Norbert Hounkonnou
email: norbert.hounkonnou@cipma.uac.bj

Partha Guha
email: parthaguh@gmail.com

Tudor Ratiu
email: tudor.ratiu@epfl.ch

Mahouton Norbert Hounkonnou

Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties

Keywords: Virasoro algebra, Left-symmetric algebras, Quasi-associativity, Coboundary operators, Nonlinear systems of hydrodynamic type. Mathematics Subject Classification (MSc) 2010: 17B68, 17B66

published or not. The documents may come

Introduction

The Virasoro algebra, also known as centrally extended Witt algebra, is probably one of the most important algebra studied by physicists and mathematicians in last few decades. It has a profound impact on mathematical and physical sciences. It appears naturally in problem with conformal symmetry and where the essential space-time is one or two dimensional and space is compactified to a circle. For more details, see [START_REF] Auslander | Simply transitive groups of affine motions[END_REF]- [START_REF] Burde | Left-invariant affine structures on reductive Lie groups[END_REF], [START_REF] Di Francesco | Conformal field theory[END_REF], [START_REF] Fuchs | Cohomologies of Lie algebra of vector fields on the circle[END_REF], [START_REF] Goddard | Kac -Moody and Virasoro algebras[END_REF], [START_REF] Green | Superstring theory[END_REF], [START_REF] Kac | Bombay lectures on highest weight representations of infinite-dimensional Lie algebras[END_REF], [START_REF] Kong | Classification of graded left-symmetric algebraic structures on Witt and Virasoro algebras[END_REF]- [START_REF] Palcoux | Panorama around the Virasoro algebra[END_REF], [START_REF] Virasoro | Subsidiary conditions and ghosts in dual-resonance models[END_REF], [START_REF] Wassermann | Lecture notes on the Kac-Moody and Virasoro algebras[END_REF] but also references therein.

We deal with one of the most important infinite dimensional Lie algebra, the Witt algebra W and its universal central extension. The Witt algebra is defined as the complex Lie algebra of derivations of the algebra C[θ , θ -1 ] of complex Laurent polynomials. The elements of Witt algebra W are defined as d n = ie inθ d dθ , n ∈ Z, so

W = ⊕ n∈Z Cd n .
The Lie-bracket of elements of W yields [d m , d n ] = (mn)d m+n .

The Virasoro algebra is constructed from the Witt algebra W by non-trivial central extension, called the Gelfand-Fuchs cocycle.

Recently, Kuperschmidt [START_REF] Kupershmidt | On the nature of the Virasoro algebra[END_REF] investigated the Virasoro algebra with the multiplication [e p , e q ] := e p e qe q e p = (pq)e p+q + θ (p 3p)δ p+q , p, q ∈ Z, [θ , e p ] = 0, (1.1) in a quasiassociative algebra endowed with the product e p e q = -q(1 + εq) 1 + ε(p + q) e p+q + 1 2 θ [p 3p + (εε -1 )p 2 ]δ 0 p+q , e p θ = θ e p = 0.

(1.2)

He focussed his analysis on the centerless quasiassociative multiplication e p e q = -q(1 + εq)

1 + ε(p + q) e p+q . (1.3)
He verified that this multiplication satisfies the quasiassociativity property e p (e q e r ) -(e p e q ) e r = e q (e p e r ) -(e q e p ) e r , p, q, r ∈ Z, (1.4) and re-interpreted, in the language of 2-cocycle, the property of a bilinear form to provide a central extension of a quasiassociative algebra. His study led to a complex on the space of cochains and its generalization. Besides, Kuperschmidt discussed the homology and performed the differentialvariational versions of the main results for the case when the centerless Virasoro algebra is replaced by the Lie algebra of vector fields on the circle. This paper addresses a generalization of the algebra (1.1), denoted by (A , [., .]), endowed with the multiplication [e x i , e x j ] = g(x i , x j )e x i +x j , (1.5) coming from the commutator [e x i , e x j ] = ae x i e x jbe x j e x i , (1.6) where (a, b) ∈ R × R + , (x i , x j ) ∈ Z 2 . We give the necessary and sufficient condition for this algebra to be a quasiassociative algebra with the multiplication e x i e x j = f (x i , x j )e x i +x j .

(1.7)

Let us immediately mention that such a generalization of the algebra (1.1) can lead to various classes of nonassociative algebras [START_REF] Schafer | An introduction to nonassociative algebras[END_REF] such as alternative algebras, Jordan algebras, and so on, as well as to their various extensions, depending on the defining functions f and g, but also on the real constants a and b. However, without loss of generality, in the sequel, the functions f and g are assumed to be defined as follows: f , g : Z × Z -→ Z, (x i , x j ) → f (x i , x j ), g(x i , x j ).

(1.8)

Moreover, in this work, we are only interested in the class of left symmetric algebras, also called quasi-associative algebras.

The main results obtained in this work can be summarized in the four following theorems:

Theorem 1.1. For the multiplication e x i e x j := R k i j e x k , the quasi-associativity condition (e x i e x j ) e x le x i (e x j e x l ) = (e x j e x i ) e x le x j (e x i e x l ), (1.9)

is expressed by the Nijenhuis-torsion free relation:

R k i j -R k ji R m kl -R k jl R m ik + R k il R m jk = 0.
(1.10)

Theorem 1.2. The hereditary condition for a linear map Φ : A -→ A associated with the generalized Virasoro algebra (1.5) makes into:

• For Φ : e x i → e x i +x 0 for some fixed x 0 ∈ Z, ∀x i ∈ Z, g(x i + x 0 , x j ) + g(x i , x j + x 0 )g(x i , x j ) = g(x i + x 0 ,

x j + x 0 ), (1.11)
or, equivalently,

T x 0 + L x 0 -T x 0 L x 0 -1 g(x i , x j ) = 0, (1.12)
where the right and left translation operators T x 0 and L x 0 are defined, respectively, by

T x 0 g(x i , x j ) := g(x i , x j + x 0 ), L x 0 g(x i , x j ) := g(x i + x 0 , x j ); (1.13) • For Φ : e x i → R k i e x k , g(x m , x j )R m i R s m+ j + g(x i , x n )R n j R s n+i -g(x i , x j )R t i+ j R s t e x s = g(x m , x n )R m i R n j e x m +x n ; (1.14) • For Φ : e x i → e x i +1 , [e x i , e x j ] := R k i j e x k , R k (i+1) j + R k ( j+1)i -R k i j -R k (i+1)( j+1) = 0. (1.15)
Theorem 1.3. For a map ρ : A -→ A , e x i -→ e x i +x 0 with some fixed x 0 ∈ Z, ∀x i ∈ Z, the ρ-compatibility equation is equivalent to the invariance of the operator (1 -E ) under the action of the right translation operator T x 0 , i.e.,

T x 0 (1 -E ) = (1 -E ), (1.16)
where the right translation and exchange operators T x 0 and E are defined, respectively, by

T x 0 f (x i , x j ) = f (x i , x j + x 0 ), E f (x i , x j ) = f (x j , x i ).
(1.17)

Theorem 1.4. The universal identity, known for nonassociative algebras [START_REF] Sokolov | Deformations of nonassociative algebras and integrable differential equations[END_REF], turns out to be in the following form for the generalized algebra introduced with the multiplication (1.7):

f (x k , x j ) -L x i (x j , x k + x j )(1 -E )( f (x i , x j ) + [[ f (x j , x k + x s ), f (x i , x k + x j )]] -L x i +x j f (x k , x s ) -L x j (x i , x k )(1 -E )( f (x i , x j ) + [[ f (x j , x k ), f (x i , x k )]] -T x i +x j f (x k , x j ) -L x j (x i , x j )(1 -E )( f (x i , x j ) + [[ f (x j , x s ), f (x i , x s )]] = 0 (1.18)
where

[[ f (x j , x l ), f (x i , x l )]] = f (x j , x l )T x s f (x i , x l ) -f (x i , x l )T x i f (x j , x l ), (1.19)
and L u , T v are the usual left and right translation operators, respectively.

