [ε u ] ∂u σ k -δ i j σ i ∂k , (4.7)
and

x i ∂j ∂k l.d.t. = x i a,b R ab kj ∂b ∂a l.d.t. = a,b R ab kj [-ε i ]   c,d R ci db [ε c ] ∂c x d -δ i b σ i   ∂a l.d.t. = - a,b,c R ab kj [-ε i ] R ci ab [ε c ] ∂c σ a - a R ai kj [-ε i ]σ i ∂a . (4.8)
Comparing the resulting expressions in (4.7) and (4.8) and collecting coefficients in ∂u , we find the necessary and sufficient condition for the resolvability of the ambiguity x i ∂j ∂k :

R ui kj [ε u ]σ k [ε u ] + δ i j δ u k σ i = a,b R ab kj [-ε i ] R ui ab [ε u ]σ a [ε u ] + R ui kj [-ε i ]σ i , (4.9)
i, k, j, u = 1, . . . , n.

Shifting by -ε u and using the property (1.3) together with the ice condition (1.4), we rewrite (4.9) in the form

R ui kj (σ k -σ i [-ε u ]) + δ i j δ u k σ i [-ε u ] = a,b R ab kj R ui ab σ a . (4.10)
For j = k the system (4.10) contains no equations. For j = k we have two cases:

• u = j and i = k. This part of the system (4.10) reads explicitly (see (1.2))

σ k -σ k [-ε j ] = 1 hkj (σ k -σ j ).
This is the system (3.7).

• u = k and i = j. This part of the system (4.10) reads explicitly

1 hkj (σ k -σ j [-ε k ]) + σ j [-ε k ] = 1 h2 kj σ k + h2 kj -1 h2 kj σ j ,
which reproduces the same system (3.7).

General solution of the system (3.8).

Proofs of Theorem 3.3 and Lemma 3.5

We shall interpret elements of Ū(n) as rational functions on h * with possible poles on hyperplanes hij + a = 0, a ∈ Z, i, j = 1, . . . , n, i = j. Let M be a subset of {1, . . . , n}. The symbol R M Ū(n) denotes the subring of Ū(n) consisting of functions with no poles on hyperplanes hij + a = 0, a ∈ Z, i, j ∈ M, j = i. The symbol N M Ū(n) denotes the subring of Ū(n) consisting of functions which do not depend on variables hi , i ∈ M. We shall say that an element f ∈ Ū(n) is regular in hj if it has no poles on hyperplanes hjm + a = 0, a ∈ Z, m = 1, . . . , n, m = j. 1. Partial fraction decompositions. We will use partial fraction decompositions of an element f ∈ Ū(n) with respect to a variable hj for some given j. The partial fraction decomposition of f with respect to hj is the expression for f of the form

f = P j (f ) + reg j (f ),
where the elements P j (f ) and reg j (f ) have the following meaning. The "regular" part reg j (f ) is an element, regular in hj . The "principal" in hj part P j (f ) is

P j (f ) = k : k =j P j;k (f ),
where

P j;k (f ) = a∈Z νa∈Z >0 u kaνa hjk -a νa , (4.11)
with some elements u kaνa ∈ N j Ū(n); the sums are finite. The fact that the ring Ū(n) admits partial fraction decompositions (that is, that the elements u kaνa and reg j (f ) belong to Ū(n)) is a consequence of the formula

1 hjk -a hjl -b = 1 hkl + a -b 1 hjk -a - 1 hjl -b .
2. Let D be a domain (a commutative algebra without zero divisors) over K. Let f be an

element of D ⊗ K Ū(n). Set Y ij (f ) := ∆ i ∆ j hij f .
(4.12)

Lemma 4.1. If Y ij (f ) = 0 for some i and j, i = j, then f can be written in the form

f = A hij + B, (4.13)
with some A, B ∈ D ⊗ K R i,j Ū(n).

Proof . We write f in the form

f = A hij -a 1 ν 1 hij -a 2 ν 2 • • • hij -a M ν M + B,
where

a 1 < a 2 < • • • < a M , ν 1 , ν 2 , . . . , ν M ∈ Z >0 , A, B ∈ D ⊗ K R i,j Ū(n)
and the element A is not divisible by any factor in the denominator. There is nothing to prove if A = 0. Assume that

A = 0. Then 0 = Y ij (f ) = hij A hij -a 1 ν 1 • • • hij -a M ν M - hij -1 A[-ε i ] hij -a 1 -1 ν 1 • • • hij -a M -1 ν M (4.14) - hij + 1 A[-ε j ] hij -a 1 + 1 ν 1 • • • hij -a M + 1 ν M + hij A[-ε i -ε j ] hij -a 1 ν 1 • • • hij -a M ν M + Y ij (B).
The denominator hij -a M -1 appears only in the second term in the right hand side of (4.14).

It has therefore to be compensated by hij -1 in the numerator. Hence the only allowed value of a M is a M = 0 and moreover we have ν M = 1. Similarly, the denominator hij -a 1 + 1 appears only in the third term in the right hand side of (4.14) and has to be compensated by ( hij + 1) in the numerator. Hence the only allowed value of a 1 is a 1 = 0 and we have ν 1 = 1. The inequalities a 1 < a 2 < • • • < a M imply that M = 1 and we obtain the form (4.13) of f .

3. Let f ∈ D ⊗ K Ū(n).
We shall analyze the linear system of finite-difference equations Y ij (f ) = 0 for all i, j = 1, . . . , n, (

where Y ij are defined in (4.12). First we prove a preliminary result. We recall Definition 3.2 of the vector spaces W i , i = 1, . . . , n. We select one of the variables hi , say, h1 . Lemma 4.2. Assume that an element f ∈ D ⊗ K Ū(n) satisfies the system (4.15). Then

f = n j=2 F j + ϑ, (4.16)
where ϑ ∈ D ⊗ K U(n) and

F j = u j hj χ j ∈ D ⊗ K W j (4.17)
with some univariate polynomials u j hj , j = 2, . . . , n, with coefficients in D.

Proof . Since Y 1m (f ) = 0, m = 2, . . . , n, Lemma 4.1 implies that the partial fraction decomposition of f with respect to h1 has the form

f = n m=2 β m hm1 + ϑ, (4.18)
where

β m ∈ D ⊗ K N 1 Ū(n), m = 2, . . . , n, and ϑ ∈ D h1 ⊗ K N 1 Ū(n).
Substituting the expression (4.18) for f into the equation Y 1j (f ) = 0, j = 2, . . . , n, we obtain

0 = Y 1j (f ) = ∆ 1 ∆ j   m : m =1,j h1j β m hm1 -β j + h1j ϑ   = ∆ 1 ∆ j   m : m =1,j h1j β m hm1 + h1j ϑ   (4.19) = m : m =1,j h1j β m hm1 - h1j + 1 β m [-ε j ] hm1 - h1j -1 β m hm1 + 1 + h1j β m [-ε j ] hm1 + 1 + ∆ 1 ∆ j h1j ϑ .
We used that β m ∈ D ⊗ K N 1 Ū(n) in the third and fourth equalities. For any m = 1, j, the terms containing the denominator hm1 in the expression (4.19) for Y 1j (f ) read

1 hm1 h1j β m -h1j + 1 β m [-ε j ] .
Therefore, h1j β m -h1j + 1 β m [-ε j ] is divisible, as a polynomial in h1 , by hm1 , or, what is the same, the value of h1j

β m -h1j + 1 β m [-ε j ] at h1 = hm is zero. This means that 0 = hmj β m -hmj + 1 β m [-ε j ] = ∆ j hmj β m .
Therefore, the element hmj β m does not depend on hj for any j > 1. We conclude that

β m = u m hm k : k =1,m
hmk with some univariate polynomial u m . We have proved that the element f has the form (4.16) where F j , j = 2, . . . , n, are given by (4.17) and the element ϑ is regular in h1 .

A direct calculation shows that for any j = 2, . . . , n, the element F j , given by (4.17), is a solution of the linear system (4.15). Therefore the regular in h1 part ϑ by itself satisfies the system Y ij (ϑ) = 0. It is left to analyze the regular part ϑ.

We use induction in n. For n = 2, the element ϑ is, by construction, a polynomial in h1 and h2 . This is the induction base. We shall now prove that ϑ is a polynomial, with coefficients in D, in all n variables h1 , . . . , hn .

For arbitrary n > 2 we have ϑ ∈ D h1 ⊗ K Ū (n -1) where we have denoted by Ū (n -1) the subring N 1 Ū(n) of Ū(n) consisting of functions not depending on h1 . Since Y ij (ϑ) = 0 for i, j = 2, . . . , n, we can use the induction hypothesis with n -1 variables h2 , . . . , hn over the ring D = D h1 .

We now select the variable h2 . It follows from the induction hypothesis that

ϑ = m : m =1,2 γ m ( hm ) l : l =1,m hml + ϑ , ( 4
= m : m =1,2    h12 γ m hm2 l : l =1,2,m hml - h12 -1 γ m [-ε 1 ] hm2 l : l =1,2,m hml - h12 + 1 γ m hm2 + 1 l : l =1,2,m hml + h12 γ m [-ε 1 ] hm2 + 1 l : l =1,2,m hml    + Y 12 (ϑ ). (4.22)
The terms containing the denominator hm2 in (4.22) read

1 hm2 l : l =1,2,m hml h12 γ m -h12 -1 γ m [-ε 1 ] . Therefore, the expression h12 γ m -h12 -1 γ m [-ε 1 ] is divisible, as a polynomial in h2 , by h2m = h2 -hm , so 0 = h1m γ m -h1m -1 γ m [-ε 1 ] = ∆ 1 h1m γ m .
Thus the product h1m γ m , m = 3, . . . , n, does not depend on h1 . Since γ m , m = 3, . . . , n, is a polynomial, this implies that γ m = 0. We conclude that ϑ = ϑ and is therefore a polynomial in all variables h1 , . . . , hn .

4. Now we refine the assertion of Lemma 4.2. We shall, at this stage, obtain the general solution of the system (4.15) in a form which does not exhibit the symmetry with respect to the permutations of the variables h1 , . . . , hn .

We recall Definition 3.4 of the vector space H. We now prove the assertion (4.24). We first study the case n = 2. Let p ∈ D h1 , h2 be a polynomial such that Y 12 (p) = 0. Since ∆ 1 ∆ 2 h12 p = 0 we have ∆ 2 ( h12 p) ∈ D h2 .

