Francisco Kordon
email: franciscokordon@gmail.com

Thierry Lambre
email: thierry.lambre@uca.fr

CENTER AND LIE ALGEBRA OF OUTER DERIVATIONS FOR ALGEBRAS OF DIFFERENTIAL OPERATORS ASSOCIATED TO HYPERPLANE ARRANGEMENTS

We compute the center and the Lie algebra of outer derivations of a familiy of algebras of differential operators associated to hyperplane arrangements of the affine space A 3 . The results are completed for 4-braid arrangements and for reflection arrangements associated to the wreath product of a cyclic group with the symmetric group S 3 . To achieve this we use tools from homological algebra and Lie-Rinehart algebras of differential operators.

Introduction

Let V be a finite-dimensional k-vector space over a field k of characteristic zero, S the algebra of coordinates on V and A a central hyperplane arrangement in V . We assume throughout the article that A is a free arrangement in the sense given by K. Saito in [START_REF] Saito | Theory of logarithmic differential forms and logarithmic vector fields[END_REF]: we require that the Lie algebra Der A of derivations of S tangent to A is a free S-module. The algebra Diff A of differential operators tangent to A, as seen by F. J. Calderon Moreno in [START_REF] Calderón-Moreno | Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor[END_REF] and by M. Suárez-Álvarez in [S Á18], is the subalgebra of End(S) generated by Der A and S. Our results concern the center and the Lie algebra of outer derivations of Diff A.

The first and simplest example of a free arrangement is the case of a central line arrangement in V = k 2 . This case is studied by the first author and M. Suárez-Álvarez in [KS Á18] when there are at least 5 lines: a description of the Hochschild cohomology HH • (Diff A), including its cup product and Gerstenhaber bracket, is given explicitly in detail through a calculation independent of the methods that we now use. The second and most important family of examples is that of the braid arrangements B n , given for n ≥ 2 by the hyperplanes H ij = {x ∈ k n : x i = x j } with i = j: these arrangements are free and have served historically as a proxy to obtain general results, for instance in V. I. Arnold's classical article [START_REF] Arnol | The cohomology ring of the colored braid group[END_REF].

In virtue of the freeness of A the algebra Diff A is isomorphic to the enveloping algebra of a Lie-Rinehart algebra (S, L) -see L. Narvaez Macarro's [START_REF] Macarro | Linearity conditions on the Jacobian ideal and logarithmic-meromorphic comparison for free divisors[END_REF] and the first author's thesis [START_REF] Kordon | Hochschild cohomology of algebras of differential operators associated with hyperplane arrangements[END_REF]and then the spectral sequence introduced by both authors in [START_REF] Kordon | Lie-Rinehart and Hochschild cohomology for algebras of differential operators[END_REF] permits the computation of HH • (Diff A) in terms of the Hochschild cohomology H • (S, Diff A) of S with values on Diff A and the Lie-Rinehart cohomology of L. This was successfully applied to arrangements of three lines in [START_REF] Kordon | Lie-Rinehart and Hochschild cohomology for algebras of differential operators[END_REF], and, ultimately, to A = B 3 -see Corollary 1.29.

The homological approach described above allows us to compute the center of Diff A under the hypothesis that the Saito's matrix of the arrangement A is triangular: more generally, we can state this result resorting to the hypothesis of triangularizability of Lie-Rinehart algebras that we give in Definition 1.12.

Theorem A (Theorem 3.4). Let (S, L) be a triangularizable Lie-Rinehart algebra with enveloping algebra U . The center of U is k.

Let A r , r ≥ 1, be the arrangement in C 3 defined by 0 = xyz(z r -y r )(z r -x r )(y r -x r ). This arrangement is B 4 when r = 1. When r ≥ 2, it is the reflection arrangement of the wreath product of the cyclic group of order r and the symmetric group S 3 . The homological method yields the following result.

Theorem B (Corollary 5.5). Let r ≥ 1. For each hyperplane H in A r let f H be a linear form with kernel H and ∂ H the derivation of Diff A r determined by

∂ H (g) = 0 if g ∈ k[x 1 , x 2 , x 3 ]; ∂ H (θ) = θ(f H )/f H if θ ∈ Der A r .
The Lie algebra of outer derivations of Diff A r together with the commutator is an abelian Lie algebra of dimension 3r + 3, the numbers of hyperplanes of A r , and is generated by the classes of the derivations ∂ H with H ∈ A r .

In the pursuit of HH • (U ) a key step is the computation of H • (S, U ). We succeeded in its calculation when • = 0, 1 for a family of Lie-Rinehart algebras that generalizes Der A r . The result in Corollary 4.6 relates H 1 (S, U ) to the cokernel of the Saito's matrix -this is an important object of the theory with a rich algebraic structure studied, for instance, by M. Granger, D. Mond and M. Schulze in [START_REF] Granger | Partial normalizations of Coxeter arrangements and discriminants[END_REF].

There are several ways in which the calculations performed in this article can be continued. In particular, following the methods of J. Alev and M. Chamarie in [START_REF] Alev | Dérivations et automorphismes de quelques algèbres quantiques[END_REF] our findings on the algebra of outer derivations of Diff A can lead to a description of Aut(Diff A) as in [KS Á18, §7] and M. Suárez-Álvarez and Q. Vivas' [START_REF] Suárez-Alvarez | Automorphisms and isomorphisms of quantum generalized Weyl algebras[END_REF].

The first author is currently a CONICET postdoctoral fellow and received support from BID PICT 2019-00099. We thank the Université Clermont Auvergne for hosting the first author in a postdoctoral position at the Laboratoire de Mathématiques Blaise Pascal during the year 2019-2020.

Unadorned Hom and End are taken over k. The set of natural numbers N is that of nonnegative integers. If n and m are positive integers, we denote by n, m the set of integers k such that n ≤ k ≤ m, and m := 1, m . 1. Generalities 1.1. Hyperplane arrangements. Definition 1.1. A central hyperplane arrangement A in a finite dimensional vector space V is a finite set {H 1 , . . . , H } of subspaces of codimension 1. Choosing a basis of V we may identify the algebra S(V * ) of coordinates of V with S = k[x 1 , . . . , x n ]: for each i ∈ let λ i ∈ S be a linear form with kernel H i . Up to a nonzero scalar, the defining polynomial

Q = λ 1 • • • λ ∈ S depends only on A.
Definition 1.2. The set of derivations tangent to the arrangement A is

Der A := {θ ∈ Der(S) : λ i divides θ(λ i ) for every i ∈ } .
This is a Lie-subalgebra and a sub-S-module of the Lie algebra of derivations Der(S) of S.

M (θ 1 , . . . , θ ) :=     θ 1 (x 1 ) • • • θ 1 (x ) . . . . . . θ (x 1 ) • • • θ (x )     is a nonzero scalar multiple of Q.
The notion of freeness connects arrangements of hyperplanes with commutative algebra, algebraic geometry and combinatorics. While not a property of generic hyperplane arrangements, many of the motivating examples of hyperplane arrangements are free. Saito's criterion in Theorem 1.4 is perhaps the most practical way to prove freeness, though there are other methods to prove this condition -see A. Bigatti, E. Palezzato and M. Torielli's [START_REF] Bigatti | New characterizations of freeness for hyperplane arrangements[END_REF] for a discussion on the state of the art.