Finding a general solution f satisfying this universal identity remains an open issue. There exist, however, known particular solutions such as those investigated by Kupershmidt in [START_REF] Kupershmidt | On the nature of the Virasoro algebra[END_REF], i. e., (1.3) and f (x i , x j ) = λx j , λ = const, valid only in the case a = b = 1 and g(x i , x j ) = x ix j .

In the sequel, we give a full characterization of this generalized Virasoro algebra for which the quasi-associativity condition and the criteria making it a Lie algebra are discussed. We deduce the hereditary operator and its generalization to the corresponding 3-ary bracket. Further, we deduce the so-called ρ-compatibility equation, and investigate a phase-space extension. Finally, concrete relevant particular situations are analyzed.

The paper is organized as follows. Section 2 deals with some preliminaries on left symmetric algebras (also called quasi-associative algebras). In section 3, we discuss the main properties of the generalized Virasoro algebra. The case of left-alternative algebra structure and its link to some classes of nonlinear systems of differential equations are also recalled. Coboundary operators and a 3-ary bracket are defined and discussed. We define the hereditary operator, and generalize it to the case of 3-ary bracket. Then, we derive the associated ρ-compatibility equation. Phasespace extension is also discussed. In section 4, we investigate the full centrally extended Virasoro algebra in the framework of the considered formalism. In section 5, we analyze the case of an infinite dimensional Lie algebra of polynomial vector fields on the real line R 1 and deduce some remarkable identities. We end, in section 6, by some concluding remarks.

Preliminaries on left symmetric algebras

The ordinary centerless Virasoro-Witt algebra belongs to the class of quasi-associative algebras, known, in the literature [START_REF] Kong | Classification of graded left-symmetric algebraic structures on Witt and Virasoro algebras[END_REF] (and references therein), under the name of left-symmetric algebras (LSAs), arising in many areas of mathematics and physics. The LSAs were initially introduced by Caley in 1896 in the context of rooted tree algebras and in recent years Vinberg and Koszul re-introduced them in the context of convex homogeneous cones. LSAs have also independently appeared in the works by Gerstenhaber. As a consequence, perhaps, LSAs are known under different names. They are also called Vinberg algebras, Koszul algebras, or quasi-associative algebras, Gerstenhaber algebras, or pre-Lie algebras. See [START_REF] Burde | Left-invariant affine structures on reductive Lie groups[END_REF], [START_REF] Wassermann | Lecture notes on the Kac-Moody and Virasoro algebras[END_REF] and [START_REF] Winterhalder | Linear Nijenhuis-tensors and the construction of integrable systems[END_REF] (and references therein).

Geometrically, they are also connected to the theory of affine manifolds and affine structures on Lie groups. See [START_REF] Auslander | Simply transitive groups of affine motions[END_REF], [START_REF] Milnor | On fundamental groups of complete affinely flat manifolds[END_REF], [START_REF] Burde | Left-invariant affine structures on reductive Lie groups[END_REF] and references therein. Recall a smooth manifold which admits a linear connection whose torsion and curvature tensor vanish, is called an affinely flat (or simply affine in short) manifold. By a well known theorem of differential geometry, such a manifold is locally equivalent to an open subset of Euclidean space with the standard connection, i. e., for each point of the manifold, there are a neighborhood and a coordinate map into the Euclidean space which is an affine equivalence. In fact, the torsion and curvature are exactly the obstructions to the existence of such a map. In general, a connection on a Lie group is completely determined by the action on the left invariant vector fields, i.e., by X Y for X,Y ∈ g using the Leibniz rule. is left-invariant if and only if X Y ∈ g whenever X,Y ∈ g. To perceive the problem algebraically, denote X Y by X.Y for a left-invariant connection and vector fields X,Y ∈ g. Then having a left-invariant connection on G is the same as having an algebra structure on g. In this way, the geometric problems involving left-invariant connection become algebraic ones.

A left-invariant connection on G is said to be bi-invariant if it is also right-invariant. As usual, this holds if and only if is adjoint invariant. We can characterize bi-invariant connections using the associated algebra structure X.Y = X Y as follows [START_REF] Kim | The geometry of left-symmetric algebra[END_REF]:

Proposition 2.1. The following statements are equivalent: (i) A left-invariant connection on G is bi-invariant. (ii) Ad g is an algebra automorphism on (g, .) for all g ∈ G.

(iii) Ad X is an algebra derivation on (g, .) for all X ∈ g, i.e.,

Ad X (Y.Z) = Ad X (Y ).Z +Y.Ad X (Z) or [X,Y.Z] = [X,Y ].Z +Y.[X, Z], X,Y, Z ∈ g. (2.1)
Proof. See [START_REF] Kim | The geometry of left-symmetric algebra[END_REF].

If furthermore the connection is torsion free, then the algebra automorphism becomes a Lie algebra automorphism since [X,Y ] = X.Y -Y.X, X,Y ∈ g. Suppose further that is affinely flat so that it has vanishing torsion and curvature tensor. Then, using small letters for elements of g and xy for X.Y = X Y , the torsion-free condition and the flatness of become algebraically:

xy -yx = [x, y] (2.2) x(yz) -y(xz) -[x, y]z = 0, (2.3)
respectively, for all x, y, z ∈ g. This leads to the following definition.

Definition 2.1. Let A be a vector space over a field K equipped with a bilinear product (x, y) → xy.

A is called a left symmetric algebra, or, equivalently, a quasi-associative algebra, if for all x, y, z ∈ A the associator x, y, z = x(yz) -(xy)z is symmetric in x, y, i.e.,

x, y, z = y, x, z , or (xy)zx(yz) = (yx)zy(xz).

(2.4)

Hence finding a left invariant affinely flat connection on G is the same as finding a leftsymmetric algebra structure on g which is compatible with Lie algebra structure of g in the sense of (2.2).

Left symmetric algebras are Lie-admissible algebras (cf. [START_REF] Perea | Flat left-invariant connections adapted to the automorphism structure of a Lie group[END_REF]). Let A be a LSA, then for any x ∈ A , denote L x the left multiplication operator, L x (y) = xy for all y ∈ A . By setting [x, y] := xy-yx a Lie bracket defines a Lie algebra G (A ), known as the sub-adjacent Lie algebra of A . Thus A is called a compatible left symmetric algebra structure on the Lie algebra G (A ).

Let G (A ) → gl(A ) with x → L x . Then (L, A ) gives a representation of the Lie algebra G (A ), i.e., [L x , L y ] = L [x,y] for all x, y ∈ A . But this is neither sufficient nor necessary condition to give compatible LSA on any Lie algebra. These are given as follows. Let G be a Lie algebra with a representation ρ : G → gl(V ), then a one-cocycle q : G → V be a linear map associated to (ρ, q) such that q[x, y] = ρ(x)q(y)ρ(y)q(x) for all x, y ∈ G . It has been shown in [START_REF] Kong | Classification of graded left-symmetric algebraic structures on Witt and Virasoro algebras[END_REF] that there is a compatible LSA structure if and only if there exists a bijective one-cocycle of G .