It is a standard fact that the operator ∆ 2 is surjective on D h2 . This can be seen, for example, by noticing that the set

h↑m 2 := h2 h2 + 1 • • • h2 + m -1 , m ∈ Z ≥0 ,
is a basis of D h2 over D, and

∆ 2 h↑m 2 = m h↑m-1 2 .
The surjectivity of ∆ 2 implies that ∆ 2 h12 p = ∆ 2 w h2 for some polynomial w h2 ∈ D h2 . Then ∆ 2 h12 p -w h2 = 0 so h12 p -w h2 = v h1 for some polynomial v h1 ∈ D h1 . Therefore

p = v h1 + w h2 h12 = v h1 -v h2 h12 + v h2 + w h2 h12 .
Since p is a polynomial we must have w = -v. Thus

p = v h1 -v h2 h12 ,
that is, p is a D-linear combination of complete symmetric polynomials in h1 , h2 . For arbitrary n, our polynomial ϑ is symmetric since, by the above argument, it is symmetric in every pair hi , hj of variables. Moreover, considered as a polynomial in a pair hi , hj , it is a D-linear combination of complete symmetric polynomials in hi , hj . It is then immediate that ϑ is a D-linear combination of complete symmetric polynomials in h1 , . . . , hn .

To finish the proof of the statement that the formula (4.23) gives the general solution of the system (4.15) it is left to check that the complete symmetric polynomials H L , L = 0, 1, . . . , in the variables h1 , . . . , hn satisfy the system (4.15). Let s be an auxiliary variable and

H(s) = ∞ L=0 H L s L = k 1 1 -s hk (4.25)
be the generating function of the elements H L , L = 0, 1, . . . It is sufficient to show that the formal power series (4.25) satisfies the system (4.15). Fix i, j ∈ {1, . . . , n}, i = j, and let

ζ ij = 1 (1-hi s)(1-hj s) . The element ∆ i hij 1 -hi s 1 -hj s = 1 1 -hj s hij 1 -hi s - hij -1 1 -hi -1 s = 1 1 -hi τ 1 -hi -1 τ
does not depend on hj so Y ij (ζ ij ) = 0. Therefore Y ij (H(s)) = 0 since the factors other than ζ ij in the product in the right hand side of (4.25) do not depend on hi and hj .

(ii) Finally, the summands in (4.23) are uniquely defined since (4.23) is a partial fraction decomposition of the element f in h1 .

5. Proof of Lemma 3.5(i). Let t be an auxiliary indeterminate. Multiplying by t -L-1 and taking sum in L, we rewrite (3.9) in the form

n j=1 1 t -hj 1 χ j = 1 n j=1 t -hj .
The left hand side is nothing else but the partial fraction decomposition, with respect to t, of the product in the right hand side. (iii) For j = 1 formula (3.12) is the uniqueness statement of Lemma 4.3. In the proof of Lemma 4.3 we could have selected any hj instead of h1 .

System (3.7)

We proceed to the study of the system (3.7), that is, the system of equations

Z ij = 0, i, j = 1, . . . , n, (4.26)
where

Z ij = hij ∆ j σ i -σ i + σ j = -∆ j hji + 1 σ i + σ j .
for the n-tuple σ 1 , . . . , σ n ∈ Ū(n).

1. We use the equations Z 1j , j = 2, . . . , n, to express the elements σ j , j = 2, . . . , n, in terms of the element σ 1 :

σ j = ∆ j hj1 + 1 σ 1 = hj1 ∆ j (σ 1 ) + σ 1 . (4.27)
Substituting the expressions (4.27) into the equations Z i1 , i = 2, . . . , n, we find

hi1 ∆ 1 hi1 ∆ i σ 1 + σ 1 -∆ i σ 1 = 0.
Simplifying by hi1 we obtain

W i = 0, i = 2, . . . , n, (4.28)
where

W i = ∆ 1 hi1 ∆ i σ 1 + σ 1 -∆ i σ 1 = ∆ i ∆ 1 hi1 + 1 σ 1 -σ 1 = ∆ i hi1 σ 1 -hi1 + 2 σ 1 [-ε 1 ] .
Substituting the expressions (4.27) into the equations Z ij , i, j = 2, . . . , n, we find

hi1 hij ∆ i ∆ j σ 1 + ∆ j σ 1 -∆ i σ 1 = 0.
Simplifying by hi1 , we obtain, with the notation (4.12),

Y ij (σ 1 ) = 0, i, j = 2, . . . , n. (4.29)
This is our first conclusion which we formulate in the following lemma.

Lemma 4.4. If σ 1 , . . . , σ n ∈ Ū(n) is a solution of the system (4.26) then the element σ 1 satisfies the equations (4.28) and (4.29). Conversely, if an element σ 1 ∈ Ū(n) satisfies the equations (4.28) and (4.29) then we reconstruct a solution of the system (4.26) with the help of the formulas (4.27).

2. We shall now analyze the consequences imposed by the equations (4.28) on the partial fraction decomposition of the element σ 1 with respect to h1 . The full form of the expression W i reads

W i = hi1 σ 1 -hi1 + 2 σ 1 [-ε 1 ] -hi1 -1 σ 1 [-ε i ] + hi1 + 1 σ 1 [-ε 1 -ε i ].
(4.30)

We write the element σ 1 in the form (keeping the notation of Section 4.2)

σ 1 = A hi1 -a 1 ν 1 • • • hi1 -a L ν L , (4.31)
where

a 1 < a 2 < • • • < a M , ν 1 , ν 2 , . . . , ν M ∈ Z ≥0 and A ∈ R 1,i Ū(n)
is not divisible by any factor in the denominator. Substitute the expression (4.31) into the equation W i = 0. The denominator hi1 -a L -1 is present only in term hi1 -1 σ 1 [-ε i ] in (4.30). It has therefore to be compensated by hi1 -1 . Hence the only allowed value of a L is a L = 0 and we have ν L ≤ 1. Similarly, the denominator hij -a 1 + 1 appears only in the term hi1 + 2 σ 1 [-ε 1 ] in (4.30). It has to be compensated by hi1 + 2 . Hence the only allowed value of a 1 is a 1 = 0 and we have ν 1 ≤ 1.

It follows that the partial fraction decomposition of the element σ 1 with respect to h1 reads

σ 1 = n k=2 A k hk1 + A k hk1 + 1 + B, (4.32)
where A k , A k , k = 2, . . . , n, do not depend on h1 and B is regular in h1 .

3. The equations (4.28) impose further restrictions on the constituents of the decomposition (4.32) of the element σ 1 . Substitute the decomposition (4.32) into the equation W i = 0. The terms which have denominators of the form hi1 + m, m ∈ Z, in (4.30) are

hi1 A i hi1 + A i hi1 + 1 -hi1 + 2 A i hi1 + 1 + A i hi1 + 2 -hi1 -1 A i [-ε i ] hi1 -1 + A i [-ε i ] hi1 + hi1 + 1 A i [-ε i ] hi1 + A i [-ε i ] hi1 + 1 . (4.33)
In the expression (4.33), the terms with the denominator hi1 + 1 read

hi1 A i -hi1 + 2 A i + hi1 + 1 A i [-ε i ] hi1 + 1 = - A i + A i hi1 + 1 + A i -A i + A i [-ε i ].
Therefore,

A i + A i = 0, i = 2, . . . , n.
With this condition, the expression (4.33) vanishes. We conclude that

σ 1 = n k=2 A k hk1 - A k hk1 + 1 + B. (4.34)
does not depend on h2 . In the notation of paragraph 1 in Section 4.2, the part P 2;j , j = 3, . . . , n, of this expression is

1 l : l =12,j
hjl h21 u j hj , h1 -h21 + 2 u j hj , h1 -1

hj2 .

Therefore, h21 u j hj , h1 -h21 + 2 u j hj , h1 -1 is divisible, as polynomial in h2 , by h2j . So the value of h21 u j hj , h1 -h21 + 2 u j hj , h1 -1 at h2 = hj is zero,

hj1 u j hj , h1 -hj1 + 2 u j hj , h1 -1 = 0. (4.36)

Set

u j = β j hj1 hj1 + 1 . (4.37)
Then equation (4.36) becomes

β j hj1 + 1 + β j [-ε 1 ]
hj1 + 1 = 0, or ∆ 1 (β j ) = 0, so β j depends only on hj . But then if β j = 0, the formula (4.37) shows that u j cannot be a polynomial in h1 .

We conclude that the principal part of the element B with respect to h2 vanishes, and B = C is a polynomial in all its variables.

6. We claim that C is a K-linear combination of the elements ∆ 1 (H L ), L = 1, 2, . . . , where H L are the complete symmetric polynomials in h1 , . . . , hn .

Consider first the case n = 2. Set

C = ξ h21 h21 + 1 ,
where ξ is some polynomial in h1 and h2 . With this substitution the equation W 2 (C) = 0 becomes

∆ 2 1 h21 + 1 ∆ 1 (ξ) = 0, that is, 1 h21 + 1 ∆ 1 (ξ) = µ,
where µ does not depend on h2 . Note that by construction, the polynomial ξ is divisible by h21 h21 + 1 , which implies that µ is a polynomial in h1 . Since ∆ 1 is surjective on polynomials, we can write µ = ∆ 2 1 z h1 for some univariate polynomial z, that is

∆ 1 (ξ) = h21 + 1 ∆ 2 1 z h1 .
We have

h21 + 1 ∆ 2 1 z h1 = ∆ 1 h21 ∆ 1 z h1 + z h1 . Therefore, ∆ 1 ξ -h21 ∆ 1 z h1 -z h1 = 0, or ξ = h21 ∆ 1 z h1 + z h1 + w h2 ,
where w h2 is a polynomial in h2 . That is,

C = ∆ 1 z h1 h21 + 1 + z h1 + w h2 h21 h21 + 1 . (4.38)
Since the element C is a polynomial, the denominator h21 in the second term in the right hand side of (4.38) shows that w = -z. Therefore,

C = h21 ∆ 1 z h1 + z h1 -z h2 h21 h21 + 1 = ∆ 1 z h1 -z h2 h21 ,
as claimed.

The claim for arbitrary n follows since for any j > 2 the element C is a linear combination of ∆ 1 H L h1 , hj , L = 1, 2, . . . 7. We summarize the results of this section in the following proposition.