Example 1.5. Let n ≥ 2, E = A n the affine space with coordinate ring S = k[x 1 , . . . , x n ]. The braid arrangement B n in E has hyperplanes H ij with equation x i -x j = 0, 1 ≤ i < j ≤ n, so that the defining polynomial is Q = 1≤i<j≤n (x j -x i ).

Consider the derivations θ 1 , . . . , θ n of S defined for k ∈ n by

θ 1 (x k ) = 1, θ i (x k ) = (x k -x 1 ) . . . (x k -x i-1 ) if i ≥ 2.
These derivations satisfy (x k -x j ) | θ i (x k -x j ) for any i, j, k ∈ n and therefore belong to Der B n . The Saito's matrix (θ i (x k )) is triangular and its determinant is Q. By Saito's Criterion, Der B n is a free S-module with basis {θ 1 , . . . , θ n }.

Example 1.6. Let n ≥ 1 and E = A n+1 be the affine space with coordinate ring S = k[x 0 , x 1 , . . . , x n ].

As in Example 1.5 above, the arrangement B n+1 in E has equation 0≤i<j≤n (x i -x j ). Consider the subspace V = {0} × A n of E, defined by the equation x 0 = 0, and the hyperplanes Hij of V defined by x i -x j = 0 for 1 ≤ i < j ≤ n. We call Bn the arrangement formed by these hyperplanes, so that Bn is defined by equation

x 1 . . . x n 0≤i<j≤n (x i -x j ) = 0. The derivations α 1 , . . . , α n of S = k[x 1 , . . . , x n ] defined for k ∈ n by α 1 (x k ) = x k , α i (x k ) = 0 if i > k; x k j<i (x k -x j ) if i ≤ k if i ≥ 2
belong to Der Bn . Thanks to Saito's Criterion, (α 1 , . . . , α n ) is a basis of Der Bn .

Example 1.7. Let V be a finite-dimensional vector space. We say that σ ∈ GL(V ) is a pseudoreflection if σ is of finite order and fixes a hyperplane H σ of V , and it is a reflection if this order is 2. A finite subgroup G of the group of automorphisms of V is a (pseudo-) reflection group if it is generated by (pseudo-) reflections, and the set of reflecting hyperplanes A(G) of a reflection group G is the reflection arrangement of G. It is a result by H. Terao in [START_REF] Terao | Free arrangements of hyperplanes and unitary reflection groups[END_REF] that every reflection arrangement over k = C is free. Consider the nth braid arrangement B n of Example 1.5: identifying the reflection with respect to the plane x i -x j = 0 with the permutation (ij) ∈ S n we see that B n = A(S n ).

Example 1.8. Let r, n ≥ 1 and consider the arrangement

A n r in V = k n defined by 0 = x 1 . . . x n 1≤i<j≤n (x r j -x r i ).
Taking r = 1 we see that A n 1 = Bn for every n. When r ≥ 2, let G = C r S n be the wreath product of the cyclic group C r of order r and the symmetric group S n . We see that

A n r is the reflection arrangement of G, this is, A n r = A(C r S n ). There is a well-known basis of Der A n r in [OT92, §B] that consists of the derivations θ 1 , . . . , θ n of S = k[x 1 , . . . , x n ] defined for 1 ≤ k, m ≤ n by θ m (x k ) = x (m-1)r+1 k . Consider the derivations α 1 , . . . , α n of S defined for 1 ≤ k ≤ n and 2 ≤ m ≤ n by α 1 (x k ) = x k , α m (x k ) = x k m-1 i=1 (x r k -x r i ). (1)
These derivations belong to Der A n r : evidently

α 1 = θ 1 , and if m ≥ 2 then α m = θ m -s 1 θ m-1 + . . . + (-1) m-1 s m-1 θ 1 , where s j = 1≤i1<...<ij ≤m-1 x r i1 • • • x r ij is the jth elementary symmetric polynomial in variables x r 1 , . . . , x r m-1 for 1 ≤ j ≤ m -1. For 1 ≤ k ≤ n θ m -s 1 θ m-1 + . . . + (-1) m-1 s m-1 θ 1 (x k ) = x (m-1)r+1 k -s 1 x (m-2)r+1 k + . . . + (-1) m-1 s m-1 x k = x k m-1 i=1 (x r k -x r i ),
which equals α m (x k ). Saito's matrix (α m (x k )) is diagonal and its determinant is

n k=1 α k (x k ) = n k=1 x k k-1 i=1 (x r k -x r i ) = x 1 . . . x n 1≤i<k≤n (x r k -x r i ).
It follows from Saito's criterion that α 1 , . . . , α n is a basis of Der A n r .

Example 1.9. Let r ≥ 1. The arrangement A r := A 3 r is defined by the nullity of If S is a commutative algebra and L is a Lie-subalgebra of the Lie algebra of derivations Der S that is at the same time an S-submodule then L is an Lie-Rinehart algebra over S. This applies to our situation of interest: Proposition 1.11. Let A be a hyperplane arrangement in a vector space V . The Lie algebra of derivations Der A of A is a Lie-Rinehart algebra over the algebra of coordinates of V . Definition 1.12. Let n ≥ 1, S = k[x 1 , . . . , x n ] and L be a subset of derivations of S such that (S, L) is a Lie-Rinehart algebra.

Q(A r ) := x 1 x 2 x 3 (x r 2 -x r 1 )(x r 3 -x r 1 )(x r 3 -x r 2 ). The basis of Der A r in (1) consist in this case of the derivations α 1 , α 2 , α 3 of S = k[x 1 , x 2 , x 3 ] with Saito's matrix    x 1 x 2 x 3 0 x 2 (x r 2 -x r 1 ) x 3 (x r 3 -x r 1 ) 0 0 x 3 (x r 3 -x r 2 )(x r 3 -x r 1 )    1.2.
(i) We call L triangularizable if L is a free S-module that admits a basis given by derivations α 1 , . . . , α n satisfying the two conditions

α i (x j ) = 0 if i > j, α 1 (x 1 ) • • • α n (x n ) = 0.
(ii) We say that L satisfies the Bézout condition if in addition for each k in n -1 , the element α k (x k ) of S is coprime with the determinant of the matrix

        α k (x k+1 ) α k+1 (x k+1 ) 0 • • • 0 . . . . . . . . . . . . . . . α k (x n-2 ) α k+1 (x n-2 ) • • • α n-2 (x n-2 ) 0 α k (x n-1 ) α k+1 (x n-1 ) • • • • • • α n-1 (x n-1 ) α k (x n ) α k+1 (x n ) • • • • • • α n-1 (x n )        
(x 2 ) = x 2 (x r 2 -x r 1 ) is coprime with α 2 (x 3 ) = x 3 (x r 3 -x r 1 )
, and the determinant det

α 1 (x 2 ) α 1 (x 3 ) α 2 (x 2 ) α 2 (x 3 ) = det x 2 x 3 x 2 (x r 2 -x r 1 ) x 3 (x r 3 -x r 1 ) = x 2 x 3 (x r 3 -x r 2 )
is coprime with α 1 (x 1 ) = x 1 .