If (ρ, q) is a bijective one-cocycle of G then x * y = q -1 ρ(x)q(y) defines a LSA structure on G , where as for a LSA A the identity transformation is a one-cocycle of G (A ) associated with the regular representation L.

It is also worth noticing that left-symmetric structures appear of natural way in the theory of integrable systems of hydrodynamic type (see [START_REF] Kupershmidt | Left-symmetric algebras in hydrodynamics[END_REF], [START_REF] Dubrovin | Differential geometry of strongly integrable systems of hydrodynamic type[END_REF], [START_REF] Tsarev | On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type[END_REF] (and references therein)) and the generalized Burgers equation. Indeed, it was proved by Sokolov and co-workers [START_REF] Habibullin | Multi-component integrable systems and nonassociative structures[END_REF], (see also [START_REF] Sokolov | Deformations of nonassociative algebras and integrable differential equations[END_REF] and references therein), the following theorem giving the link between LSAs and multicomponent generalizations of Burgers equations.

Theorem 2.1. If C i jk are structure constants of any LSA, then the system of Burgers equations,

u i t = u i xx + 2C i jk u k u j x + A i jkm u k u j u m , where i, j, k, m = 1, . . . , N (2.5)
is integrable iff the following relations hold:

A i jkm = 1 3 C i jr C r km +C i kr C r m j +C i mr C r jk -C i r j C r km -C i rk C r m j -C i rm C r jk C i jr C r km -C i kr C r jm = C r jk C i rm -C r k j C i rm . (2.6)
In this case, let e 1 , . . . , e N be a basis of a LSA A , and u = u i e i . Then, the integrable system can be written as

u t = u xx + 2u • u x + u • (u • u) -(u • u) • u, (2.7)
where • denotes the multiplication in A .

Generalized Virasoro algebra: quasi-associativity, hereditary operator and ρ-compatibility equation

In this section, we discuss the main properties of the generalized Virasoro algebra. The case of left-alternative algebra structure and its link to some classes of nonlinear systems of differential equations are also recalled. A 3-ary bracket is defined and the relation between the functions f and g is given. We define the hereditary operator, and generalize it to the case of 3-ary bracket. Then, we derive the associated ρ-compatibility equation.

Skew-symmetry, Jacobi identity, coboundary operators and derivation property

• Skew-symmetry The skew-symmetry property [e x i , e x j ] = ae x i e x jbe x j e x i = g(x i , x j )e x i +x j = -[e x j , e x i ] =ae x j e x ibe x i e x j = -g(x j , x i )e x i +x j (3.1)

induces the conditions

a = b or g(x i , x j ) = -g(x j , x i ). (3.2)
• Jacobi identity criterion The Jacobi identity

[e x i , e x j ], e x k + [e x j , e x k ], e x i + [e x k , e x i ], e x j = 0 (3.3)
reduces to a condition similar to the Bianchi's identity

J k i j + J i jk + J j ki = 0 (3.4)
where

J k i j := g(x i , x j )g(x i + x j , x k ). (3.5)
Exploiting the relation between the functions f and g, i.e.,

g(x i , x j ) = a f (x i , x j ) -b f (x j , x i ). (3.6)
we can re-express the criterion for the Jacobi identity by the following result:

a 2 T k i j + T i jk + T j ki + b 2 G k ji + G j ik + G i k j -ab G k i j + G i jk + G j ki + T i k j + T k ji + T j ik = 0 (3.7)
where

T k i j := f (x i , x j ) f (x i + x j , x k ), G k i j =: f (x i , x j ) f (x k , x i + x j ). (3.8)
The criteria (3.2) and (3.4) or (3.7) confer the Lie algebra structure to the algebra (A , [., .]).

For a = b, skew-symmetric functions f , (i. e. f (x i , x j ) =f (x j , x i ) for all x i , x j ∈ Z), are solutions of (3.7), which is consistent with the above skew-symmetry conditions (3.2). Note, however, that the search for a general solution to this functional equation remains an open issue.

• Coboundary operators, 2-cocycle and second cohomology

In addition to the above criteria, we consider an associated Lie module M over A , and a k-cochain, i.e. an alternating K-multilinear map

ψ : A × A × . . . × A (k copies of A ) -→ M .
The most important A -modules are the trivial module K, i.e., the action reads e x n •λ = 0 for all λ ∈ K and all e x n ∈ A , and adjoint module A , i.e., A acts on A by the adjoint action.

Denote the vector space of k-cochains by C k (A , A ) and define the coboundary operators

δ k : C k (A , A ) -→ C k+1 (A , A ), k ∈ N, with δ k+1 • δ k = 0.
(3.9) The second cohomology of A with values in the adjoint representation is

Definition 3.1. A k-cochain ψ is called a k-cocycle if it
H 2 (A , A ) = Ker δ 2 /Im δ 1 ,
whereas H 2 (A , K) with values in the trivial module is related to the central extension of A . It is worth saying that deformations of the Lie algebra A are related to the Lie algebra cohomology and H 2 (A , A ) classifies infinitesimal deformations [START_REF] Gerstenhaber | On the deformation of rings and algebras I, II, III[END_REF]. If H 2 (A , A ) = 0, then A is infinitesimally formally rigid. An elementary and direct calculation of the vanishing second Lie algebra cohomology of the Witt and Virasoro algebras with values in the adjoint module is given by Schlichenmaier [START_REF] Schlichenmaier | An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module[END_REF]. In 1989, A. Fialowski showed by explicit calculations the vanishing of the second Lie algebra cohomology of the Witt algebra (in an unpublished manuscript). She also gave statements of the rigidity of the Witt and Virasoro algebras [START_REF] Fialowski | Deformations of some infinite-dimensional Lie algebras[END_REF] without proof.

Write the 0-cocycle as ψ(e x i , e x j ) = ψ x i ,x j e x i +x j .

(3.12)

If it is a coboundary, then it can be given as a coboundary of a linear form of degree 0, i.e.,

φ (e x i ) = φ x i e x i . (3.13)
Then the following result holds. leads to the functional relation:

g(x i , x j )ψ x i +x j ,x k + g(x j , x k )ψ x j +x k ,x i + g(x k , x i )ψ x i +x k ,x j -g(x i , x j + x k )ψ x j ,x k + g(x j , x i + x k )ψ x i ,x k -g(x k , x i + x j )ψ x i ,x j = 0. (3.15)
Besides, there results from the expression (3.11):

(δ 1 φ )(x i , x j ) = g(x i , x j ) φ x i +x j -φ x i -φ x j e x i +x j . (3.16)
Hence, ψ is a coboundary if and only if there exists a system of φ x k ∈ C, x k ∈ Z, such that

ψ x i ,x j = g(x i , x j ) φ x i +x j -φ x i -φ x j . ( 3
f (x j , x k )g(x i , x j + x k ) = g(x i , x k ) f (x j , x i + x k ) + g(x i , x j ) f (x i + x j , x k ) (3.19)
or, equivalently, using the relation (3.6) between f and g, to

f (x j , x k ) a f (x i , x j + x k ) -b f (x j + x k , x i ) -f (x j , x i + x k ) a f (x i , x k ) -b f (x k , x i ) -f (x i + x j , x k ) a f (x i , x j ) -b f (x j , x i ) = 0 (3.20)
which can be simply rewritten as

a G i jk -G j ik -T k i j + b G j ki + T k ji -T i jk = 0. (3.21)
Let us extend now the notion of skew-symmetry and Jacobi identity to the vector space B = A ⊕ A with the multiplication (e x p , e x r ), (e x q , e x s ) := [e x p , e x q ], e x p e x se x q e x r (3.22) where (., .) is the ordinary dot product. Explicitly, this new bracket gives (e x p , e x r ), (e x q , e x s ) = g(x p , x q ) e x p +x q , f (x p , x s )e x p +x sf (x q , x r )e x q +x r ,

or, equivalently, (e x p , e x r ), (e x q , e x s ) = g(x p , x q ) f (x p , x s ) e x p +x q , e x p +x sf (x q , x r ) e x p +x q , e x q +x r . (3.24)

Therefore, the skew-symmetry criterion reads g(x p , x q )g(x q , x p ) f (x p , x s ) e x p +x q , e x p +x sf (x q , x r ) e x p +x q , e x q +x r = 0.