Proposition 4.5. The general solution of the system (4.28) and (4.29) has the form

σ 1 = n i=2 1 hi1 - 1 hi1 + 1 α i hi l : l =1,i hil + ∆ 1 (ν), ν ∈ H (4.39)
and α 2 , . . . , α n are univariate polynomials. The elements α 2 , . . . , α n and ν are uniquely defined.

4.4 Potential. Proof of Proposition 3.6

First proof. We rewrite the formula (4.39) in the form

σ 1 = ∆ 1 (σ), where σ = n i=2 α i hi χ i + ν ∈ W.
Then the expressions for the elements σ j , j = 2, . . . , n, see (4.27), read

σ j = ∆ j hj1 + 1 n i=2 α i hi hi1 + 1 χ i + hj1 + 1 ∆ 1 (ν) . (4.40) Since, for ν ∈ H, ∆ 1 ∆ j hj1 ν = 0, we find that ∆ j hj1 + 1 ∆ 1 (ν) = ∆ j (ν).
The term with i = j in the sum in the right hand side of (4.40) is simply

α j hj χ j .
where u jaνa ∈ N 1 Ū(n) and the sums are finite. Then

∆ 1 ∆ 2 (P 1;j (f )) = a∈Z νa∈Z >0 ∆ 2 (u jaνa ) 1 h1j -a νa - 1 h1j -a -1 νa . ( 4

.43)

We prove that the elements u jaνa do not depend on h2 . Indeed, if this is not true then there is a minimal a ∈ Z for which ∆ 2 (u jaνa ) = 0 for some ν a . But then the denominator ( h1j -a) νa in the right hand side in (4.43) cannot be compensated. We conclude that f = f 2,0 + g where f 2,0 = j>2 P 1;j (f ) does not depend on h2 and g is regular in h1 . We decompose g with respect to h2 . As above, the part P 2;1 (g) vanishes and the calculation, parallel to (4.43), shows that P 2;j (g), j > 2, does not depend on h1 . Now we have

f = f 2,0 + f 1,0 + f + ,
where f 1,0 = j>2 P 2;j (g) does not depend on h1 and f + is regular in h1 and h2 .

We use the decomposition (4.41) for the regular part f + and write f

+ = f + 1 + f + 2 , where f + 1
does not depend on h1 and f + 2 does not depend on h2 . This leads to the required decomposition (4.42) with

f 1 = f 1,0 + f + 1 and f 2 = f 2,0 + f + 2 . Lemma 4.7. Let σ 1 , . . . , σ k , k ≤ n, be a k-tuple of elements in Ū(n) such that ∆ a (σ b ) = ∆ b (σ a ), a, b = 1, . . . , k.
Assume that σ a belongs to the image of ∆ a for all a = 1, . . . , k, that is, there exist elements f 1 , . . . , f k ∈ Ū(n) for which σ a = ∆ a (f a ), a = 1, . . . , k. Then there exists a potential f ∈ Ū(n) such that σ a = ∆ a (f ) = 0, a = 1, . . . , k.

Proof . For k = 1 there is nothing to prove. Let now k > 1. We use the induction in k. By the induction hypothesis, there exist elements F, G ∈ Ū(n) such that

σ a = ∆ a (F ) for a = 1, 3, . . . , k and σ b = ∆ b (G) for b = 2, 3, . . . , k. Then ∆ c (F ) = ∆ c (G) for c = 3, . . . , k and ∆ 1 ∆ 2 (G) = ∆ 2 ∆ 1 (F ).
The element F -G does not depend on hc , c = 3, . . . , k, and ∆ 1 ∆ 2 (F -G) = 0. According to Lemma 4.6, there exist two elements u, v ∈ Ū(n) such that u does not depend on h2 , v does not depend on h1 , and

F -G = u -v. Then f := F + v = G + u
is the desired potential.

Second proof of Proposition 3.6. The symmetric, in i and j, part of the equation (3.7) is ∆ i σ j = ∆ j σ i .

(4.44)

The system (4.44) by itself does not imply the existence of a potential. However, the equation (3.7) can be written in the form σ j = ∆ j hji + 1 σ i . So for each j = 1, . . . , n the element σ j belongs to the image of the operator ∆ j . Then, according to Lemma 4.7, there exists σ ∈ Ū(n) such that σ j = ∆ j (σ).

4.5 Polynomial potentials. Proof of Lemma 3.7

The operator qi defined by (3.14) can be an automorphism of the ring Diff h,σ (n) only if qi (σ j ) = σ s i (j) = ∆ s i (j) (σ), i, j = 1, . . . , n. (4.45)

On the other hand, qi (σ j ) = qi (∆ j (σ)) = ∆ s i (j) (q i (σ)), i, j = 1, . . . , n. As before, we use the symbol u l.d.t. to denote these lower degree terms in an expression u. We have

x j c(t) -c(t)x j l.d.t. = - e(t) 1 + hj t σ j -ρ(t)[-ε j ] + ρ(t) x j .
Thus the element c(t) commutes with the generators x j , j = 1, . . . , n, if and only if the polynomial ρ(t) satisfies the system (3.16). The use of the anti-automorphism (4.6) shows that the element c(t) then commutes with the generators ∂j , j = 1, . . . , n, as well.

(ii) We check the case j = 1. The calculation for σ ∈ W j is similar. Since the combination e(t) 1+ h1 t does not depend on h1 , we have, for ρ(t) = e(t) 1+ h1 t σ, , where x •i := x i ψ i , i = 1, . . . , n, generates the localized ring S -1

∆
x Diff h,σ (n). Moreover, the complete set of the defining relations for the generators from the set B D does not remember about the potential σ. It reads hi hj = hj hi , hi x •j = x •j hi + δ j i , x •i x •j = x •j x •i , i, j = 1, . . . , n, c i are central, i = 1, . . . , n.

The proof is the same as for the ring Diff h (n), see [START_REF] Ogievetsky | Rings of h-deformed differential operators[END_REF]. The isomorphism is now clear.

(ii) Assume that ι : Diff h,σ (n) → Diff h,σ (n) is an isomorphism of filtered rings over Ū(n). To distinguish the generators, we denote the generators of the ring Diff h,σ (n) by x i and ∂ i . The ε i -weight subspace E i of the ring Diff h,σ (n) consists of elements of the form θx i where θ is a polynomial in the elements Γ j , j = 1, . . . , n, with coefficients in Ū(n). Since the space of the elements of E i of filtration degree ≤ 1 is Ū(n)x i , we must have ι :

x i → µ i x i , ∂i → ∂ i ν i (4.48)
with some invertible elements µ i , ν i ∈ Ū(n), i = 1, . . . , n. Let γ i := µ i ν i , i = 1, . . . , n. The defining relation (4.4) and the corresponding relation for the ring Diff h,σ (n) shows that the formulas (4.48) may define an isomorphism only if

γ i = γ j [ε j ], i, j = 1, . . . , n, (4.49)
and

γ i σ i = σ i , i = 1, . . . , n. (4.50)
The condition (4.49) implies that γ i = γ for some γ ∈ K. The condition (4.50) then becomes γσ i = σ i and the assertion follows.

4.8 Lowest weight representations. Proof of Proposition 3.10

We need the following identity (see [START_REF] Ogievetsky | Rings of h-deformed differential operators[END_REF]Lemma 5]):

j 1 hj + t -1 Q + j = 1 - e(t
σ kαγ = σ iαγ [-ε k ],
which implies the assertion (3.20).

A direct calculation, with the help of the properties (1.3), (1.4) and (1.5) of the operator R, shows that the condition (3.20) implies the equalities (4.61) and (4.62) as well as all the remaining conditions for the flatness of the deformation.

Proposition 3 . 10 .

 310 ) define the D n -module structure on M λ . The lowest weight representation of lowest weight λ is the induced representation Ind Diff h,σ (n) Dn M λ . We describe the values of the central polynomial c(t), see (3.15), on the lowest weight representations. The element c(t) acts on Ind Diff h,σ (n) Dn M λ by the multiplication on the scalar -ρ(t)[-ε],

2 4. 1

 21 Poincaré-Birkhof f-Witt property. Proof of Proposition 3.1

  .20) where γ m hm , m = 3, . . . , n, are univariate polynomials, with coefficients in D , and the element ϑ is a polynomial, with coefficients in D , in the variables h2 , . . . , hn . We rewrite the equality(4.20) in the formϑ = m : m =1,2 γ m hm , h1 l : l =1,m hml + ϑ ,(4.21)with some polynomials γ m , m = 3, . . . , n, in two variables, with coefficients in D; the element ϑ is a polynomial in all variables h1 , . . . , hn with coefficients in D. The equation Y 12 (ϑ) = 0 for ϑ given by (4.21) reads 0

Lemma 4. 3 .F

 3 (i) The general solution of the linear system (4.15) for an element f ∈ D ⊗ K Ū(n) has the formf = n j=2 j + ϑ,(4.23)whereF j ∈ D ⊗ K W j and ϑ ∈ D ⊗ K H.(4.24)(ii) The elements F j , j = 2, . . . , n, and ϑ are uniquely defined.Proof . (i) In Lemma 4.2 we have established the decomposition (4.23) with ϑ ∈ D ⊗ K U(n).

6 .

 6 Proof of Theorem 3.3. The assertion of the Theorem follows immediately from the decomposition (4.23) in Lemma 4.3 and the identity (3.9). 7. Proof of Lemma 3.5(ii) and (iii). (ii) The formula (3.10) follows from the uniqueness of the decomposition (4.23) in Lemma 4.3. The element f of the form (4.23) is S n -invariant if and only if f ∈ H and the assertion (3.11) follows.

4. 7

 7 Rings of fractions. Proof of Lemma 3.9(i) The set B D := hi , x •i , c i n i=1

  [ε c ] ∂cγ x dα -δ i b σ iαγ   ∂aβ l.d.t. =a,b,c R ab kj [-ε i ] R ci ab [ε c ] ∂cγ σ aαβa R ai kj [-ε i ]σ iαγ ∂aβ . (4.60)Take β = γ. Equating the coefficients in ∂uβ , u = 1, . . . , n, in (4.59) and (4.60), we findR ui kj [ε u ]σ kαγ [ε u ] = R ui kj [-ε i ]σ iαγ , i, k, j, u = 1, . . . , n.(4.61)Equating the coefficients in ∂uγ , u = 1, . . . , n, in (4.59) and (4.60), we findδ i j δ u k σ iαβ = a,b R ab kj [-ε i ] R ui ab [ε u ]σ aαβ [ε u ], i, k, j, u = 1, . . . , n.(4.62)Shifting by -ε u and using the property (1.3) we rewrite the equality (4.61) in the formR ui kj (σ kαγ -σ iαγ [-ε u ]) = 0. (4.63) Setting u = k and j = i (with arbitrary i, k = 1, . . . , n) in (4.63), we obtain

  and (2.8) show that the ambiguities of the forms xxx and ∂ ∂ ∂ are resolvable. It remains to check the ambiguities It follows from the properties (2.5) and (2.8) that the choice of the order for the generators with indices j and k in (4.5) is irrelevant. Besides, it can be verified directly that the ring Diff

	x i	∂j ∂k	and	x j x k	∂i .	(4.5)

h (σ 1 , . . . , σ n ), with arbitrary σ 1 , . . . , σ n ∈ Ū(n) admits an involutive anti-automorphism , defined by hi

  which implies that σ is S n -invariant. The assertion now follows from Lemma 3.5(ii).