Diff(A) = n≥1 I Diff(S) (Q n Diff(S)).
As seen in [START_REF] Calderón-Moreno | Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor[END_REF] for k = C or in [S Á18] for k of characteristic zero, if A is free then the algebra Diff A coincides with the sub-associative algebra of End(S) generated by Der A and the set of maps given by left multiplication by elements of S.

Example 1.17. The arrangement A = B2 in k 2 with equation 0 = xy(y -x) admits, by [KL21, §5], a presentation of Diff A adapted from [KS Á18]: the two derivations

E = x∂ x + y∂ y , D = y(y -x)∂ y of k[x, y]
form a basis of Der A, and the algebra Diff A is generated by the symbols x, y, D and E subject to the relations

[y, x] = 0, [D, x] = 0, [D, y] = y(y -x), [E, x] = x, [E, y] = y, [E, D] = D.
Given a Lie-Rinehart algebra (S, L), a Lie-Rinehart module -or (S, L)-module-is a vector space M which is at the same time an S-module and an L-Lie module in such a way that if s ∈ S, α ∈ L and m ∈ M then

(sα) • m = s • (α • m), α • (s • m) = (sα) • m + α S (s) • m.
Theorem 1.18 ([Hue90, §1]). Let (S, L) be a Lie-Rinehart algebra.

(i) There exists an associative algebra U = U (S, L), the universal enveloping algebra of (S, L), endowed with a morphism of algebras i : S → U and a morphism of Lie algebras

j : L → U satisfying, for s ∈ S and α ∈ L, i(s)j(α) = j(sα), j(α)i(s) -i(s)j(α) = i(α S (s)).
(ii) The algebra U is universal with these properties.

(iii) The category of U -modules is isomorphic to the category of (S, L)-modules.

Example 1.19. If S = k[x 1 , . . . , x n ] then the full Lie algebra of derivations L = Der S is a Lie-Rinehart algebra and its enveloping algebra is isomorphic to the algebra of differential operators Diff(S) = A n , the nth Weyl algebra.

The following result -[NM08, §12] when k = C and [Kor19, Theorem 2.19] for k of characteristic zero-is our motivation to consider Lie-Rinehart algebras in the algebraic aspects of hyperplane arrangements.

Theorem 1.20. Let A be a free hyperplane arrangement on a vector space V and let S be the algebra of coordinate functions on V . There is a canonical isomorphism of algebras

U (S, Der A) ∼ = Diff(A).
Proposition 1.21. For n ≥ 1 there is an isomorphism of algebras

Diff B n+1 ∼ = A 1 ⊗ Diff Bn . Proof. Let n ∈ N, S = k[x 0 , x 1 , . . . , x n ], T = k[y 1 , . . . , y n ] and observe that the unique morphism of algebras k[z] ⊗ T → S given by z → x 0 and y k → x k -x 0 if k ≥ 1 is an isomorphism -we are identifying z with z ⊗ 1 and y k with 1 ⊗ y k .
The derivations in the basis of Der B n+1 given in Example 1.5 induce derivations θ1 , . . . , θn+1 on k[z] ⊗ T . For 1 ≤ i ≤ n + 1 and 1 ≤ k ≤ n these derivations satisfy θ1 :

z → 1;

y k → 0; θ2 : z → 0; y k → y k ; θi : z → 0; y k → y k i-1 j=1 (y k -y j ) if i ≥ 3.
The Lie algebra Der S is isomorphic to the Lie algebra product Der k[z] × Der Bn : the derivations θi with i ≥ 2 correspond to the α i 's in Example 1.6. It follows that the enveloping algebra of the

Lie-Rinehart pair (S, Der B n+1 ) is isomorphic to the product U (k[z], Der k[z]) × U (T, Der Bn ).
The result is now a consequence of Theorem 1.20 and Example 1.19.

Definition 1.22. Let n ≥ 1, S = k[x 1 , . . . ,
x n ] and L a triangularizable Lie-Rinehart algebra over S with basis (α 1 , . . . , α n ). We say that L satisfies the orthogonality condition if there exists a family (u 1 , . . . , u n ) of elements of U and

f i k ∈ S for 1 ≤ k ≤ n and 1 ≤ i ≤ n -1 such that u k = α n + n-1 i=1 f i k α i , [u k , x l ] = 0 if k = l.
Example 1.23. Consider for n ≥ 2 the Lie-Rinehart algebra Der B n from Example 1.5. The family (u 1 , . . . , u n ) of elements of U defined for k ∈ n by

u k = n i=k (-1) n-i n j=i+1 (x j -x k ) θ i is such that [u k ,
x l ] = 0, whence the orthogonality condition is satisfied. The Lie-Rinehart algebra Der Bn from Example 1.6 also satisfies this condition with a similar choice of orthogonal elements.

Let r ≥ 1 and A r = A(C r S 3 ). Let S = k[x 1 , x 2 , x 3 ] and L = Der A r be the Lie-Rinehart algebra associated to A r . The derivations {α 1 , α 2 , α 3 } given in Example 1.9 make of L a triangular Lie algebra that satisfies the Bézout condition. We identify the universal enveloping algebra of L with Diff A r .

Proposition 1.24. The Lie-Rinehart algebra associated to A r together with the family {u 1 , u 2 , u 3 } of elements of Diff A r defined by

u 1 = α 3 -(x r 3 -x r 1 )α 2 + (x r 3 -x r 1 )(x r 2 -x r 1 )α 1 , u 2 = α 3 -(x r 3 -x r 2 )α 2 , u 3 = α 3
satisfies the orthogonality condition.

Proof. The condition [u k , x l ] = 0 if k, l ∈ 3 and l = k holds true whenever l < k, so we suppose that l > k. If k = 3 there is nothing to see; the case k = 2 amounts to the verification that

[u 2 , x 3 ] = α 3 (x 3 ) -(x r 3 -x r 2 )α 2 (x 3 ) = x 3 (x r 3 -x r 2 )(x r 3 -x r 1 ) -(x r 3 -x r 2 )x 3 (x r 3 -x r 1 ) = 0
and for or k = 1 we have

[u 1 , x 2 ] = α 3 (x 2 ) -(x r 3 -x r 1 )α 2 (x 2 ) + (x r 3 -x r 1 )(x r 2 -x r 1 )α 1 (x 2 ) = -(x r 3 -x r 1 )x 2 (x r 2 -x r 1 ) + (x r 3 -x r 1 )(x r 2 -x r 1 )x 2 = 0 and [u 1 , x 3 ] = α 3 (x 3 ) -(x r 3 -x r 1 )α 2 (x 3 ) + (x r 3 -x r 1 )(x r 2 -x r 1 )α 1 (x 3 ) = x 3 (x r 3 -x r 2 )(x r 3 -x r 2 ) -(x r 3 -x r 1 )x 3 (x r 3 -x r 1 ) + (x r 3 -x r 1 )(x r 2 -x r 1 )x 3 = 0.
1.4. Cohomology. Given an associative algebra A the (associative) enveloping algebra A e is the vector space A ⊗ A endowed with the product

• defined by (a 1 ⊗ a 2 ) • (b 1 ⊗ b 2 ) = a 1 b 1 ⊗ b 2 a 2 ,
so that the category of left A e -modules is equivalent to that of A-bimodules. The Hochschild cohomology of A with values on an A e -module M is

H • (A, M ) := Ext • A e (A, M ).
When M = A we write HH • (A) := H • (A, M ). C. Weibel's book [START_REF] Weibel | An introduction to Homological algebra[END_REF] may serve as general reference on this subject.