(3.25)

The Jacobi identity property

[(e x p , e x r ), (e x q , e x s )], (e x t , e x u ) + [(e x q , e x s ), (e x t , e x u )], (e x p , e x r )

+ [(e x t , e x u ), (e x p , e x r )], (e x q , e x s ) = 0 (3.26)

gives here the following functional relation:

g(x p , x q )g(x p + x q , x t ) f (x p + x q , x u ) -g(x q , x t )g(x q + x t , x p ) f (x q , x u ) f (x p , x q + x u )
+g(x t , x p )g(x t + x p , x q ) f (x p , x u ) f (x q , x p + x u ) e x p +x q +x t , e x p +x q +x u g(x p , x q )g(

x p + x q , x t ) f (x p , x s ) f (x t , x p + x s ) -g(x q , x t )g(x q + x t , x p ) f (x t , x s ) f (x p , x t + x s )
-g(x t , x p )g(x t + x p , x q ) f (x p + x t , x s ) e x p +x q +x t , e x p +x t +x s + g(x p , x q )g(x p + x q , x t ) f (x q , x r ) f (x t , x q + x r ) + g(x q , x t )g(x q + x t , x p ) f (x q + x t , x r )

-g(x t , x p )g(x t + x p , x q ) f (x t , x r ) f (x q , x r + x t ) e x p +x q +x t , e x q +x r +x t = 0, or, equivalently, J t qp L x q f (x p , x u ) -J p qt G p qu + J q t p G q pu e x p +x q +x t , e x p +x q +x u + J q t p L x t f (x p , x s ) + J p qt G p ts -J t pq G t ps e x p +x q +x t , e x p +x t +x s + J t pq G t qr + J p qt L x q f (x t , x r ) -J q t p G q tr e x p +x q +x t , e x q +x r +x t = 0, (

where L x t stands for the left-translation operator acting as L x t f (x q , x r ) = f (x q + x t , x r ).

Quasi-associativity condition

From the definition of the quasiassociativity w.r.t. the multiplication rule (1.7), we infer the following relation

f (x i , x j ) -f (x j , x i ) f (x i + x j , x k ) = f (x j , x k ) f (x i , x j + x k ) -f (x i , x k ) f (x j , x i + x k ) (3.28)
expressing the necessary and sufficient property which confers a space phase structure to the subadjacent Lie algebra [START_REF] Kupershmidt | On the nature of the Virasoro algebra[END_REF].

Setting the ansatz

f (x i , x j ) := φ x i (x j ) (3.29)
for some linear map φ : Z → End(Z), φ x = φ (x), then the quasiassociativity condition (3.28) reads

φ x i (x j ) -φ x j (x i ) φ x i +x j (x k ) = φ x j (x k )φ x i (x j + x k ) -φ x i (x k )φ x j (x i + x k ), (3.30)
which can be further simplified into the expression

R j i -R i j R k i+ j -R k j R j+k i + R k i R i+k j = 0 (3.31)
by defining φ x i (x j ) := R j i . The relation (3.31) can be given an interpretation in terms of the null value of the Nijenhuis-torsion. Indeed, we have the following result. Proposition 3.2. Let e x i e x j := R k i j e x k . Then the quasi-associativity condition, i.e., (e x i e x j ) e x le x i (e x j e x l ) = (e x j e x i ) e x le x j (e x i e x l ),

is expressed by the Nijenhuis-torsion free relation:

R k i j -R k ji R m kl -R k jl R m ik + R k il R m jk = 0. (3.33)
This well agrees with the result exposed in [START_REF] Winterhalder | Linear Nijenhuis-tensors and the construction of integrable systems[END_REF]. In this case, as well pointed out by Kuperschmidt [START_REF] Kupershmidt | Left-symmetric algebras in hydrodynamics[END_REF] (see also references therein), all associated hydrodynamic systems are diagonalizable in Riemann invariants whenever they are hyperbolic. Moreover, a diagonal N-component hyperbolic hydrodynamic system whose Nijenhuis torsion is zero, is isomorphic to N noninteracting scalar equations. Definition 3.2 (Alternative algebra). An alternative algebra U over a field F is an algebra defined with the two identities [START_REF] Schafer | An introduction to nonassociative algebras[END_REF] x 2 y = x(xy) for all x, y ∈ U (3.34)

and

yx 2 = (yx)x for all x, y ∈ U , (3.35
respectively. Therefore, the relations (3.31) and (3.33) are trivially satisfied, 0 = 0, for i = j, i.e., in the case of left-alternative algebra. In accordance with the modified Riccati scheme introduced by Kazakov [START_REF] Ya | Riccati scheme for integrating nonlinear systems of differential equations[END_REF], such a left-alternative algebra is associated with the following vector system of equations:

Y t = -Y Y + ε, (3.38)
where the vector function ε(t) takes value in the considered left alternative algebra (LAA). The nonlinear system (3.38) can be reduced to linear problems by means of this LAA. Notice that, in contrast to systems of hydrodynamic type, the nonlinear systems of equations generated by leftalternative algebras do not, in general, have integrals of the motion. For more details, see [START_REF] Ya | Riccati scheme for integrating nonlinear systems of differential equations[END_REF]. It is also worth mentioning the well-known Burgers vectorial equation [START_REF] Sokolov | Deformations of nonassociative algebras and integrable differential equations[END_REF]:

u t = u xx + 2u u x + u (u u) -(u u) u, (3.39)
which is one of the important examples of equations associated with LSAs. In the case of the generalized Virasoro algebra examined in this work, theproduct is defined by the multiplication law (1.7). New examples of nonlinear systems may exist, but their full investigations remain totally open and may be the core of our forthcoming works.

3-ary bracket

Ternary algebra plays an important role in the construction of the world volume theories of multiple M2 branes [START_REF] Bagger | Gauge symmetry and supersymmetry of multiple M2-branes[END_REF]. The ternary bracket was introduced by Nambu [START_REF] Nambu | Generalized Hamiltonian dynamics[END_REF] and developed by Filippov [START_REF] Filippov | n-Lie algebras[END_REF].

Several authors [START_REF] Curtright | Classical and quantum Nambu mechanics[END_REF] studied Kac-Moody and centerless Virasoro (or Virasoro-Witt) 3-algebras and demonstrated some of their applications to the Bagger-Lambert-Gustavsson theory. The su(1,1) enveloping algebra was used by Curtright et al [START_REF] Curtright | Ternary Virasoro-Witt algebra[END_REF] to construct ternary Virasoro-Witt algebra. Motivated by this work, we study the ternary algebra of the generalized Virasoro algebra defined by the following 3-ary bracket:

[e x i , e x j , e x k ] := e x i [e x j , e x k ] + e x j [e x k , e x i ] + e x k [e x i , e x j ].