				.46)
	Comparing (4.45) and (4.46) we obtain
	∆ j (σ -qi (σ)) = 0,	i, j = 1, . . . , n,
	4.6 Central elements. Proof of Proposition 3.8
	i	e(t) 1 + hi t	Γ

(i) To analyze the relation x j c(t) -c(t)x j = 0, we shall write the expression x j c(t) -c(t)x j in the ordered form, in the order ∂xx. The element c 0 (t) = i is central in the homogeneous ring Diff h,0 (n), see the calculation in

[START_REF] Ogievetsky | Rings of h-deformed differential operators[END_REF] Proposition 3

]. Hence we have to track only those ordered terms whose filtration degree, see

(3.6)

, is smaller than 3.

  Substituting the obtained expression for ξ into (4.56) and taking into account (4.58) we conclude that Several copies. Proof ofLemma 3.11 Assume that, say, N > 1. Repeating the calculations (4.7) and (4.8) for one copy in Section 4.1, we find, for i, j, k = 1, . . . , n,x iα ∂jβ ∂kγ l.d.t. = R ui vj [ε u ] ∂uβ x vα -δ i j σ iαβ ∂kγ l.d.t.

	so
	ξ = -	σ[-ε] 1 + t h1 -1		+	1 1 + t h1	σe(t) e(t)[-ε]	.
	ω(t) = e(t)[-ε] -	σ[-ε] 1 + t h1 -1	+	1 1 + t h1	σe(t) e(t)[-ε]	-	e(t) 1 + t h1	σ
		= -	e(t)[-ε] 1 + t h1 -1	σ[-ε] = -ρ(t)[-ε],
	as stated.
	4.9 u,v
										)[-ε] e(t)	(4.51)
	j	1 hjm + 1	Q + j = 1.	(4.52)
	Then,
	i	1 1 + t hi	1 hik + 1	Q + i =	1 1 + t hk -1 i	1 hik + 1	-	t 1 + t hi	Q + i
										=	1 1 + t hk -1	e(t)[-ε] e(t)	.	(4.53)
	i	1 1 + t hi -1	1 hki + 1	Q -i =	1 1 + t hk	e(t) e(t)[-ε]	.	(4.54)

and its several consequences. At t = 1 -hm -1 , m = 1, . . . , n, the equality (4.51) becomes

We used (4.51) and (4.52) in the last equality. The substitution hi hi + 1, i = 1, . . . , n, and t -t into (4.53) gives

=u R ui kj [ε u ] ∂uβ σ kαγ -δ i j σ iαβ ∂kγ ,

(4.59)

x iα ∂jβ ∂kγ l.d.t. = x iα a,b

R ab kj ∂bγ ∂aβ l.d.t.

Acknowledgements

The work of O.O. was supported by the Program of Competitive Growth of Kazan Federal University and by the grant RFBR 17-01-00585.

4. Now we substitute the obtained expression (4.34) for σ 1 into the equation W j = 0 with j = i and follow the singularities of the form hi1 + m, m ∈ Z. The singular terms are

hj1

In the expression (4.35), the terms with the denominator hi1 read

hj1

Therefore, the numerator, as a polynomial in h1 , must be divisible by the denominator hi1 . The polynomial remainder of this division equals

Therefore, for any j = 2, . . . , n, j = i, the combination hij A i does not depend on hj . It follows that

where each α i is a univariate polynomial.

For the moment, we have found that

where the element B is regular in h1 and

A direct calculation shows that the element σ hjl + C, where u j hj , h1 , j = 3, . . . , n, is a polynomial in hj , h1 and C is a linear combination of complete symmetric polynomials in h2 , . . . , hn with coefficients in K

we can rewrite the term with i = j in the right hand side of (4.40) in the form

Therefore,

The proof of Proposition 3.6 is completed.

Second proof. Let p ∈ U(n) be a polynomial such that ∆ 1 ∆ 2 (p) = 0. Thus, ∆ 2 (p) does not depend on h1 so, by surjectivity of ∆ 2 on polynomials in h2 , there exists a polynomial p 1 which does not depend on h1 and ∆ 2 (p) = ∆ 2 (p 1 ). The polynomial p 2 := p -p 1 does not depend on h2 . The next lemma generalizes this decomposition

to the ring Ū(n).

then there exist elements f 1 , f 2 ∈ Ū(n) such that f 1 does not depend on h1 , f 2 does not depend on h2 , and

Proof . Decompose f into partial fractions with respect to h1 . We have P 1;2 (f ) = 0. Indeed, write P 1;2 (f ) in the form

is not divisible by any factor in the denominator. Assume that u = 0. Then

The factor h12 -a L -1 appears only in the denominator of the second term in the right hand side and cannot be compensated by the numerator. Thus P 1;2 (f ) = 0 (the consideration of the factor h12 -a 1 + 1 in the denominator of the third term proves the claim as well). Now we write the part P 1;j (f ), j > 2, in the form (4.11),

Proof of Proposition 3.10. Since the element c(t) is central, it is sufficient to calculate its value on the vector | . Denote

We have

where Ψ is the skew inverse of the operator R, see (1.6) (we refer, e.g., to [START_REF] Ogievetsky | Uses of quantum spaces[END_REF]Section 4.1.2] for details on skew inverses). Since the generators ∂i , i = 1, . . . , n, annihilate the vector | , see (3.18), we find, in view of (4.55), that

We have used (4.54) in the last equality. Note that


D Grasso

E Tassi
email: tassi@cpt.univ-mrs.fr

H M Abdelhamid

P J Morrison
email: morrison@physics.utexas.edu

Structure and computation of two-dimensional incompressible extended MHD

A comprehensive study of a reduced version of Lüst's equations, the extended magnetohydrodynamic (XMHD) model obtained from the two-fluid theory for electrons and ions with the enforcement of quasineutrality, is given. Starting from the Hamiltonian structure of the fully three-dimensional theory, a Hamiltonian two-dimensional incompressible four-field model is derived. In this way energy conservation along with four families of Casimir invariants are naturally obtained. The construction facilitates various limits leading to the Hamiltonian forms of Hall, inertial, and ideal MHD, with their conserved energies and Casimir invariants. Basic linear theory of the four-field model is treated, and the growth rate for collisionless reconnection is obtained. Results from nonlinear simulations of collisionless tearing are presented and interpreted using, in particular normal fields, a product of the Hamiltonian theory that gives rise to simplified equations of motion.

I. INTRODUCTION

Out of necessity for practicable computation, for many decades researchers have produced reduced fluid models for describing aspects of laboratory and naturally occurring plasmas.

Applications of such models include the exploration of MHD kink modes by means of reduced MHD, 1 subsequently extended to include, among other effects, Hall and gyro physics 2 as well as parallel compressibility and diamagnetic effects. 3 Further applications range from the investigation of drift-waves 4 to low-frequency turbulence [5][6][7][8] and magnetic reconnection. [9][10][11][12][13][14][15][16] The various models have been obtained by various means: in some cases rigorous asymptotics were employed, while other models were built on intuition, or using the device of effecting closure by constraining to match a desired linear theory (e.g. Ref. 2). Based on the noncanonical Hamiltonian formalism introduced for MHD in Ref. 17 (see e.g. Refs. 18 and 19 for review) it was advocated in a series of papers [20][21][22][23] that retention of Hamiltonian form can serve as a derivational aide or as a filter for selecting out good theories in the ideal limit.

By ideal limit, is meant the limit of the model where all dissipative terms, such as collisions, Landau damping, and dissipative anomalous transport terms are neglected. Subsequently there have been many papers by many authors that have adopted this point of view.

In the present work we consider the two-dimensional (2D) incompressible reduction of the XMHD model derived by Lüst in Ref. 24. This model is simply a reduced case of a two-fluid model in which the charge quasineutrality condition is invoked, the displacement current is ignored, and the smallness of the electron-ion mass ratio is taken to the first order approximation. The value of the XMHD model resides in its ability to capture and describe the main two-fluid effects, e.g., Hall drift and electron inertia. Unlike its parent 3D version, 25,26 the 2D incompressible reduction of XMHD (RXMHD) has not yet been explored, however, from the above mentioned Hamiltonian perspective. The Hamiltonian approach can indeed be particularly fruitful in this context because of the richness of the Casimir invariants that typically emerge in 2D models. These invariants, which are associated with the Hamiltonian structure, provide information on the dynamics. Identifying the Hamiltonian structure of RXMHD and providing its Casimir invariants is one of the goals of this paper. A further related issue that we treat, is that of investigating how the conservation laws related to the Casimir invariants in RXMHD, which properly accounts for both ion and electron physics corrections, compare with those of submodels such as ideal reduced, Hall and inertial MHD, where some of these effects are neglected.

In addition to the investigation of the Hamiltonian structure of the model, we also present an application of RXMHD where the Hamiltonian approach plays a role. Given that RXMHD is a model that extends 2D incompressible Hall MHD by properly accounting for electron physics, a natural application for RXMHD is to 2D magnetic reconnection driven by electron inertia. Such magnetic reconnection has already been studied by means of a model very similar to RXMHD in Ref. 27. These authors considered a weakly dissipative model and identified the fundamental mechanisms of two-fluid collisionless reconnection, in particular with regard to the role of the Hall term and of the electron MHD governing the dynamics at scales below the ion skin depth. In the present manuscript we carry out an investigation of purely non-dissipative magnetic reconnection by means of RXMHD, with an approach that is somewhat complementary to that adopted in Ref. 27. We provide an analytical expression for the linear growth rate of reconnecting perturbations and check it against numerical solutions, and we take advantage of the Hamiltonian formulation to compare the evolution of the physical fields, in terms of which the model was originally formulated, with normal fields, an alternative set of variables. Normal fields are associated with the Casimir invariants and express a simpler dynamics. This approach was used in previous studies of collisionless reconnection in Hamiltonian models (see, e.g. Refs. 10, 28-31). Also, the Hamiltonian formulation provides the correct expression for the total energy, which we exploit in order to investigate the redistribution of magnetic energy into different forms. We also remark that, in a recent publication, 32 a model very similar to RXMHD was adopted to investigate numerical reconnection rates and the conservation of three invariants during reconnection.