Definition 1.25 [START_REF] Rinehart | Differential Forms on General Commutative Algebras[END_REF]). Let (S, L) be a Lie-Rinehart algebra with enveloping algebra U and let N be an U -module. The Lie-Rinehart cohomology of (S, L) with values on N is

H • S (L, N ) := Ext • U (S, N ). Remark 1.26 ([Rin63]
). In the setting of Definition 1.25 above, suppose that L is S-projective and let Λ • S L denote the exterior algebra of L over S. The complex Hom S (Λ

i (A 1 ) ∼ = k if i = 0.
Corollary 1.29. The Hilbert series of the Hochschild cohomology of Diff B 3 is

h(t) = 1 + 3t + 6t 2 + 4t 3 .
Proof. Proposition 1.28 particularizes to HH • (Diff B 3 ) ∼ = HH • (Diff B2 ), and then [KL21, Corollary 5.8] reads h HH * (Diff B3) = h HH * (Diff B2) = 1 + 3t + 6t 2 + 4t 3 .

Combinatorics of the Koszul complex

We let n ≥ 1 and assume throughout this section that (S, L) is a Lie-Rinehart algebra with S = k[x 1 , . . . , x n ] and L a free S-module with basis (α 1 , . . . , α n ). Let U = U (S, L) be its Lie-Rinehart enveloping algebra. To compute the Hochschild cohomology of S we use the Koszul resolution of S available in [Wei94, §4.5].

Lemma 2.1. Let W be the subspace of S with basis (x 1 , . . . , x n ). The complex

P • = S e ⊗ Λ • W with differentials b • : P • → P •-1 defined for s, t ∈ S, q ∈ n and 1 ≤ i 1 < • • • < i q ≤ n by b q (s|t ⊗ x i1 ∧ • • • ∧ x iq ) = q j=1 (-1) j+1 [(sx ij |t) -(s|x ij t)] ⊗ x i1 ∧ • • • ∧ xij ∧ • • • ∧ x iq
and augmentation ε : S e → S given by ε(s|t) = st is a resolution of S by free S e -modules. The notation is the usual one: the symbol | denotes the tensor product inside S e and xij means that x ij is omitted.

Through a classical adjunction, the complex Hom S e (P • , U ) is isomorphic to

Hom(Λ • W, U ) ∼ = U ⊗ Hom(Λ • W, k) =: X • .
(2)

We compute the Hochschild cohomology H • (S, U ) from the complex (X • , d • ). For each q in 0, n the basis {x k1 ∧ . . . ∧ xkq : 1 ≤ k 1 < . . . < k q ≤ n} of Hom(Λ q W, k) dual to the basis {x k1 ∧ . . . ∧ x kq } of Λ q W induces a basis of X q as a U -module. Write α I := α in n . . . α i1 1 for each n-tuple of nonnegative integers I = (i n , . . . , i 1 ), and call

|I| = i n + . . . + i 1 the order of I. A result à la Poincaré-Birkhoff-Witt in [Rin63, §3] assures that the set α I : I ∈ N n (3)
is an S-basis of U . Moreover, U is a filtered algebra, with filtration (F p U : p ≥ 0) given by the order of differential operators:

F p U = f α I : f ∈ S, |I| ≤ p for each p ≥ 0. Proposition 2.2. Let q ∈ {0, . . . , n}. (i) The set formed by α I xk1 ∧ • • • ∧ xkq with I ∈ N n and 1 ≤ k 1 < . . . < k q ≤ n is an S-basis of X q . (ii)
There is a filtration (F p X q : p ≥ 0) of vector spaces on X q determined for each p ≥ 0 by

F p X q = f α I xk1 ∧ • • • ∧ xkq : f ∈ S, 1 ≤ k 1 < . . . < k q ≤ n, I ∈ N n such that |I| ≤ p .
Proof. In view of (2), for each q the U -module X q admits xk1 ∧ • • • ∧ xkq : 1 ≤ k 1 < . . . < k q ≤ n as a basis. The claim follows from this and the S-basis of U in (3) above.

The differentials d q : X q → X q+1 induced by b • : P • → P •-1 satisfy for q = 0, 1

d 0 : u → n k=1 [u, x k ]x k , d 1 : n k=1 u k xk → 1≤k<l≤n ([u k , x l ] -[u l , x k ]) xk ∧ xl .
Given m ∈ n we denote by e m the n-tuple whose components are all zero except for the (n-m)th, where there is a 1.

Lemma 2.3. Let a = |I|=p f I α I for f I ∈ S with I ∈ N n . If k ∈ n and J = (j n , . . . , j 1 ) ∈ N n has order p -1 then the component of [a, x k ] in α J is n m=1 (j m + 1)α m (x k )f J+em .
Proof. If I = (i n , . . . , i 1 ) ∈ N n has order p then

[f I α I , x k ] ≡ i n f I α n (x k )α I-en + . . . + i 1 f I α 1 (x k )α I-e1 mod F p-2 U.
If there exists a monomial in this expression belonging to Sα J then there exists m ∈ n such that I -e m = J. This happens when the component of [f

I α I , x k ] in α J is i m α m (x k )f I = (j m + 1)α m (x k )f J+em ,
and therefore [a,

x k ] ≡ |J|=p-1 n m=1 (j m + 1)α m (x k )f J+em α J modulo F p-2 U .
For g 1 , . . . , g n ∈ S and

f 1 = (f 1 1 , . . . , f 1 n ), . . . , f n = (f n 1 , . . . , f n n ) ∈ S ×n we let Ω 0 (g n , . . . , g 1 ) := n i=1 g i α i ∈ F 1 U, Ω 1 (f n , . . . , f 1 ) := n l=1 n i=1 f i k α i xk ∈ F 1 X 1 . Proposition 2.4. Let p ≥ 0, u ∈ F p U and ω ∈ F p X 1 . (i) If {f I : I ∈ N n , |I| = p} ⊂ S is such that u ≡ |I|=p f I α I mod F p-1 U then d 0 (u) ≡ |J|=p-1 d 0 Ω 0 (j n + 1)f J+en , (j n-1 + 1)f J+en-1 , . . . , (j 1 + 1)f J+e1 α J modulo F p-2 X 1 . (ii) If ω ≡ n l=1 |I|=p f I l α I xi mod F p-1 X 1 for f I l : I ∈ N n , |I| = p, l ∈ n ⊂ S then d 1 (ω) ≡ |J|=p-1 d 1 Ω 1 (j n + 1)f J+en , (j n-1 + 1)f J+en-1 , . . . , (j 1 + 1)f J+e1 α J modulo F p-2 X 2 .
Proof. To prove (i) it suffices to see that the desired equality holds in each coefficient of the S-basis α J xk : J ∈ N n , k ∈ n of X 1 given in Proposition 2.2. Let then J = (j n , . . . , j 1 ) ∈ N n of order p -1 and k ∈ n . Thanks to Lemma 2.3 the component in

α J xk of d 0 (u) = n l=1 [u, x l ]x l is n m=1 (j m + 1)α m (x k )f J+em . ( 4
)
On the other hand, given f n , . . . , f 1 ∈ S a direct calculation shows that the component in xk of

d 0 Ω 0 f n , . . . , f 1 is n i=1 f i α i (x k ).
It follows that the component of d 0 Ω 0 (j n + 1)f J+en , . . . , (j 1 + 1)f J+e1 in xk is equal to (4), which is tantamount to what we wanted to see. The proof of (ii) is completely analogous.