Putting

E k i j := g(x i , x j ) f (x k , x i + x j )e x i +x j +x k (3.41)
yields the following expression for the 3-ary bracket:

[e x i , e x j , e x k ] = E k i j + E i jk + E j ki . (3.42)
In the other hand, using the relations (1.6) and (1.7) the same bracket can also be evaluated in terms of the real a and b. In this case, denoting by

F k i j := a f (x i , x j ) -b f (x j , x i ) f (x k , x i + x j )e x i +x j +x k , (3.43)
we obtain

[e x i , e x j , e x k ] = F k i j + F i jk + F j ki .

(3.44)

The relations (3.42) and (3.44) can therefore be used to define a 3-algebra generalization of the generalized Virasoro algebra proposed by Kupershmidt [START_REF] Kupershmidt | On the nature of the Virasoro algebra[END_REF]. Indeed, such a formulation of the 3-ary bracket, cyclic in the indices i, j, k, reminiscent of the Bianchi's identity for curvature tensor in differential geometry, is quite relevant for detailed analysis of 3-algebras as will be developed in a forthcoming paper.

Hereditary operator

The hereditary operator is defined as follows [START_REF] Fuchssteiner | Compatibility in absract algebraic structures[END_REF]:

Definition 3.

(Hereditary operator).

A linear map Φ :

(A , •) → (A , •) defined as [a, b] Φ := (Φa) • b + a • (Φb) -Φ(a • b), (3.45)
where • is some binary bilinear operator on A , is called hereditary if

Φ[a, b] Φ = (Φa) • (Φb), (3.46)
or, equivalently,

Φ 2 (a • b) + (Φa) • (Φb) = Φ (Φa) • b + a • (Φb) . (3.47)
Next, let us introduce the map L k such that

L k (Φ)(b) := k • Φ(b) -Φ(b) • k -Φ(k • b) + Φ(b • k) (3.48) for all b ∈ A . Then Φ is k-invariant iff L k Φ = 0.
Formally, let us write

L a b = a • b -b • a. (3.49)
Then, the Leibniz rule reads

L a (Φ • b) = L a (Φ) • b + Φ(L a b). (3.50)
Therefore, the map Φ is hereditary iff

Φ(L a Φ) = L Φ(a) Φ. (3.51)
Three particular cases give rise to interesting simpler conditions as follows.

Proposition 3.3. The hereditary condition for the generalized Virasoro algebra (1.5) makes into:

(i) For Φ : e x i → e x i +x 0 for some fixed x 0 , g(x i + x 0 , x j ) + g(x i , x j + x 0 )g(x i , x j ) = g(x i + x 0 , x j + x 0 ), (3.52)

or, equivalently,

T x 0 + L x 0 -T x 0 L x 0 -1 g(x i , x j ) = 0, (3.53)
where the right and left translation operators T x 0 and L x 0 are defined by T x 0 g(x i , x j ) := g(x i , x j + x 0 ), L x 0 g(x i , x j ) := g(x i + x 0 , x j );

(3.54)

(ii) For Φ : e x i → R k i e x k , g(x m , x j )R m i R s m+ j + g(x i , x n )R n j R s n+i -g(x i , x j )R t i+ j R s t e x s = g(x m , x n )R m i R n j e x m +x n ; (3.55) (iii) For Φ : e x i → e x i +1 , [e x i , e x j ] := R k i j e x k , R k (i+1) j + R k ( j+1)i -R k i j -R k (i+1)( j+1) = 0.
Besides, defining the action of Φ on a 3-ary bracket in the simple case of Φ : e x i → e x i +x 0 as: where

Φ e x i ,
P k i j := g(x i + x 0 , x j ) + g(x i , x j + x 0 ) -g(x i , x j ) -g(x i + x 0 , x j + x 0 ) f (x k + x 0 , x i + x j + 2x 0 ) (3.59)
or, equivalently, in operator form,

P k i j := T x 0 + L x 0 -T x 0 L x 0 -1 g(x i , x j )T 2x 0 +x j L x 0 f (x k , x i ). (3.60)
Hereditary operators play an important role in the field of nonlinear evolution equations. Indeed, as showed in [START_REF] Fuchssteiner | The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional abelian symmetry groups[END_REF], they generate on a systematic level many new classes of nonlinear dynamical systems which possess infinite dimensional abelian groups of symmetry transformations. Their so-called permanence properties, given by Fuchssteiner [START_REF] Fuchssteiner | The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional abelian symmetry groups[END_REF], allow to construct new hereditary operators out of given ones.

ρ-compatibility equation

Following [START_REF] Kupershmidt | On the nature of the Virasoro algebra[END_REF], define a new multiplication e x i e x j = e x i e x j + εh e x i ,e x j , ε 2 = 0 (3.61)

with some (bi-)linear map h : A ⊗ A → A , such that it makes A into a left-symmetric algebra as well. The new associator (e x i , e x j , e x k ) can be expressed in terms of old one as follows:

(e x i , e x j , e x k ) = (e x i , e x j , e x k ) + εT e x i ,e x j ,e x k (3.62)

where T e x i ,e x j ,e x k = h e x i ,e x j e x kh e x i e x j , e x k + e x i h e x j ,e x kh e x i ,e x j e x k .

(3.63)

Therefore, the left symmetric condition for (A , ) is provided by the relation T e x i ,e x j ,e x k = T e x i ,e x j ,e x k -T e x j ,e x i ,e x k = h e x i ,e x j e x kh e x j ,e x i e x kh [e x i ,e x j ],e x k + e x i h e x j ,e x ke x j h e x i ,e x k

h e x i ,e x jh e x j ,e x i e x k = 0.

(3.64)

Write here also h e x i ,e x j := φ e x i (e x j ) (3.65)

for some linear map φ : A → A , φ e x i = φ (e x i ).

Then the quasiassociativity condition (3.62) takes the following form:

φ e x i (e x j e x k )φ e x j (e x i e x k )φ [e x i ,e x j ] (e x k ) + e x i φ e x j (e x k ) e x j φ e x i (e x k ) -[φ e x i (e x j )φ e x j (e x i )] e x k = 0.

(3.66)

The relation (3.66) can be rewritten in an operator form as:

φ e x i
, L e x jφ e x j , L e x i = L φ ex i (e x j )-φ ex j (e x i )+φ [ex i ,ex j ] .

(3.67) Suppose now that φ e x i is of a special form:

φ e x i = L ρ(e x i ) , ρ ∈ End(A ), (3.68)
with some operator ρ : A → A . Since for LSAs, = -e x j ρ(e x i ) + e x i ρ(e x j )ρ e x i e x je x j e x i (3.71)

[L u , L v ] = L [u,v] , ∀u, v ∈ A (3.
The ρ-compatibility equation then reads E(e x i , e x j ) = 0 ⇔ e x i ρ(e x j )e x j ρ(e x i ) = ρ e x i e x je x j e x i

:= ρ f (x i , x j )e x i +x j -f (x j , x i )e x i +x j , (3.72)
instead of the weaker deformation condition (3.70). The operator ρ is called a strong deformation.

For the particular case, when ρ : e x i -→ e x i +x 0 for fixed x 0 ∈ Z, the ρ-compatibility equation (3.72) turns out to be a simpler difference equation

f (x i , x j + x 0 ) -f (x i , x j ) -f (x j , x i + x 0 ) -f (x j , x i ) = 0. (3.73)
Define the right translation and exchange operators T x 0 and E , respectively, by

T x 0 f (x i , x j ) = f (x i , x j + x 0 ), E f (x i , x j ) = f (x j , x i ). (3.74)
Then, the relation (3.73) reads

T x 0 (1 -E ) f (x i , x j ) = (1 -E ) f (x i , x j ). (3.75)
Therefore the following result holds.