Our paper is organized as follows. In Sec. II, XMHD is reviewed and the Hamiltonian form of RXMHD, a four-field model, is obtained from that of the full model. 25,26 Consequences of this reduction are explored in Sec. III where it is shown that in addition to the energy, the RXMHD system posses four infinite families of invariants, the Casimir invariants. In addition, the so-called normal fields are obtained and it is observed that the equations of motion take a particularly simple form when expressed in terms of them (compare Eqs. ( 25)- (28) to Eqs. (50)-( 51)). Section IV treats various limits of RXMHD leading to reduced Hall MHD (RHMHD), reduced inertial MHD (RIMHD) and reduced MHD (RMHD). Numerical solution of RXMHD is treated in Sec. V. Here the dispersion relation is plotted for the basic modes of the system, the collisionless tearing instability growth rate is identified and numerically verified, and nonlinear simulations of collisionless tearing are performed, which reveal how energy migrates from field into flow. Finally, in Sec. VI we summarize our results and draw conclusions.

II. DERIVATION OF REDUCED EXTENDED

MAGNETOHYDRODYNAMICS

A. Extended magnetohydrodynamics

The governing equations of extended magnetohydrodynamics (XMHD) are the continuity equation

∂ρ ∂t = -∇ • (ρV) , (1)
the force law,

ρ ∂V ∂t + (V.∇) V = -∇p + J × B -d 2 e (J • ∇) J ρ , (2)
and the generalized Ohm's law

E + V × B = - d i ρ ∇p e + d i J ρ × B +d 2 e ∂ ∂t J ρ + (V • ∇) J ρ + J ρ • ∇ V -d i d 2 e J ρ • ∇ J ρ . ( 3
)
Here ρ is the total mass density, V is the center of mass velocity, B is the magnetic field, E is the electric field, J is the current density and p = p i +p e is the total pressure, with p i being the ion pressure and p e the electron pressure. The system is normalized to the standard Alfvénic units with d e = c/(ω pe L) and d i = c/(ω pi L), corresponding to the normalized electron and ion skin depths, respectively, where ω pe and ω pi are the electron and ion plasma frequencies, and L is the system size. Equations ( 1)-( 3) are coupled with the pre-Maxwell equations

∇ × E = - ∂B ∂t and ∇ × B = J, (4)
and for this paper the systems will be closed by assuming a barotropic equation of state, i.e., the pressure p is assumed to depend only on the density ρ.

Upon using ρ -1 ∇p = ∇h (ρ), which follows from the barotropic assumption, where h (ρ)

is the enthalpy, and using the pre-Maxwell equations (4), one can obtain, from Eqs. ( 2) and

(3), the following system:

∂V ∂t = -(∇ × V) × V + ρ -1 (∇ × B) × B * -∇ h + V 2 /2 + d 2 e (∇ × B) 2 /2ρ 2 , ( 5
) ∂B * ∂t = ∇ × (V × B * ) -∇ × ρ -1 (∇ × B) × B * + d 2 e ∇ × ρ -1 (∇ × B) × (∇ × V) , (6)
where

B * = B + d 2 e ∇ × ρ -1 (∇ × B) . (7)
Equations ( 1), ( 5) and ( 6) with the total energy, 33

H := d 3 x ρ V 2 2 + U (ρ) + B • B * 2 , (8)
as Hamiltonian, and the Poisson bracket

{F, G} = -d 3 x [F ρ ∇ • G V + F V • ∇G ρ ] (9)
- (∇ × V) ρ • F V × G V - B * ρ • F V × (∇ × G B * ) - B * ρ • (∇ × F B * ) × G V +d i B * ρ • (∇ × F B * ) × (∇ × G B * ) -d 2 e (∇ × V) ρ • (∇ × F B * ) × (∇ × G B * )
constitute a noncanonical Hamiltonian system in which the phase space is spanned by the dynamical variables ρ, V, and B * . In (9) F ξ := δF/δξ denote the functional derivative of the functional F with respect to the dynamical variable ξ. The full Poisson bracket of Eq. ( 9) and a proof of the Jacobi identity were first given in Ref. 25, with further properties and a simplified proof of the Jacobi identity given in Ref. 26. The bracket of Eq. ( 9) is an extension of the MHD bracket first given in Ref. 17, amended by the inclusion of two additional terms, one proportional to d i , accounting for the Hall effect, and one proportional to d 2 e , accounting for electron inertia. The Poisson bracket (9) has three independent Casimir invariants,

C 1 = d 3 x B * • V - d i 2d 2 e A * , (10)
C 2 = d 3 x B * • A * + d 2 e V • (∇ × V) , (11)
C 3 = d 3 x ρ. (12)
Combining C 1 and C 2 , produces the "canonical helicities"

C ± = 1 2 d 3 x P ± • ∇ × P ± , (13)
where

P ± = V + λ ± A * , with λ ± = -d i ± d 2 i + 4d 2 e 2d 2 e . (14)
B. Reduced extended MHD

Direct reduction

In the incompressible limit, the reduced extended magnetohydrodynamics (RXMHD) can be obtained by writing V and B in the Clebsch-like forms

B (x, y, t) = ∇ψ (x, y, t) × z + b (x, y, t) z, (15)
V (x, y, t) = -∇φ (x, y, t) × z + v (x, y, t) z , (16)
where ψ and φ are the flux and stream functions, respectively, and b and v are z-components of these fields. From (15), the current density J is seen to be given by

J = ∇ × B = ∇b × z -∇ 2 ψ z, (17)
Upon setting ρ = 1 and using ( 15) and ( 16), the z-component of Eq. ( 2) yields

∂v ∂t = -[φ, v] + [b, ψ] -d 2 e b, ∇ 2 ψ , (18)
where [f, g] = ∇f × ∇g • z, is the standard canonical Poisson bracket with x and y as canonically conjugate coordinates. Similarly, operating with z • ∇× on (2) yields

∂∇ 2 φ ∂t = -φ, ∇ 2 φ -∇ 2 ψ, ψ -d 2 e b, ∇ 2 b , (19)
and the ẑ-component of ( 3) is

- ∂ψ ∂t + [ψ, φ] = d i [b, ψ] -d 2 e ∂ ∂t ∇ 2 ψ + d 2 e ∇ 2 ψ, φ +d 2 e [v, b] -d i d 2 e b, ∇ 2 ψ , (20)
where we made use of the relation E z = -∂ψ/∂t. Finally operating with z • ∇× on (3) gives

- ∂b ∂t + [v, ψ] -[φ, b] = d i ψ, ∇ 2 ψ -d 2 e ∂ ∂t ∇ 2 b +d 2 e ∇ 2 φ, b + d 2 e ∇ 2 b, φ -d i d 2 e b, ∇ 2 b . (21)
Therefore, with the definitions

ω = ∇ 2 φ (22)
ψ * = ψ -d 2 e ∇ 2 ψ (23) b * = b -d 2 e ∇ 2 b , (24)
the RXMHD equations can be written as follows:

∂ψ * ∂t = -[φ, ψ * ] -d i [b, ψ * ] + d 2 e [b, v] , (25)
∂ω ∂t = -[φ, ω] -∇ 2 ψ, ψ -d 2 e b, ∇ 2 b , (26)
∂b * ∂t = -[φ, b * ] + d i ∇ 2 ψ, ψ + [v, ψ] + d 2 e [b, ω] + d i d 2 e b, ∇ 2 b , ( 27
) ∂v ∂t = -[φ, v] + [b, ψ * ] . (28)
The Hamiltonian (energy) (8) in terms of the new variables becomes

H := 1 2 d 2 x -φω -∇ 2 ψψ * + bb * + v 2 , (29)
which can be shown by direct calculation to be conserved by the RXMHD system of ( 25)-( 28).

Reduction via chain rule

Another way to obtain RXMHD is by Hamiltonian reduction. With this method the Poisson bracket ( 9) is rewritten in terms of the new variables via the functional chain rule (see e.g. Refs. 34 and 35 where this is done for MHD). This method has the advantage of yielding directly the Hamiltonian structure of RXMHD.

The chain rule proceeds by assuming functionals obtain their dependence on V and B * through the new variables ω, v, ψ * , and b * , i.e.

F [V, B * ] = F [ω, v, ψ * , b * ] ( 30
)
Varying both sides of (30) gives

d 2 x F V • δV = d 2 x Fω δω + Fv δv , (31)
while variation of the velocity field V of ( 16) gives

δV = z × ∇δφ + δv z. (32)
From (32) we obtain

δv = z • δV (33)
while z × δV = -∇δφ. Thus using δω = ∇ • ∇δφ, we obtain

δω = ∇ • δV × z . (34)
Upon inserting Eqs. ( 32) and ( 34) into Eq. ( 31), performing an integration by parts, and using the arbitrariness of δV we obtain

F V = ∇ Fω × ẑ + Fv . (35)
In a similar way we obtain

∇ × F B * = ∇ Fb * × ẑ + Fψ * ẑ. ( 36
)
Now we are in position to use ( 35) and ( 36) to reduce the Poisson bracket of ( 9) to one in terms of the reduced variables. This calculation gives

{F, G} = d 2 x ψ * [F ω , G ψ * ] + [F ψ * , G ω ] + [F v , G b * ] + [F b * , G v ] (37)
-d i ([F ψ * , G b * ] + [F b * , G ψ * ]) + ω [F ω , G ω ] + d 2 e [F b * , G b * ] + b * [F ω , G b * ] + [F b * , G ω ] -d i [F b * , G b * ] + v [F ω , G v ] + [F v , G ω ] + d 2 e [F ψ * , G b * ] + [F b * , G ψ * ] ,
where, consistent with the representation ( 16), we have removed the ρ dependence and used the relation

d 2 x f [g, h] = d 2 x h [f, g] = d 2 x g [h, f ] , (38)
valid for generic functions f , g and h and appropriate boundary conditions. Here and henceforth, we drop the bars on the functionals.