Cohomologies in degree zero and centers

In this section (S, L) is a triangularizable Lie algebra: S = k[x 1 , . . . , x n ] for some n ≥ 1 and L is a sub-S-module of derivations of S with a basis given by derivations α 1 , . . . , α n that satisfy α i (x j ) = 0 if i > j and α i (x i ) = 0 for every i ∈ n . Let U be the enveloping algebra of (S, L).

The cohomology of S with values on U . The Hochschild cohomology H

• (S, U ) is the cohomology of the complex (X • , d • ) of (2). Lemma 3.1. The restriction of d 0 : X 0 → X 1 to F 1 X 0 has kernel F 0 X 0 . Proof. It is evident that F 0 X 0 = S is contained in ker d 0 . Let u ∈ F 1 U and f 1 , . . . , f n ∈ S such that u ≡ n i=1 f i α i modulo S. We examine the equations d 0 (u)(1|x l |1) = 0, that is, [u, x l ] = 0 for each 1 ≤ l ≤ n. We first observe that 0 = [u, x 1 ] = n i=1 f i α i (x 1 ) = f 1 α 1 (x 1 ),
and then f 1 = 0. Proceeding inductively on k, we assume that u =

n i=k f i α i and compute 0 = [u, x k ] = f k α k (x k ) + . . . + f n α n (x k ) = f k α k (x k ).
We deduce that f k = 0 and conclude that u ∈ S. d Ω 0 (j n + 1)f J+en , . . . , (j 1 + 1)f J+e1 α J mod F p-2 X 1 .

We deduce that 0 = d 0 Ω 0 (j n + 1)f J+en , . . . , (j 1 + 1)f J+e1 for each J with |J| = p -1, provided that p -1 ≥ 0. Thanks to Lemma 3.1 we deduce that 0 = f J+e N -2 = . . . = f J+e-1 . Since we can write every I ∈ N with |I| = p as I = J + e m for some m ∈ n we conclude that if p -1 ≥ 0 then f I = 0 for every I with |I| = p. Proposition 3.3. The inclusion S → U = X 0 induces an isomorphism of graded U -modules H 0 (S, U ) = S Proof. Let us write u = u 0 + . . . + u p with u q ∈ F q U \ F q-1 U and p ≥ 0 maximal among those q such that u q = 0. As d 0 (u) = 0 and d 0 (u q ) ∈ F q-1 X 1 for every q ∈ 0, p we have that d(u p ) ≡ 0 mod F p-2 X 1 , and we may use Proposition 3.2 to see that if p > 0 then u p = 0. We conclude then that p = 0, so that actually u ∈ S. We obtain the result with the evident observation that every element of S is a 0-cocycle in X • .

3.2. The cohomology of U . Our recent calculation of H 0 (S, U ) leaves us just one step away from the zeroth Hochschild cohomology space of U . Theorem 3.4. Let (S, L) be a triangularizable Lie-Rinehart algebra with enveloping algebra U . There is an isomorphism of vector spaces HH 0 (U ) ∼ = k.

Proof. As a consequence of the immediate degeneracy of the spectral sequence of Theorem 1.27 there is an isomorphism of vector spaces HH 0 (U ) ∼ = H 0 S (L, H 0 (S, U )). In view of Proposition 3.3, this isomorphism amounts to

HH 0 (U ) ∼ = H 0 S (L, S) ∼ = {f ∈ S : α i (f ) = 0 if i ∈ n } Since α i (f ) = n j=1 α i (x j )∂ j f for f ∈ S, the condition that α i (f ) = 0 if i ∈ n means that (∂ 1 f, . . . , ∂ n f ) belongs to the kernel of the Saito's matrix M = (α i (x j )) n i,j=1
. As this matrix is triangular and its determinant is nonzero, the condition α i (f ) = 0 for all i ∈ n is equivalent to ∂ j f = 0 for all j ∈ n , which is to say that f ∈ k.

The first cohomology space H 1 (S, U )

We now restrict our attention to the case in which n = 3. Let then (S, L) be a Lie-Rinehart algebra with S = k[x 1 , x 2 , x 3 ] and L the free S-module generated by the subset of derivations {α 1 , α 2 , α 3 } in Der S. We suppose that (S, L) is triangularizable, this is, α i (x j ) = 0 if i > j and α 1 (x 1 )α 2 (x 2 )α 3 (x 3 ) = 0, and that (S, L) satisfies the Bézout condition:

• the polynomials α 2 (x 2 ) and α 2 (x 3 ) are coprime;

• the polynomials α 1 (x 1 ) and det α1(x2) α1( x3)

α2(x2) α2(x2) are coprime. Lemma 4.1. Let f i l : i ∈ {1, 2} , l ∈ 3 ⊂ S and write ω = 3 l=1 f 2 l α 2 + f 1 l α 1 xl ∈ X 1 .
If ω is a cocycle then there exist unique elements g 11 , g 12 , g 22 of S such that g 11 α 1 (x 1 ) = f 1 1 , g 12 α 1 (x 1 ) = f 2 1 and g 22 α 2 (x 2 ) = f 2 2 -g 12 α 1 (x 2 ). These elements satisfy

ω ≡ d( 1 2 g 11 α 2 1 + g 12 α 2 α 1 + 1 2 g 22 α 2 2 ) mod F 0 X 1 .
Proof. The components in x1 ∧x 2 and x1 ∧x 3 of dω = 0 tell us that α 1 (x j )f 1 1 +α 2 (x j )f j 1 = α 1 (x 1 )f 1 j for j ∈ {2, 3}. We can arrange these two equations as

α 1 (x 2 ) α 2 (x 2 ) α 1 (x 3 ) α 2 (x 3 ) f 1 1 f 2 1 = α 1 (x 1 ) f 1 2 f 1 3 (5)
and then Cramer's rule tells us that if i ∈ {1, 2} then f i 1 det M = α 1 (x 1 ) det Mi , where M is the matrix on the left hand of (5) and Mi is the matrix obtained by replacing the ith column of M by

f 1 2 f 1 3 . It follows that α 1 (x 1 ) divides f i
1 -because it is coprime with det M in view of the Bézout hypothesis-and then there exist g 11 and g 12 in S such that g 1i α 1 (x 1 ) = f i 1 .