Proposition 3.4. Let ρ : e x i -→ e x i +x 0 for some fixed x 0 ∈ Z. Then the ρ-compatibility equation (3.72) is equivalent to the invariance of the operator (1-E ) under the action of the right translation operator T x 0 , i.e.,

T x 0 (1 -E ) = (1 -E ).
(3.76)

Universal identity

All nonassociative algebras naturally arising in connection with integrable systems satisfy a universal identity [START_REF] Sokolov | Deformations of nonassociative algebras and integrable differential equations[END_REF], i.e.,

[[e x i , e x j , e x k e x s ]] -[[e x i , e x j , e x k ]] e x se x k [[e x i , e x j , e x s ]] = 0, (

where

[[x, y, z]] = (x, y, z) -(y, x, z). (3.78)
Here we get

[[e x i , e x j , e x k e x s ]] = f (x k , x s ) f (x j , x k + x s ) f (x i , x j + x k + x s ) -f (x i , x k + x s ) f (x j , x i + x k + x s ) -f (x i + x j , x k + x s ) f (x i , x j ) -f (x j , x i ) (3.79) [[e x i , e x j , e x k ]] e x s = f (x i + x j + x k , x s ) f (x i + x j , x k ) f (x j , x i ) -f (x i , x j ) + f (x j , x k ) f (x i , x j + x k ) -f (x i , x k ) f (x j , x i + x k ) (3.80) e x k [[e x i , e x j , e x s ]] = f (x k , x i + x j + x s ) f (x i + x j , x s ) f (x j , x i ) -f (x i , x j ) + f (x j , x s ) f (x i , x j + x s ) -f (x i , x s ) f (x j , x i + x s ) = 0. (3.81)
Therefore, Proposition 3.5. The universal identity (3.77) turns out to be of the following form for the generalized quasiassociative algebra defined with the multiplication (1.7):

f (x k , x j ) -L x i (x j , x k + x j )(1 -E )( f (x i , x j ) + [[ f (x j , x k + x s ), f (x i , x k + x j )]] -L x i +x j f (x k , x s ) -L x j (x i , x k )(1 -E )( f (x i , x j ) + [[ f (x j , x k ), f (x i , x k )]] -T x i +x j f (x k , x j ) -L x j (x i , x j )(1 -E )( f (x i , x j ) + [[ f (x j , x s ), f (x i , x s )]] = 0 (3.82)
where

[[ f (x j , x l ), f (x i , x l )]] = f (x j , x l )T x s f (x i , x l ) -f (x i , x l )T x i f (x j , x l ), (3.83)
and L u , T v are the usual left and right translation operators, respectively.

From the definition of the algebra product (1.7), the functions f can be regarded as the algebra structure constants. Therefore, the form (3.82) of the universal identity (3.77) could be linked to the integrability condition of the differential equations associated with the considered generalized algebra. However, at this stage of our study, such an assertion deserves further investigations.

Phase space extension

As known from [START_REF] Kupershmidt | Left-symmetric algebras in hydrodynamics[END_REF], the category of LSAs is closed with respect to the operation of phase-space extension, unlike the smaller category of associative algebras: if A is LSA then so is T A = A ⊕ A : e x i e x i e x j e x j = e x i e x j e x i e x je x j e x i , e x i , e x j ∈ A , e x i , e x j ∈ A ,

where, e x i e x j , e x k =e x j , e x i e x k , e x i , e x k ∈ A , e x j ∈ A .

The integrability of the hydrodynamical systems of the type [START_REF] Kupershmidt | Left-symmetric algebras in hydrodynamics[END_REF]:

u t = ρu e x i + u e x i u (3.86)
is preserved under such phase-space extensions. In (3.86), ρ is the operator whose matrix elements are ρ x i x j :

ρ(e x j ) = ∑

x i

ρ x i x j e x i , (3.87)
{e x i } being a basis of A , whose structure constants are C x i x j x k :

e x j e x k = ∑ x i C x i x j x k e x i . (3.88)
Assume ρ satisfies the following ρ-compatibility equation (3.72), i. e., e x i ρ(e x j )e x j ρ(e x i ) = ρ e x i e x je x j e x i := ρ f (x i , x j )e x i +x jf (x j , x i )e

(x i , x j ) -f (x j , x i ) e x i +x j +x 0 0 = f (x i , x j + x 0 ) -f (x j , x i + x 0 ) e x i +x j +x 0 0 , (3.99)
which is equivalent to

1 -E + T x 0 (E -1) f (x i , x j ) 0 = 0 0 (3.100) giving T x 0 (1 -E ) = (1 -E ). (3.101)
We thus recover the ρ-compatibility condition (3.76).

The phase-space extension of the system (3.86) reads:

u u t =

Centrally extended Virasoro algebra

In this section we aim at investigating the quasi-associativity condition, the 3-ary bracket and the fundamental identity for the above generalized algebra in the particular case when a = b = 1 and

f (x i , x j ) = - x j (1 + εx j ) 1 + ε(x i + x j ) + 1 2 θ x i 3 -x i + (ε -ε -1 )x i 2 δ 0 x i +x j , (4.1)
g(x i , x j ) = (x i -x j ) + θ (x i 3 -x i )δ x i +x j (4.2)
This algebra corresponds to the Virasoro algebra, also called central extension of the Witt algebra with the multiplication:

[e x i , e x j ] = g(x i , x j )e x i +x j , e x i θ = θ e x i = 0 (4.3) coming from the commutator [e x i , e x j ] = e x i e x je x j e x i e x i e x j = f (x i , x j )e x i +x j (4.4)

where (x i , x j ) ∈ Z 2 .

Proposition 4.1. For the centrally extended Virasoro algebra, (i) The skew-symmetry condition (3.2) is equivalent to the system of equations is identified to the condition

   x i + x j + ε x 2 i + x 2 j = 0 x 3 i + x 3 j -(x i + x j ) + ε -ε -1 x 2 i + x 2 j = 0; (4.5) (
J k i j + J i jk + J j ki = 0 (4.7)
where

J k i j := (x i -x j ) + θ (x i 3 -x i )δ x i +x j (x i + x j -x k ) + θ (x i + x j ) 3 -(x i + x j ) δ x i +x j +x k ; (4.8)
(iii) The derivation property, i.e.,

[e x i , e x j e x k ] := e x j [e x i , e x k ] + [e x i , e x j ] e x k (4.9) leads to

- x k (1 + εx k ) 1 + ε(x j + x k ) (x i -(x j + x k ))e x i +x j +x k + θ (x i 3 -x i )δ 0 x i +x j +x k = 1 1 + ε(x i + x j + x k ) × -(x 2 i -x 2 k ) -x k (x i -x j ) -ε[(x 2 i -x 2 k )(x i + x k ) + x 2 k (x i -x j )] e x i +x j +x k + 1 2 θ (x j 3 -x j )(x i -x k ) + (x i -x j )[(x i + x j ) 3 -(x i + x j )] + (ε -ε -1 )[x j 2 (x i -x k ) + (x i -x j )(x i + x j ) 2 ] δ 0 x i +x j +x k . ( 4
.10)

Quasi-associativity condition

We answer the question: Does it exist a necessary and sufficient condition for this algebra to be a quasi-associative algebra with the multiplication

e x i e x j = - x j (1 + εx j ) 1 + ε(x i + x j ) e x i +x j + 1 2 θ x i 3 -x i + (ε -ε -1 )x i 2 δ 0 x i +x j e x i θ = θ e x i = 0? (4.11)
Theorem 4.1. The algebra defined with the multiplication rule (4.11) is neither associative, nor left-symmetric.