The above bracket (37) with the Hamiltonian (29) produces the equations of motion ( 25)-

C. Jacobi identity

As a further check that the set of equations ( 25)-( 28) with the Hamiltonian (29) constitutes a noncanonical Hamiltonian system with Poisson bracket (37), we verify the following requisite bracket properties: antisymmetric

• antisymmetry {F, G} = -{G, F } , • Leibniz property {F G, H} = F {G, H} + G {F, H} ,
• Jacobi identity

{F, {G, H}} + {H, {F, G}} + {G, {H, F }} = 0,
Assuming boundary conditions such that surface terms vanishes, as would be the case for periodic boundary conditions, we can easily demonstrate the first two properties. However, the proof of Jacobi identity is more difficult. A direct proof is tedious, but instead we can follow the general theory of Ref.

{F, G} = d 2 x[F µ , F ν ]W µν γ ζ γ , (39)
where F µ = δF/δζ µ and the quantities W µν γ are symmetric in their upper indices. Considering the W µν γ as a family of matrices indexed by ν, the Jacobi identity is satisfied if and only if the following matrices pairwise commute:

W (ω) =        1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1        , W (ψ * ) =        0 0 0 0 1 0 -d i 0 0 0 0 0 0 0 d 2 e 0        , W (b * ) =        0 0 d 2 e 0 0 -d i 0 1 1 0 -d i 0 0 d 2 e 0 0        , W (v) =        0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0       
, which follows from a relatively easy calculation. Consequently, the Poisson bracket (37) satisfies the Jacobi identity.

D. Remarkable transformations

In Ref. 26 it was shown that the Poisson bracket (9) follows from a remarkable sequence of variable and parameter transformations of a basic bracket for Hall MHD. This led a dramatically simplified calculation for the Jacobi identity and quite naturally to the Casimir invariants. We will show that the reduced Poisson bracket of (37) possesses analogous transformations.

Specifically, the bracket (9) maps into the Poisson bracket of Hall MHD in terms of the field B λ ± , when one carries out the transformation

B λ ± = B * + λ -1 ± ∇ × V, (40)
The analogous transformation in our 2D case would then be of the form,

B λ ± = ∇ψ λ ± × ẑ + b λ ± ẑ (41)
and this suggests the change of variables

ψ λ ± = ψ * + v/λ ± , b λ ± = b * + ω/λ ± . (42)
With this change of variables, the bracket (37) becomes

{F, G} = d 2 x ψ λ ± 2 λ ± -d i F ψ λ ± , G b λ ± (43) + F b λ ± , G ψ λ ± + F ψ λ ± , G ω + F ω , G ψ λ ± + F v , G b λ ± + F b λ ± , G v + ω [F ω G ω ] + v [F ω , G v ] + [F v , G ω ] + b λ ± F ω , G b λ ± + F b λ ± , G ω + 2 λ ± -d i F b λ ± , G b λ ± .
Note that one obtains the bracket (43) for either choice of the values of λ ± in Eq. ( 14). Also, note that the bracket ( 43) is identical to the Poisson bracket identified by Eqs. ( 43)- (44) in Ref. 23 if one replaces, in the latter bracket, β with -1 and 2δβ with 2/λ ± -d i . We have thus shown that the bracket (37) can be transformed, by means of an invertible change of variables, into a known Poisson bracket for which the Jacobi identity has already been proven. Consequently, this serves as an alternative verification that the bracket (37) satisfies the Jacobi identity. We remark that the model in Ref. 23 (in the 2D cold ion limit with no magnetic curvature), is isomorphic to 2D incompressible Hall MHD, which is consistent with the above mentioned general result of Ref. 26.

Given the relationship to the results of Ref. 23 we can immediately identify the Casimir invariants and normal fields, a special class of field variables, which we consider next.

III. NORMAL FIELDS AND CASIMIR INVARIANTS A. Normal fields

The four-field bracket of ( 43) is complicated, as one might expect considering the physics described by the RXMHD model. However, as described in Ref. 36, noncanonical brackets can be mapped by coordinate changes into special simplified forms. For systems of four fields, there are only a few such simplified forms. The fields in which the bracket is simplified are called normal fields -for the present case they are given by

ψ + =ψ λ ± , ψ -= ψ λ ± - 2 λ + -d i v, b + =b λ ± , b -= b λ ± - 2 λ + -d i ω. (44)
In terms of these normal fields the bracket (43) becomes

{F, G} = 2 λ + -d i d 2 x ψ + [F ψ + , G b + ] + [F b + , G ψ + ] + b + [F b + , G b + ] -b -[F b -, G b -] -ψ -[F ψ -, G b -] + [F b -, G ψ -] , (45)
a form that is the direct sum of two semidirect product brackets (see Ref. 36). In terms of the normal fields ψ ± , b ± the corresponding Casimirs for this bracket are known to be

C 1,2 = d 2 x F ± (ψ ± ) , C 3,4 = d 2 x b ± G ± (ψ ± ) , (46)
with F ± and G ± arbitrary functions.

We remark that, in the 2D incompressible limit, the Casimir invariants C 1,2 of XMHD reduce to

d 2 x B * • V - d i 2d 2 e A * = d 2 x ωψ * + vb * - d i 2d 2 e ψ * b * , (47)
and

d 2 x B * • A * + d 2 e V • (∇ × V) = d 2 x ψ * b * + d 2 e vω , (48)
respectively. Such Casimir invariants indeed correspond to linear combinations of the Casimir invariants C 3,4 of Eq. ( 46), for the particular choice G ± = ψ ± . This shows how the Casimir invariants of XMHD are related to those of RXMHD.

We remark that, in Ref. 32, a system isomorphic to RXMHD was studied but only three out of the infinite number of invariants of the model were presented.

Noting that 2/λ + -d i = 1/λ + -1/λ -, we find that the normal fields are related to the original variables by

ψ ± = ψ * + v λ ± and b ± = b * + ω λ ± . ( 49
)
In terms of the normal fields the RXMHD system obtains the perspicuous form

∂ψ ± ∂t + [φ ± , ψ ± ] = 0, ( 50
)
∂b ± ∂t + [φ ± , b ± ] = λ ± [ψ ± , ψ], (51)
where We remark that, by expanding in the limit d 2 e /d 2 i → 0, one can obtain the following relations:

φ ± = φ -d 2 e λ ± b
ψ + ψ + d i v iz , ψ -ψ - d 2 e d i v ez , (52) ẑ
× ∇φ + v i⊥ , ẑ × ∇φ -v e⊥ , (53) b
+ b + d i ω i , b -b - d 2 e d i ω e , (54)
φ φ + + d 2 e d 2 i φ -, b φ --φ + d i , (55)
where v iz and v ez are the z-components of the ion and electron fluid velocities, (so that The Hamiltonian (29) can be expressed in terms of the normal fields by making use of the following transformations:

∇ 2 ψ = (v ez -v iz )/d i and v v iz + (d 2 e /d 2 i )v ez ), v
ψ * = λ + ψ + -λ -ψ - λ + -λ - , v = ψ + -ψ - d 2 e (λ + -λ -) , ( 56
) b * = λ + b + -λ -b - λ + -λ - , ω = b + -b- d 2 e (λ + -λ -) , (57)
and introducing the linear operator L, such that ψ * = Lψ and b * = Lb. Assuming this operator is invertible, one can then write

ψ = L -1 λ + ψ + -λ -ψ - λ + -λ - , b = L -1 λ + b + -λ -b - λ + -λ - , (58)
and replace these expressions in (29). The resulting functional is

H = 1 2 d 2 x - b + -b- d 2 e (λ + -λ -) ∇ -2 b + -b- d 2 e (λ + -λ -) + λ + b + -λ -b - λ + -λ - L -1 λ + b + -λ -b - λ + -λ - - λ + ψ + -λ -ψ - λ + -λ - ∇ 2 L -1 λ + ψ + -λ -ψ - λ + -λ - + ψ 2 + -2ψ + ψ -+ ψ 2 - d 4 e (λ + -λ -) 2 . ( 59
)
Because the Hamiltonian of (59) is complicated it may be more straightforward to consider that of (29) in terms of the original variables. An approximate form can be obtained by neglecting again d 2 e /d 2 i when compared to terms of order unity, and making use of the relations (53), (55), and the relation ∇ 2 ψ = (v ez -v iz )/d i . This leads to the following approximate expression for the Hamiltonian:

H 1 2 d 2 x |∇ψ| 2 + b 2 + v 2 i⊥ + v 2 iz + d 2 e d 2 i v 2 e⊥ + v 2 ez . (60)
The expression (60) shows that the Hamiltonian is nearly given by the sum of magnetic energy (the first two terms on the right-hand side of (60)), with the ion kinetic energy (third and fourth terms) and the electron kinetic energy (fifth and sixth terms).

IV. LIMITS OF RXMHD A. 2D incompressible Hall MHD

If we set d e = 0 in Eqs. ( 25)-( 28), we obtain the 2D incompressible Hall MHD system

∂ψ ∂t = -[φ, ψ] -d i [b, ψ], (61)
∂ω ∂t = -[φ, ω] -[∇ 2 ψ, ψ], (62)
∂b ∂t = -[φ, b] + d i [∇ 2 ψ, ψ] + [v, ψ], (63)
∂v ∂t = -[φ, v] + [b, ψ]. (64)
As anticipated above, this model is also Hamiltonian, with Hamiltonian functional

H = 1 2 d 2 x |∇φ| 2 + |∇ψ| 2 + b 2 + v 2 , (65)
and Poisson bracket

{F, G} = d 2 x ψ [F ω , G ψ ] + [F ψ , G ω ] + [F v , G b ] + [F b , G v ] -d i ([F ψ , G b ] + [F b , G ψ * ]) + ω [F ω , G ω ] + v [F ω , G v ] + [F v , G ω ] + b [F ω , G b ] + [F b , G ω ] -d i [F b , G b ] . (66)
This system, which previously appeared in Ref. 23, has the Casimirs

C 1 = d 2 x K (ψ) , (67)
C 2 = d 2 x b S (ψ) , (68)
C 3 = d 2 x T (ψ H ) , (69)
C 4 = d 2 x b H R (ψ H ) , (70)
where b H = b + d i ω, ψ H = ψ + d i v and K, S, T and R are arbitrary functions. In particular, for S = ψ and R = ψ H one retrieves the 2D incompressible versions of the functionals

d 3 x A • B, (71)
and

d 2 x (A + d i V) • (B + d i ∇ × V) , (72)
respectively, which are Casimir invariants for 3D Hall MHD, corresponding to magnetic helicity and to a generalized magnetic helicity.