Let u 1 := 1 2 g 11 α 2 1 + g 12 α 2 α 1 and ω := ω -d(u 1 ), and write ω =

3 l=1 f 2 l α 2 + f 1 l α 1 xl . Since d(u 1 ) : x 1 → [u, x 1 ] ≡ g 11 α(x 1 )α 1 + g 12 α 1 (x 1 )α 2 mod S = f 2 1 α 1 + f 2 1 α 2 , x 2 → [u, x 2 ] ≡ g 11 α 1 (x 2 )α 1 + g 12 α 2 (x 2 )α 1 + g 12 α 1 (x 2 )α 2 mod S (6)
we have f 1 1 = f 2 1 = 0. Now, the equation dω = 0 in x1 ∧ x2 and x1 ∧ x3 tells us, as in (5), that f 1 2 = f 1 3 = 0, and in x2 ∧ x3 that α 2 (x 2 ) f 2 3 = α 2 (x 3 ) f 2 2 . Thanks to the Bézout condition there exists g 22 ∈ S such that g 22 α 2 (x 2 ) = f 2 2 ; in view of (6), f 2 2 is equal to f 2 2 -g 12 α 1 (x 2 ). Put u 2 := 1 2 g 22 α 2 2 . We see that d(u 2 )(x 1 ) = 0 and that

d(u 2 )(x 2 ) = [u 2 , x 2 ] ≡ g 22 α 2 (x 2 )α 2 mod S = f 2 2 α 2 .
The difference ω := ω -d(u 2 ) is therefore a coboundary with no component modulo S in x1 nor in x2 , so we can write ω ≡ f 1 α 1 + f 2 α 2 x3 mod F 0 X 1 . Now, the equations that come from ω being a coboundary are 0 = f 1 α 1 (x 1 ) in x1 ∧ x3 , from which f 1 = 0, and 0 = f 2 α 2 (x 2 ) in x2 ∧ x3 , whence finally ω ∈ F 0 X 1 . We have in this way obtained that ω ≡ d(u 1 + u 2 ) mod F 0 X 1 , as desired.

Proposition 4.2. Let p ≥ 0 and ω ∈ F p X 1 , and let f

(i3,i2,i1) l : l ∈ 3 , i 3 , i 2 , i 1 ≥ 0 ⊂ S such that ω ≡ 3 l=1 i1+i2+i3=p f (i3,i2,i1) l α (i3,i2,i1) xl mod F p-1 X 1 . ( 7
)
If ω is a cocycle and f

(p,0,0) 1 = f (p,0,0) 2 = f (p,0,0) 3 = 0 then ω ∈ F p-1 X 1 .
Proof. Let us prove by descending induction on i from p to 0 that the cocycle ω is cohomologous modulo F p-1 X 1 to a cocycle of the form (7) with f

(i3,i2,i1) l = 0 if l ∈ 3 and i 3 ≥ i. ( 8
)
Our hypotheses give us the truth of (8) for i = p. Suppose now that (8) is true for p, . . . , i and assume, without loss of generality, that ω is of the form (7) with f

(i3,i2,i1) l = 0 if l ∈ 3 , i 1 + i 2 + i 3 = p and i 3 ≥ i. Lemma 4.3. Let q ∈ {0, . . . , p -i + 1}. The cocycle ω is cohomologous modulo F p-1 X 1 to a cocycle of the form (7) with f (i-1,p-i+1,0) l = . . . = f (i-1,p-i+1-q,q) l = 0 and f (i3,i2,i1) l = 0 if i 1 + i 2 + i 3 = p and i 3 ≥ i for every l ∈ 3 .
The auxiliary result above implies at once the truth of the inductive step of the proof of (8), thus demonstrating Proposition 4.2.

Proof of Lemma 4.3. Suppose that q = 0. Equation dω = 0 in its component (i -1, p -i, 0) reads, thanks to Proposition 2.4,

0 = d Ω 1 (if (i,p-i,0) , (p -i + 1)f (i-1,p-i+1,0) , f (i-1,p-i,1) )
and the inductive hypothesis (8) tells us that f (i,p-i,0) = 0. Applying now Lemma 4.1 in we obtain that there are g 11 , g 12 , g 22 ∈ S such that 0) and write ω = ω -d(v), so that there exists f (i3,i2,i1)

g 11 α 1 (x 1 ) = f (i-1,p-i,1) 1 , g 12 α 1 (x 1 ) = (p -i + 1)f (i-1,p-i+1,0) 1 , g 22 α 2 (x 2 ) = (p -i + 1)f (i-1,p-i+1,0) 2 -g 12 α 1 (x 2 ) Let v = ( 1 2 g 11 α 2 1 + 1 (p-i+1) g 12 α 2 α 1 )α (i-1,p-i,
l ⊂ S such that ω ≡ n l=3 i1+i2+i3=p f (i3,i2,i1) l α (i3,i2,i1) xl mod F p-1 X 1 . Recall that d(v) : x l → [v, x l ] for l ∈ 3 . Since [v, x 1 ] ≡ (g 11 α 1 (x 1 )α 1 + 1 (p-i+1) g 12 α 1 (x 1 )α 2 )α (i-1,p-i,0) mod F p-1 U = f (i-1,p-i,1) 1 α (i-1,p-i,1) + f (i-1,p-i+1,0) 1 α (i-1,p-i+1,0) , we have that f (i-1,p-i,1) 1 = f (i-1,p-i+1,0) 1 = 0. Moreover, as [v, x 2 ], [v, x 3 ] ∈ i3<i Sα (i3,i2,i1) the coefficients f (i3,i2,i1) l are equal to f (i3,i2,i1) l
and therefore to zero if i 1 + i 2 + i 3 = p, i 3 ≥ i and l ∈ 3 .

We now look at equation dω = 0, again in its coefficient of α (i-1,p-i,0) to obtain that 0

= d Ω 1 (0, (p -i + 1) f (i-1,p-i+1,0) , f (i-1,p-i,1) ) . This equation in its component in x1 ∧ x2 tells us, thanks to (5), that f (i-1,p-i,1) 2 = f (i-1,p-i,1) 3 = 0. On the other hand, applying Lemma 4.1 we get g ∈ S such that gα 2 (x 2 ) = (p -i + 1) f (i-1,p-i+1,0) 2 . Let now λ = 1/(p -i + 2)(p -i + 1) and ṽ = λgα (i-1,p-i+2,0) . Since [ṽ, x 1 ] = 0, [ṽ, x 2 ] ≡ λg(p -i + 2)α 2 (x 2 )α (i-1,p-i+1,0) mod F p-1 U = λ(p -i + 1) f (i-1,p-i+1,0) 2 (p -i + 2)α (i-1,p-i+1,0) = f (i-1,p-i+1,0) 2 α (i-1,p-i+1,0)
and [ṽ,

x 3 ] ∈ i3<i Sα (i3,i2,i1) , the difference ω -d(ṽ) is a cohomologous modulo F p-1 X 1 to a cocycle η = n l=3 i1+i2+i3=p h (i3,i2,i1) l α (i3,i2,i1) xl with h (i3,i2,i1) l = 0 if i 1 + i 2 + i 3 = p, i 3 ≥ i and l ∈ 3 and h (i-1,p-i,1) l = h (i-1,p-i+1,0) l = 0 if l ∈ {1, 2}
. Applying Lemma 4.1 one final time we obtain g11 , g12 , g22 ∈ S such that g11 α 1 (x 1 ) = (p -i + 1)h

(i-1,p-i+1,0) 1 , g12 α 1 (x 1 ) = h (i-1,p-i,1) 1 and g22 α 2 (x 2 ) = (p -i + 1)h (i-1,p-i,1) 2
-g12 α 1 (x 2 ) -and therefore g11 , g12 and g22 must be equal to 0-and that satisfy -1,p-i,1) = 0, and therefore that η ≡ 0 mod F p-1 X 1 . This finishes the proof of the base step of Lemma 4.3.