Proof. By direct computation, we find:

e x i (e x j e x k ) -(e x i e x j ) e x k = x k (1 + εx k ) 1 + ε(x i + x j + x k ) x k + ε (x i + x j )(x j + x k ) -x 2 j 1 + ε(x i + x j ) e x i +x j +x k - θ 2 x k (1 + εx k ) 1 + ε(x j + x k ) x i 3 -x i + (ε -ε -1 )x i 2 - x j (1 + εx j ) 1 + ε(x i + x j ) (x i + x j ) 3 -(x i + x j ) + (ε -ε -1 )(x i + x j ) 2 δ 0 x i +x j +x k (4.12)
while e x j (e x i e x k ) -(e x j e x i ) e

x k = x k (1 + εx k ) 1 + ε(x i + x j + x k ) x k + ε (x i + x k )(x i + x j ) -x 2 i 1 + ε(x i + x j ) e x i +x j +x k - θ 2 x k (1 + εx k ) 1 + ε(x i + x k ) x j 3 -x j + (ε -ε -1 )x j 2 - x i (1 + εx i ) 1 + ε(x i + x j ) (x i + x j ) 3 -(x i + x j ) + (ε -ε -1 )(x i + x j ) 2 δ 0 x i +x j +x k = e x i (e x j e x k ) -(e x i e x j ) e x k . (4.13)
It is worth noticing that for i = j, this algebra becomes a left-alternative algebra as required by the general formalism developed in the previous section.

3-ary bracket and fundamental identity

The 3-ary bracket, defined by the relation (3.40)

] = - 1 1 + ε(x i + x j + x k ) ε (x 2 j -x 2 i )x k + (x 2 i -x 2 k )x j + (x 2 k -x 2 j )x i e x i +x j +x k + θ 2 (x i 3 + x j 3 + x k 3 )(1 + ε -ε -1 ) -(x i + x j + x k ) δ 0 x i +x j +x k . ( 4
implies cumbersome functional equations:

(x 2 l -x 2 k )x m + (x 2 k -x 2 m )x l + (x 2 m -x 2 l )x k 1 + ε(x k + x l + x m ) (x 2 j -x 2 i )(x k + x l + x m ) + x 2 i -(x k + x l + x m ) 2 x j + (x k + x l + x m ) 2 -x 2 j x i = (x 2 j -x 2 i )x k + (x 2 i -x 2 k )x j + (x 2 k -x 2 j )x i 1 + ε(x i + x j + x k ) (x 2 l -(x i + x j + x k ) 2 )x m + (x i + x j + x k ) 2 -x 2 m x l + (x 2 m -x 2 l )(x i + x j + x k ) + (x 2 j -x 2 i )x l + (x 2 i -x 2 l )x j + (x 2 l -x 2 j )x i 1 + ε(x i + x j + x l ) (x i + x j + x l ) 2 -x 2 k x m + (x 2 k -x 2 m )(x i + x j + x l ) + (x 2 m -(x i + x j + x l ) 2 )x k + (x 2 j -x 2 i )x m + (x 2 i -x 2 m )x j + (x 2 m -x 2 j )x i 1 + ε(x i + x j + x m ) (x 2 l -x 2 k )(x i + x j + x m ) + (x 2 k -(x i + x j + x m ) 2 )x l + (x j + x j + x m ) 2 -x 2 l x k (4.16)
and

(x i + x j + x k ) 3 + (x j + x k + x l ) 3 + (x i + x j + x m ) 3 -(x k + x l + x m ) 3 + 2(x 3 k + x 3 l + x 3 m ) -(x 3 i + x 3 j ) (1 + ε + ε -1 ) -2(x i + x j + x k + x l + x m ) = 0. ( 4
.17)

These functional equations reduce to the simpler relation (4.17) when ε 2 = 0.

5. L k -infinite dimensional Lie algebra of polynomial vector fields on the real line R 1

Consider the algebra L kas the infinite dimensional Lie algebra of polynomial vector fields on the real line R 1 . Let us define this algebra by the infinite basis {e i } :

e i = x i+1 d dx , i ∈ N (5.1)
with the commutator [e i , e j ] = e i e je j e i = ( ji)e i+ j (5.2)

with the multiplication e p e q = (q + 1)e p+q + e p+q+1 d dx .

(5.3)

Here themultiplication is nothing but the ordinary operators product. One can easily prove by direct computation that this algebra endowed with the product (5.3) is an associative algebra, i.e., its associator is equal to zero. Indeed, e p (e q e r ) = (e p e q ) e r ) = (r + 1)(q + r + 1)e p+q+r + (q + 2r + 3)e p+q+r+1 d dx + e p+q+r+2 d 2 dx 2 . (5.4)

Further the corresponding Nambu brackets are null, i.e., [e p , e q , e r ] := e p [e q , e r ] + e q [e r , e p ] + e r [e p , e q ] = 0, (5.5) that is the Jacobi identity is automatically satisfied. Thus we have a null 3-algebra for an infinite set of non-trivial noncommuting oscillator charges. The Filippov condition is trivially satisfied in this case, i.e., e p , e q , [e r , e s , e t ] := [e p , e q , e r ], e s , e t + e r , [e p , e q , e s ], e t + e r , e s , [e p , e q , e t ] .

(5.6)

The Bremner operator, also called the associative operator, of course, perfectly works, i.e., e p , [e q , e r , e s ], e t , e u , e v := [e p , e q , e r ], [e s , e t , e u ], e v (5.7) as a consequence of the associativity. Moreover, the skew-symmetry property is obeyed by the definition of the product, i.e.,

[e i , e j ] = e i e je j e i = -[e j , e i ] = -( ji)e i+ j .

(5.8)

Provided the Jacobi identity and skew-symmetry properties are satisfied, this algebra is made into a Lie algebra structure. On the other hand, [e p , e q e r ] = (r + 1)(q + r + 1) -(p + 1)(p + r + 1) e p+q+r + (q + r -2p)e p+q+r+1 d dx , (5.9)

e q [e p , e r ] = (p + r + 1)(rp)e p+q+r + (rp)e p+q+r+1 d dx ,

[e p , e q ] e q = (r + 1)(qp)e p+q+r + (qp)e p+q+r+1 d dx .

(5.11)

Summing the relations (5.10) and (5.11), we find that [e p , e q e r ] = e q [e p , e r ] + [e p , e q ] e q , (5.12)

showing that the derivation property is satisfied, what makes (L k , [., ]) into a Poisson structure. These properties induce the following consequences:

Proposition 5.1. Let L e q and R e q be the left and right multiplication operators by e q , (for some fixed e q ∈ L k ), defined, respectively, as:

L e q (e p ) = e q e p , R e q (e p ) = e p e q , ∀e p ∈ L k .

(5.13)

Then the following relations hold, for all e p , e q ∈ L k :

• [L e p , L e q ] = L [e p ,e q ] (5.14)

• R [e p ,e q ] (.) = (.) [e p , e q ] = -(.) [R e p , R e q ] (5.15)

•

[L e p , R e q ](.) = e p [(.), e q ] = 0 (5.16)

•

[R e p , e q ](.) = e q [e p , (.)] (5.17)

•

[R e p R e q + R e p e q ](.) = (.) [R e p (e q ) + R e q (e p )].

(5.18)

Let us mention an interesting identity of general interest: e p , [e q , e r ] + e r , [e p , e q ] = e q [e p , e r ] -[e p , e r ] e q , (5. [START_REF] Nambu | Generalized Hamiltonian dynamics[END_REF] valid whatever the -product, with the usual commutator.