B. 2D incompressible inertial MHD

If we set d i = 0 in Eqs. ( 25)-( 28) while retaining d e , we obtain the 2D incompressible inertial MHD system

∂ψ * ∂t = -[φ, ψ * ] + d 2 e [b, v] , (73)
∂ω ∂t = -[φ, ω] -∇ 2 ψ, ψ -d 2 e b, ∇ 2 b , (74)
∂b * ∂t = -[φ, b * ] + [v, ψ] + d 2 e [b, ω] , (75)
∂v ∂t = -[φ, v] + [b, ψ * ] . (76)
In this limit the Hamiltonian (29) does not change, but the Poisson brackets becomes

{F, G} = d 2 x ψ * [F ω , G ψ * ] + [F ψ * , G ω ] + [F v , G b * ] + [F b * , G v ] + ω [F ω , G ω ] + d 2 e [F b * , G b * ] + b * [F ω , G b * ] + [F b * , G ω ] + v [F ω , G v ] + [F v , G ω ] + d 2 e [F ψ * , G b * ] + [F b * , G ψ * ] . (77)
We can easily proof that the above system is Hamiltonian through one of the methods discussed in Sec. II C.

It may seem odd to retain d e while dropping d i , since they scale with the mass ration, but this limit may make sense in a different ordering. 33 The Poisson bracket (77) possesses the following four families of Casimir invariants:

C 1,2 = d 2 x Y ± ψ i ± , (78)
C 3,4 = d 2 x b i ± P ± ψ i ± , (79)
where b i ± = b * ± d e ω, ψ i ± = ψ * ± d e v and Y ± and P ± are arbitrary functions. 3D inertial MHD, which previously appeared in Ref. 26, has the Casimirs

d 3 x B * • V (80)
and

d 3 x B * • A * + d 2 e V • (∇ × V) , (81)
which, in their 2D incompressible limit, become

d 2 x (ωψ * + b * v), (82)
and

d 2 x (ψ * b * + d 2 e v ω), (83)
respectively. These are linear combinations of

d 2 x b i ± ψ i ± , (84)
corresponding to the Casimir invariants C 3,4 of Eq. (79) for the choice P ± = ψ i ± .

C. 2D incompressible ideal MHD

The 2D incompressible ideal MHD system can be obtained by setting d i = d e = 0 in Eqs. ( 25)-( 28), giving

∂ψ ∂t = -[φ, ψ] , (85)
∂ω ∂t = -[φ, ω] -∇ 2 ψ, ψ , (86)
∂b ∂t = -[φ, b] + [v, ψ] , (87)
∂v ∂t = -[φ, v] + [b, ψ] . (88)
Reduced ideal MHD has energy

H = 1 2 d 2 x |∇φ| 2 + |∇ψ| 2 + b 2 + v 2 , (89)
and the Poisson bracket

{F, G} = d 2 x ψ [F ω , G ψ ] + [F ψ , G ω ] + [F v , G b ] + [F b , G v ] + ω [F ω , G ω ] + v [F ω , G v ] + [F v , G ω ] + b [F ω , G b ] + [F b , G ω ] . (90)
This reduced ideal MHD model, which previously appeared in Ref. 23, has the Casimirs

C 1 = d 2 x I (ψ) , (91)
C 2 = d 2 x b O (ψ) , (92)
C 3 = d 2 x v Q(ψ), (93)
C 4 = d 2 x (ω U(ψ) + b v U (ψ)), (94)
where I, O, Q, U are arbitrary functions and the prime symbol denotes derivative with respect to the argument of the function.

With the choices O = ψ and U = ψ one retrieves, from C 2 and C 4 in Eqs. ( 92) and (94), the 2D incompressible versions of the magnetic helicity

d 3 x A • B ( 95
)
and of the cross-helicity

d 3 x V • B ( 96
)
of 3D ideal MHD.

V. NUMERICAL RESULTS

A. Linear analysis

Before describing our nonlinear simulations, we perform a simple linear stability analysis in Sec. V A 1 to verify that RXMHD contains the basic whistler and cyclotron waves. This is followed by an investigation of collisionless tearing modes in our nonlinear simulation geometry, which serves as an introduction to our nonlinear numerical results.

Basic modes

We linearize the RXMHD equations of ( 26)-( 27) to investigate the basic modes it contains. Upon assuming a magnetostatic equilibrium state corresponding to a unit vector b0 in x -y plane, we expand all quantities as ψ = ψ exp (ik ⊥ .r ⊥ -iwt), where w is the angular frequency and k ⊥ is the perpendicular wavenumber, to obtain

φ = b0 • k ⊥ w ψ ṽ = - b0 • k ⊥ w b w 1 + d 2 e k 2 ⊥ ψ = b0 • k ⊥ φ + d i b0 • k ⊥ b w 1 + d 2 e k 2 ⊥ b = b0 • k ⊥ ṽ -d i k 2 ⊥ b0 • k ⊥ ψ .
Manipulation of the above yields

w 1 + d 2 e k 2 ⊥ ψ = b0 • k ⊥ 2 w ψ + d i b0 • k ⊥ b, w 1 + d 2 e k 2 ⊥ b = b0 • k ⊥ 2 w b + d i k 2 ⊥ b0 • k ⊥ ψ ,
whence we obtain the dispersion relation of RXMHD

w 2 1 + d 2 e k 2 ⊥ -k 2 ⊥ cos 2 θ 2 = w 2 d 2 i k 4 ⊥ cos 2 θ ,
where θ is the angle between b0 and k ⊥ .

As expected, this linear dispersion relation is coincident with the 3D nonlinear dispersion relation of XMHD. 37 In Fig. 1, the upper branch represents whistler waves, whilst the lower branch represents ion cyclotron waves. We can also observe that both branches saturate, at the electron gyrofrequency and ion gyrofrequency, respectively.

Collisionless tearing modes

The XMHD model of ( 25)-( 28) can describe various instabilities, including collisionless tearing modes induced by the presence of electron inertia, which breaks the usual MHD frozen-in condition as evidenced by Eqs. ( 25) and (27).

In order to investigate collisionless tearing, we suppose a resonant surface is located at

x = 0 and we choose an equilibrium around x = 0 given by ψ eq = -x 2 , b eq = 0, φ eq = 0, v eq = 0.

(97) Then upon linearizing the system of ( 25)-( 28) about this equilibrium, assuming solutions of the form ψ(x, y, t) = ψ(x)e iky+γt with analogous expressions for φ, v, and b, where the constants γ and k indicate the growth rate and the wave number of the perturbation, respectively, we obtain

g((1 + k 2 d 2 e ) ψ -d 2 e ψ ) = -ix( φ + d i b), ( 98
) g( φ -k 2 φ) = -ix( ψ -k 2 ψ), ( 99
) g((1 + k 2 d 2 e ) b -d 2 e b ) = ixd i ( ψ -k 2 ψ) + ixṽ, (100)
gṽ = ix b, ( 101
)
where g = γ/(2k) and the prime symbol denotes derivative with respect to the argument.

We remark then that the linear system (98)-(100) corresponds also to the linearization of the four-field model studied in Ref. 38, provided one uses ψ eq = -x 2 instead of ψ eq = -x 2 /2 and replaces the constant d β of Ref. 38 with the constant d i (which corresponds to taking the limit β → +∞). The additional terms of the model of ( 25)-( 28) that are absent in the four-field model of Ref. 38, indeed, contribute only during the nonlinear regime. We can then export the analysis carried out in Ref. 38, where a relation for the growth rate in terms of equilibrium parameters was found by asymptotic matching. In this way we obtain the following expression for the growth rate for our system of (98)-(100):

- π ∆ - π 2
g 2 d i G(g/d i ) + d e d i G(g/d i ) g = 0, ( 102
)
where ∆ is the classical tearing stability parameter 39 In Fig. 2, values of the growth rate γ, obtained from the asymptotic relation (102), are checked against values obtained from numerical simulations, for a case with large ∆ . The numerical code used for these simulations is an adaptation of the one used in Ref. 11 to solve the four-field system initialized by perturbing about the equilibrium

ψ eq = 1 cosh 2 x , b eq = φ eq = v eq = 0. ( 103
)
Note that the equilibrium (103), when expanded about x = 0, corresponds to the equilibrium (97) adopted for deriving the analytical expression for the growth rate.

The model equations are solved on a grid consisting of up to 2048×4096 grid points, depending on the scale lengths to be resolved. All the fields are split into the timeindependent equilibrium and an evolving perturbation advanced in time by a third order Adams-Bashforth algorithm. Periodic boundary conditions have been imposed along the shear equilibrium magnetic field direction, y, whereas Dirichlet conditions have been applied in the x-direction with all the perturbed fields vanishing at the boundaries. A pseudospectral method is adopted for the periodic direction, while a compact finite difference algorithm on a non-equispaced grid is used for the spatial operations along the x direction.

The tearing instability is initiated by perturbing the equilibrium with a small disturbance of the parallel current density j = -∇ 2 ψ of the form δj (x, y) = δj (x) cos(2πy/L y ), where δj (x) is a function localized within a width of order d e around the rational surface x = 0.

One can observe from the figure that the agreement between numerical and analytical values becomes better and better as the parameter g/d i decreases. This is expected, since, the relation (102) holds in the asymptotic limit g/d i 1. We remark that, in the large ∆ regime, in the limit g/d i 1, the relation (102) can be approximated by 38

g = 1 √ 2π Γ(1/4) Γ(3/4) d e d i , (104)
which also shows the accelerating role played by the Hall term, associated with the length scale d i .

B. Nonlinear numerical simulations

Having obtained a handle on the linear dynamics, we now describe our nonlinear numerical simulations. In particular, we follow the nonlinear evolution of the velocity and magnetic fields during the process of magnetic reconnection initiated by perturbing the equilibrium (103). The code employed is that of Sec. V A 2. As noted in Sec. III A, it is evident from Eqs. (52) that the normal fields ψ ± correspond to the z-components of the electron and ion canonical momenta. It is important to recall that the notion of canonical momentum originates in the Hamiltonian formalism and, consequently, that they should play a clarifying role in the present application is not surprising.