Ω 1 (0, (p-i+1)h (i-1,p-i+1,0) , h (i-1,p-i,1) ) ≡ d( 1 2 g11 α 2 1 + g12 α 2 α 1 + 1 2 g22 α 2 2 ) mod F 0 X 1 . It follows that h (i-1,p-i+1,0) = h (i
We finally deal with the inductive step of Lemma 4.3. Let q, i and ω be as in the statement. The component in α (i-1,p-i-q,q) of equation dω = 0 yields 0 = d Ω 1 if (i,p-i-q,q) , (p -i -q + 1)f (i-1,p-i-q+1,q) , (q + 1)f (i-1,p-i-q,q+1) . Now, our inductive hypotheses of (8) and of Lemma 4.3 tell us, respectively, that f (i,p-i-q,q) = 0 and that f (i-1,p-i-q+1,q) = 0, and therefore our equation above reduces to 0 = d Ω 1 0, 0, f (i-1,p-i-q,q+1) .

Applying to this situation Lemma 4.1 we obtain g ∈ S such that gα 1 (x l ) = f

(i-1,p-i-q,q+1) l for l ∈ 3 . Let v = 1 (q+2) gα (i-1,p-i-q,q+2) and write ω = ω -d(v): let f (i3,i2,i1) l ⊂ S such that ω ≡ n l=3 i1+i2+i3=p f (i3,i2,i1) l α (i3,i2,i1) xl modulo F p-1 X 1 . As [v, x 1 ] ≡ gα 1 (x 1 )α (i-1,p-i-q,q+1) = f (i-1,p-i-q,q+1) 1 α (i-1,p-i-q,q+1) mod F p-1 U and if j ∈ {2, 3} then [v, x j ] ∈ Sα (i-2,p-i-q,q+2) ⊕ Sα (i-1,p-i-q-1,q+2) ⊕ Sα (i-1,p-i-q,q+1) ⊕ F p-1 U, we obtain that f (i3,i2,i1) l = 0 whenever i 3 ≥ i and f (i-1,p-i+1,0) l = . . . = f (i-1,p-i+1-q,q) l
= 0 for every l ∈ 3 and, in addition, that f (i-1,p-i-q,q+1) 1 = 0. As a consequence of this, the component in α (i-1,p-i-q,q) of equation dω = 0 reduces to 0 = d Ω 1 0, 0, f (i-1,p-i-q,q+1) . The element g that is provided for this situation by Lemma 4.1 satisfies gα 1 (x l ) = f (i-1,p-i-q,q+1) l for l ∈ 3 : it follows that g = 0 and hence f (i-1,p-i-q,q+1) l = 0 for l ∈ 3 and ω ≡ 0 mod F p-1 X 1 . This finishes the proof of Lemma 4.3.

From this point on we demand to (S, L) that in addition it satisfy the orthogonality condition: that there be a family u 1 , u 2 , u 3 of elements of U that can be written as

u k = α 3 + h 2 k α 2 + h 1 k α 1 for some h i k : k ∈ 3 , i ∈ 2 ⊂ S and such that [u k , x l ] = 0 whenever k = l.
The idea is that that we can add to any cocycle in F p X 1 an S-linear combination of u p k xk to remove its components in the maximum power of α 3 and in this way obtain a cocycle that falls in the hypotheses of Proposition 4.2.

Corollary 4.4. Let {u 1 , u 2 , u 3 } be the family that makes (S, L) satisfy the orthogonality condition.

(i) The cochains η p k = u p k xk ∈ F p X 1 defined for p ≥ 0 and k ∈ 3 are cocycles. (ii) Every cocycle in X 1 is cohomologous to one in the S-submodule of X 1 generated by

{η p k : k ∈ 3 , p ≥ 0}.
Proof. Let us denote by Z 1 the S-module generated by {η p l : l ∈ 3 , p ≥ 0}. We prove by induction on p ≥ 0 that if ω ∈ F p X 1 is a cocycle then there exist z ∈ Z 1 and u ∈ U such that ω = d 0 (u) + z. We first observe that F 0 (X 1 ) = F 0 (Z 1 ) because xl = η 0 l , and then for p = 0 we have that ω ∈ F 0 (X 1 ) ⊂ Z 1 .

Assume now that p > 0 and let

f I l : l ∈ 3 , I ∈ N 3 ⊂ S such that ω ≡ 3 l=1 |I|=p f I l α I xl mod F p-1 X 1 . Defining z = 3 l=1 f (p,0,0) l η p
l we see that the cocycle ω := ω -z has its components in α p x1 , α p x2 , α p x3 equal to zero, and applying Proposition 4.2 we deduce that ω is a coboundary modulo F p-1 X 1 : let u ∈ U and ω ∈ F p-1 X 1 be such that ω = d 0 (u) + ω . The inductive hypothesis tells us that there exist u ∈ U and z ∈ Z 1 such that ω = d 0 (u ) + z , and thus ω = ω + z = d 0 (u + u ) + (z + z ), as we wanted.

Proposition 4.5. Let p ≥ 0. (i) Let ω ∈ F p X 1 be a cocycle, so that there exist {f 1 , f 2 , f 3 } ⊂ S and u ∈ U such that ω ≡ 3 l=1 f l η p l + du modulo F p-1 X 1 . The cocycle ω is equivalent to a coboundary modulo F p-1 X 1 if and only if 3 l=1 f l xl is a coboundary. (ii) The unique S-linear map γ p : F 0 X 1 → F p X 1 such that xl → η p l if 1 ≤ l ≤ 3 induces an isomorphism of S-modules F p H 1 (S, U )/F p-1 H 1 (S, U ) ∼ = F 0 H 1 (S, U ).
Proof. Suppose that ω 0 = n l=1 f l xl is a coboundary and let v ∈ U such that d 0 (v) = ω 0 . Thanks to Proposition 3.2 we may assume that v ∈ F 1 X 1 and write v ≡ g 3 α 3 + g 2 α 2 + g 1 α 1 mod S for some g 3 , g 2 , g 1 ∈ S. In view of Proposition 2.4 there exist f I 0 ∈ F 0 X 1 such that we may write