Concluding remarks

In this paper, we have discussed the appearence of left symmetric algebras in a generalized Virasoro algebra. We have provided the necessary and sufficient condition for this algebra to be a quasiassociative algebra. The criteria of skew-symmetry, derivation and Jacobi identity making this algebra a Lie algebra have been derived. Coboundary operators are defined, the 2-cocycle and coboundary are discussed. We have deduced the hereditary operator and its generalization to the corresponding 3-ary bracket. Further, we have derived the so-called ρ-compatibility equation, and performed a phase-space extension. Concrete relevant particular cases have also been investigated and discussed. This study brings some interesting questions to light which merit a separate in-depth treatment. For instance, new examples of nonlinear systems associated with the considered generalization of the Virasoro algebra may exist, but their full investigation remains totally open. Besides, a detailed analysis of the main properties of 3-algebras on the basis of definition (3.42) or (3.44) is also of great importance. These topics will be the core of our forthcoming works.

Proposition 3 . 1 .

 31 The 2-cocycle ψ defined by δ 2 ψ(e x i , e x j , e x k ) := 0(3.14)

  ) known, respectively, as left and right alternative laws. In terms of associators, (3.34) and (3.35) are equivalent to (x, x, y) = 0 for all x, y ∈ U (3.36) and (y, x, x) = 0 for all x, y ∈ U ,

  69) the equation (3.66) becomes L E(e x i ,e x j ) = 0 (3.70) where E(e x i , e x j ) = [ρ(e x i ), e x j ] -[ρ(e x j ), e x i ]ρ(e x i ) e x jρ(e x j ) e x iρ [e x i , e x j ]

  lies in the kernel of the coboundary operator δ k . It is called a k-coboundary if it lies in the image of the (k-1) coboundary operator. Lie algebra 2-cocycle with values in the adjoint module if δ 2 ψ(e x i , e x j , e x k ) : = ψ([e x i , e x j ], e x k ) + ψ([e x j , e x k ], e x i ) + ψ([e x k , e x i ], e x

A skew-symmetric map ψ :

A × A → A is a j ) -[e x i , ψ(e x j , e x k )] + [e x j , ψ(e x i , e x k )] -[e x k , ψ(e x i , e x j )] = 0 (3.10)

and a coboundary if there exits a linear map φ : A → A with ψ(e x i , e x j ) = (δ 1 φ )(e x i , e x j ) := φ ([e x i , e x j ]) -[e x i , φ (e x j )] + [e x j , φ (e x i )].

(3.11)

  e x j , e x k

	Φ	:= Φ(e x i ), Φ(e x j ), Φ(e x k )	(3.57)
	yields a Bianchi like identity
	P k i j + P i jk + P j ki = 0	(3.58)

  x i +x j (3.89) By formulae (3.61),(3.65),(3.68), we realize a new left-symmetric multiplication in the following way: e x i e x j := e x i + ερ(e x i ) e x j , εe x j e x i , e x i , e x j ∈ A , e x i , e x j ∈ A , (3.91) where e x i e x j , e x k =e x j , e x i e x k =e x j , e x i + ερ(e x i ) e x k = e x i + ερ(e x i ) e x j , e x k When ρ : e x i → e x i +x 0 for some fixed x 0 , the choice (3.95) reduces to a simpler relation: Proposition 3.7. Let ρ : e x i → e x i +x 0 . Then the relations (3.96) and (3.97) reduced to f

	Proof. We have:
	ρ 1	e x i e x i	1	e x j e x j	-	e x j e x j	1	e x i e x i		= ρ 1	e x i e x j -e x j e x i e x i e x j -e x j e x i
										=	ρ(e x i e x j -e x j e x i ) 0	(3.96)
		e x i e x i	1 ρ 1	e x j e x j	-	e x j e x j	1 ρ 1	e x i e x i	=		2 = 0 e x i e x i	1	ρ(e x j ) 0	(3.90)
	corresponding to the phase-space extension -e x j e x j 1 ρ(e x i ) 0	=	e x i ρ(e x j ) -e x j ρ(e x i ), 0	.	(3.97)
	Therefore, the ρ-compatibility equation (3.72) also holds in T A .
	e x i e x i We infer the following result. 1 e x j e x j	=	e x i e x j e x i e x i e x i e x j (3.92) ρ 1 : → e x i +x 0 0 . (3.98)
	implying
			e x i e x e x i e x i 1 e x j e x j =	e x i e x i	+ ερ 1	e x i e x i		1	e x j e x j	,	(3.94)
	we are in right to postulate the following choice:
	Proposition 3.6.
					ρ 1 :	e x i e x i	→	ρ(e x i ) 0			(3.95)

j = e x i + ερ(e x i ) e x j . (3.93) Provided the former relation and taking into account the natural extension : A → T A = A ⊕ A : ρ → ρ 1 should satisfy: satisfies the ρ-compatibility equation (3.72) in T A .

  It is also worth noticing that, in the case of the left-symmetric double[START_REF] Kupershmidt | Left-symmetric algebras in hydrodynamics[END_REF]:A d = A ⊕ A , we get = e x i e x j e x i e x be x j e x a = f (x i , x j )e x i +x j f (x i , x b )e x i +x bf (x j , x a )e x j +x) + e x g e x a e x µ e x i + e x ν e x a , e x g , e x µ , e x ν = constants +x 0 + e x g e x a e x µ e x i + e x ν e x a , when ρ(e x i ) := e x i +x 0 .

			ρ 2	e x i e x a	=	ρ(e x i (3.104)
	yielding
			ρ 2	e x i e x a	=	e x i (3.105)
	Furthermore, as claimed in [28], the formula (3.90) shows that, in addition to the integrable
	hydrodynamic hierarchy starting with the equation (3.86), i.e.,
							u t = ρu e x i + u e x i u	(3.106)
	we have a second hierarchy, starting with the equation
				u t = u e x i u = u e x i u + ερ(u e x i ) u, ε 2 = 0.	(3.107)
				ρ 1	u e x i u e x i	+	u e x i u e x i	1	u u	=	ρ(u e x i ) + u e x i u u e x i u	.	(3.102)
	e x i e x a	2	e x j e x b

a (3.103) and if ρ : A → A is an operator of strong deformation, then so is ρ 2 : A d → A d ,

  ii) The Jacobi identity [e x i , e x j ], e x k + [e x j , e x k ], e x i + [e x k , e x i ], e x j = 0 (4.6)

  , i.e., [e x i , e x j , e x k ] := e x i [e x j , e x k ] + e x j [e x k , e x i ] + e x k [e x i , e x j ] leads to the expression [e x i , e x j , e x k

  .14) Defining the fundamental identity, (also called Filippov identity), in this case as:x i , e x j , [e x k , e x l , e x m ] := [x i , e x j , e x k ], e x l , e x m + x k , [e x i , e x j , e x l ], e x m + x k , e x l , [e x i , e x j , e x m ] ,

Acknowledgements

The authors are grateful to anonymous referees for their useful comments which permit to improve the paper. The work of MNH is partially supported by the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) through the Office of External Activities (OEA)-Prj-15. The ICMPA is also in partnership with the Daniel Iagolnitzer Foundation (DIF), France. MNH and PG acknowledge the staff of the Centre Interfacultaire Bernoulli for the hospitality during their stay from October to December 2014 at the Ecole Polytechnique Fédérale de Lausanne, Switzerland. MNH thanks Professor Valentin Ovsienko for his useful comments and suggestions.