Because the fields ψ ± are advected by the velocities associated with φ ± (cf. Eqs. ( 50)), we examine the moduli |∇φ + | and |∇φ -| of the perpendicular velocities V ± = z × ∇φ ± that are doing the advecting. Figure 6 shows profiles of the moduli at y = 0, i.e. across the X-point.

One observes that, in a narrow region around the resonant surface at x = 0, the velocity V -, which is predominantly due to the electrons, dominates over the the velocity V + , which is predominantly due the ions and actually vanishes at x = 0. This is consistent with the behavior described in Ref. 27 where magnetic flux bundle coalescence in a high β regime was investigated. As discussed in Ref. 27, this behavior can be explained considering that, in a region with the size L ∼ d e d i around the resonant surface, the system ( 25)-( 28) reduces to 2D electron MHD. The dynamics is then essentially governed by electron motion, whereas ions are immobile. On the other hand, on scales L ≥ d i , one enters an MHD regime, where v e⊥ ≈ v i⊥ , as Fig. 6 shows with corrections due to the use of the normal fields. The global structures of the velocities V ± are revealed in the contour plots of φ + and φ -, respectively, shown in Fig. 7. As expected, upon comparing φ to φ + it is seen that the bulk velocity is mostly due to the ion velocity, which exhibits the characteristic convective cells.

The stream function φ -, associated with the corrected electron velocity, on the other hand, concentrates mainly in narrow structures along the separatrices. An analytical argument justifying such behavior of the electron velocity was provided in Ref. 27, based on the electron MHD approximation, valid on scales much smaller than d i .

Because of the Hamiltonian nature of the model the total energy of ( 29) is conserved, yet during the course of the dynamics energy may transfer from one term to another. In order to track this we write with

H = H Kp + H v + H B + H b + H Ke + H Kez (105)
H Kp = d 2 x |∇φ| 2 /2 (106)
H v = d 2 x v 2 /2 (107)
H B = d 2 x |∇ψ| 2 /2 (108)
H b = d 2 x b 2 /2 (109)
H Ke = d 2 x d 2 e |∇ 2 ψ| 2 /2 (110)
H Kez = d 2 x d 2 e |∇b| 2 /2 (111)
and track each term during the reconnection process. Here, for convenience and physical clarity, we do this in terms of the original fields that appear in the Hamiltonian as a sum of squares, rather than evaluate the expression of (59) in terms of the normal fields. In Fig. 8 To compare these results with more conventional analyses we consider the approximate Hamiltonian of (60), which although not exactly conserved can be used to proved a physically transparent interpretation of the energy redistribution process during the reconnection. We in that reference.

VI. SUMMARY AND CONCLUSIONS

In this paper we have given a comprehensive analysis of reduced extended MHD, a 2D version of extended MHD. We have derived the model starting from the Hamiltonian form of Lüst's equations by a reduction procedure that produced the Hamiltonian form of the reduced model, RXMHD. This procedure led to the physical energy, which serves as the reduced Hamiltonian, the four families of Casimir invariants, and the definitions of the normal fields (b ± , ψ ± ) in terms of which the four equations of motion take a simplified intuitive form. Further reductions of the RXMHD led in a natural way to reduced Hall MHD, inertial MHD, and ideal MHD, Hamiltonian field theories with conserved energies and associated Casimir invariants. Analyses of RXMHD revealed the natural modes of oscillation, the expected whistler and ion cyclotron waves. The analytical expression for the linear collisionless tearing growth rate was inferred and checked against numerical solutions. Nonlinear simulations of collisionless tearing revealed behavior typical of Hall and electron inertia physics, but better organized by the new normal field variables.

The content of this work opens many avenues for further study, both analytical and numerical. We mention a few. On the analytical side one can effect absolute equilibrium calculations akin to those of Refs. Having in hand the weakly 3D version of RXMHD opens the way for numerical treatment of weakly 3D collisionless tearing.

( 28 )

 28 in the form ∂ζ/∂t = {ζ, H }, where ζ = (ψ * , ω, b * , v) t denotes the dynamical variables of the system.

36 .

 36 Using ζ = (ψ * , ω, b * , v) t with each field being indexed by ζ µ , µ = 1, • • • , 4, we can write (37) in the form

.

  Here we also made use of the relation d i -1/λ ± = -d 2 e λ ± . From Eqs. (50) and (51) it emerges that ψ ± are Lagrangian invariants of the model, reminiscent of Ohm's law for reduced MHD (RMHD), whereas the equations describing b ± are reminiscent of the RMHD vorticity equation.

  i⊥ and v e⊥ are the ion and electron perpendicular fluid velocities, whereas ω i,e = ẑ • ∇ × v i,e⊥ are the z components of the corresponding vorticities. From Eqs. (52)-(54) it emerges then that ψ ± correspond to the zcomponents of the canonical momenta for ions and electrons. These are advected, according to Eqs. (50), by the perpendicular ion and electron velocities, respectively. The normal fields b ± , on the other hand, represent some generalized vorticities, analogous to the generalized vorticity of Hall-MHD.

FIG. 1 .

 1 FIG. 1. The dispersion relation (w) profiles for θ = 0, d i = 0.1 and d e = 0.0233. The upper branch corresponds to whistler waves, while the lower branch represents ion cyclotron waves. The dashed reference line corresponding to ideal Alfvén waves which in dimensionless units is w = k ⊥ .

d e g d i 1 are satisfied. 38 FIG. 2 .

 382 FIG. 2. Comparison between values of the growth rate γ obtained from numerical simulations and from the asymptotic relation (102), for different values of the parameter g/d i . Crosses and asterisks indicate analytical and numerical values, respectively, for d i = 0.5, whereas diamonds and triangles correspond to analytical and numerical values, respectively, for d i = 0.2. For all cases ∆ = 59.9.

Figure 3

 3 Figure 3 shows contour plots of the out-of-plane magnetic and vorticity fields, at times well into the nonlinear regime, for two choices of skin depths with the same mass ratios. The two times, 126τ A for the case with d e = 0.05, d i = 0.5 and 56τ A for the case with d e = 0.1, d i = 1, were chosen because they represent approximately the same nonlinear stage. In both cases the field b exhibits the characteristic quadrupolar structure, a signature of Hall reconnection (see, e.g., Refs. 41 and 42). In the case with the smaller skin depths, Figs. 3(a) and 3(b),one observes that the vorticity concentrates on a narrow region with a size of order of d e . In this region the behavior is mainly dictated by incompressible hydrodynamics and the system can eventually become prone to the Kelvin-Helmholtz instability.27 On the other hand, when increasing d e and d i , as in Figs.3(c) and 3(d), vorticity is no longer concentrated on a narrow

FIG. 3 .

 3 FIG. 3. Contour plots of the out-of-plane magnetic field b and vorticity ω at time 126τ A for d e = 0.05 and d i = 0.5 (Figs. 3(a) and 3(b)) and at time 56τ A for d e = 0.1 and d i = 1 (Figs. 3(c) and 3(d)) . The magnetic island is superimposed on the contour plots.

FIG. 4 .

 4 FIG. 4. Contour plots of the velocity field v Fig. 4(a) and flux ψ Fig. 4(b) at time 126τ A for d e = 0.05 and d i = 0.5. The magnetic island is superimposed on the contour plots.

FIG. 5 .

 5 FIG. 5. Contour plots of normal fields (b ± , ψ ± ) at time 126τ A for d e = 0.05 and d i = 0.5. The magnetic island is superimposed on the contour plots.

FIG. 6 .

 6 FIG. 6. Plots of |∇φ + | (dashed line) and |∇φ -| (solid line) at y = 0 and t = 126.

FIG. 7 .

 7 FIG. 7. Contour plots of φ (left), φ + (middle) and φ -(right) for d e = 0.05 and d i = 0.5. The magnetic island is superimposed to the contour plots..

  write the approximate Hamiltonian H as the sumH = HB + Hi + He ,(112)where HB = (1/2)d 2 x(|∇ψ| 2 + b 2 ) is the total magnetic energy, Hi = (1/2) d 2 x(v 2 i⊥ + v 2iz ) is the total ion kinetic energy and He = (1/2)(d 2 e /d 2 i ) d 2 x (v 2 e⊥ + v 2 ez) is the total electron kinetic energy. It is easy to infer from Fig.8that reconnection converts most of the magnetic energy into kinetic energy of the ion flow. This is consistent with what is heuristically mentioned in Ref.27, although the actual conserved energy was not identified

FIG. 8 .

 8 FIG. 8. Temporal plots of the terms of the Hamiltonian H of (105) relative to their values at t = 0 for parameters values d e = 0.05 and d i = 0.5. The total energy, represented by E tot , retains its initial value (solid line) throughout the simulation. The other terms, defined by (106)-(111), are seen to increase at the expense of the decreasing perpendicular magnetic energy E B (dashdot-dot-dot) with most energy going into the perpendicular kinetic energy E Kp (dash-dot). Note that times before t = 60τ A are not shown because the dynamics is still in the linear phase where variations of all terms are negligible.

  43 and 44 in order to infer energy cascades. In addition one can derive the Hamiltonian form of 3D incompressible XMHD using the Dirac constraint technique of Ref. 45 and derive the weakly 3D version of the present model, where the latter gives rise to terms linear in parallel derivatives caused by a strong guide field. The general Hamiltonian form of such weakly 3D models is available in Ref. 30 and the correct one can be obtained by aspect ratio expansion of the full XMHD model or by Hamiltonian reduction.

  the total energy H is displayed as a solid line, showing that indeed the numerics preserves it well up to time 126τ A for our example with d e = 0.05 and d i = 0.5. Instead of plotting H , we plot E tot , which is the total energy relative to the initial value of H . The other energies of (106)-(111) are plotted similarly, e.g. E Kp is the value of H Kp , relative to H . Next we observe that the energy E B decreases while getting transferred to all of the

other terms in varying amounts. The energy E Ke , which is essentially the electron kinetic energy, gains only a small amount, as does the energy E Kez , which also contains higher order derivatives. Note both these energies are referred to the left hand scale. All of the other energies grow significantly more, but by far most of the energy goes into E Kp , the

ACKNOWLEDGMENTS

ET was supported by the CNRS by means of the PICS project NEICMAR. HMA would like to thank the Egyptian Ministry of Higher Education for supporting his research activities.

PJM was supported by U.S. Dept. of Energy under contract #DE-FG02-04ER-54742. He would also like to acknowledge support from the Alexander von Humboldt Foundation and the hospitality of the Numerical Plasma Physics Division of the IPP, Max Planck, Garching.