d 0 g 3 p + 1 α p+1 3 + g 2 α p 3 α 2 + g 1 α p 3 α 1 ≡ d 0 (v) α p 3 + i3<p f (i3,i2,i1) 0 α (i3,i2,i1) mod F p-1 X 1 . It follows that the difference ω -d 0 g 3 p+1 α p+1 3 + g 2 α p 3 α 2 + g 1 α p 3 α 1 is a cochain whose components in α p 3 x1 , α p 3 x2 , α p 3 xN are zero. Applying Proposition 4.2 we see that ω is equivalent to a coboundary modulo F p-1 X 1 .
Reciprocally, let u ∈ U such that d 0 (u) = ω. Thanks to Proposition 3.2 we know that u ∈ F p+1 U : let us write u ≡ |K|=p+1 h K α K with {h K : K ∈ N 3 , |K| = p + 1} ⊂ S. Taking into account Proposition 2.4 again we see that

d 0 (u) ≡ d Ω 0 (p + 1)h (p+1,0,0) , h (p,1,0) , h (p,0,1) α p 3 + i3<p f (i3,i2,i1) 0 α (i3,i2,i1) modulo F p-1 X 1 for some f I 0 ∈ F 0 X 1 .
The equality of this to ω implies, looking at the components in α p 3 x1 , α p 3 x2 , α p 3 x3 , that d Ω 0 (p + 1)h (p+1,0,0) , h (p,1,0) , h (p,0,1) = 3 l=1 f l xl . This completes the proof of the first item. Now, the truth of the first item implies two things: first, that the composition F 0 X 1 → F p X 1 /F p-1 X 1 of γ p with the projection to the quotient descends to cohomology and, second, that the map induced in cohomology by this composition is a monomorphism. It is also surjective thanks to Corollary 4.4(ii).

Recall that the filtered S-module F • H 1 (S, U ) has a graded associated S-module Gr • H 1 (S, U ) = p≥0 Gr p H 1 (S, U ) given by Gr p H 1 (S, U ) := F p H 1 (S, U )/F p-1 H 1 (S, U ). We have just seen that Gr p H 1 (S, U ) is isomorphic as an S-module to F 0 H 1 (S, U ) for any p ≥ 0: we claim that we can make it an isomorphism of graded S-modules.

Given p ≥ 0, the map γ p : F 0 X 1 → F p X 1 induces an isomorphism of S-modules F 0 H 1 (S, U ) ∼ = Gr p H 1 (S, U ) that shifts the polynomial degree in 3(p -1): indeed, for each l ∈ 3 the class of η l ∈ X 1 , which has polynomial degree 2, is sent to the class of η p l , which has polynomial degree 3p -1. On the other hand, the morphism of S-modules γ :

F 0 (H 1 (S, U )) ⊗ k[α 3 ] → Gr H 1 (S, U ) such that [η l ] ⊗ α p 3 → [η p l ]
for l ∈ 3 and p ≥ 0 does respect the graduation and is an isomorphism because so is each γ p . In addition to this, we observe that

F 0 (H 1 (S, U )) = S ⊗ x1 , x2 , x3 Sd 0 (α 1 ) + Sd 0 (α 2 ) + Sd 0 (α 3 ) ∼ = coker M.
We summarize our findings in the following statement.

Corollary 4.6. Let S = k[x 1 , x 2 , x 3 ] and L a free S-submodule of Der S generated by derivations α 1 , α 2 , α 3 in such a way that (S, L) is a triangularizable Lie-Rinehart algebra that satisfies the Bézout and orthogonality conditions. Let U be the Lie-Rinehart enveloping algebra of L. There is an isomorphism of S-graded modules

H 1 (S, U ) ∼ = coker M ⊗ k[α 3 ],
where M is the Saito's matrix of (S, L).

Recall that the cokernel of M has a rich algebraic structure -see M. Granger, D. Mond and M. Schulze's [START_REF] Granger | Partial normalizations of Coxeter arrangements and discriminants[END_REF].

Computation of HH 1 (U )

The spectral sequence in Theorem 1.27, regardless of its degeneracy, gives us an strategy to obtain the first Hochschild cohomology space HH • (U ) of the enveloping algebra U of a Lie-Rinehart algebra (S, L): indeed, HH 1 (U ) is isomophic to H 1 S (L, H 0 (S, U )) ⊕ H 0 S (L, H 1 (S, U )). In Sections 3 and 4 we computed H 0 (S, U ) and H 1 (S, U ) when (S, L) is as in Corollary 4.6, which is the conclusion of Section 4. In this section we describe their L-module structure and use it to compute their respective Lie-Rinehart cohomology spaces for the case in which (S, L) is associated to a hyperplane arrangement of the form A r = A(C r S 3 ) as in Example 1.9. 5.1. The L-module structure on H • (S, U ). Let (S, L) be a Lie-Rinehart pair with enveloping algebra U . Let us describe the construction in [START_REF] Kordon | Lie-Rinehart and Hochschild cohomology for algebras of differential operators[END_REF] that gives an L-module structure to the Hochschild cohomology H • (S, U ) of S with values on U .

Fix α ∈ L and an S e -projective resolution ε : P • → S. Let α • be an α e S -lifting of α S : S → S to P • , that is, a morphism of complexes α • = (α q : P q → P q ) q≥0 such that ε • α 0 = α S • ε and for each q ≥ 0, s, t ∈ S and p ∈ P q α q ((s ⊗ t)

• p) = (α S (s) ⊗ t + s ⊗ α S (t)) • p + (s ⊗ t) • p.
The endomorphism α • of Hom S e (P • , U ), defined for each q ≥ 0 to be α q (φ) : p → [α, φ(p)] -φ • α q (p) whenever φ ∈ Hom S e (P q , U ) and p ∈ P q , allows us to define the map ∇ • α : H • (S, U ) → H • (S, U ) as the unique graded endomorphism such that

∇ q α ([φ]) = [α q (φ)],
where [-] denotes class in cohomology. The final result is that α → ∇ q α defines an L-module structure on H q (S, U ) for each q ≥ 0.

The liftings.

From now on we work on the Lie-Rinehart algebra associated to A r and put E := α 1 , D := α 2 and C := α 3 . The commuting relations in L are determined by the rules

[E, C] = (2r + 1)C, [E, D] = (r + 1)D, [D, C] = r(x r 3 + x r 2 -x r 1 ), (9)
as a straightforward calculation shows.

Proposition 5.1. The rules

D 1 (1|x 1 |1) = 0, D 1 (1|x k |1) = s+t=r x s k |x k |x t k - s+t=r-1 x s k |x 1 |x t 1 x k -x r 1 |x k |1 if k = 2, 3 define a D e -lifting of D : S → S. Proof. It is evident that d 1 • D 1 and D 0 • d 1 coincide at 1|x 1 |1; if k = 2, 3 then d 1 • D 1 (1|x k |1) is s+t=r x s k (x k |1 -1|x k )x t k - s+t=r-1 x s k (x 1 |1 -1|x 1 )x t 1 x k -x r 1 x k |1 + x r 1 |x k = x r+1 k -x k x r 1 |1 -1| x r+1 k -x k x r 1 , which equals D 0 • d 1 (1|x k |1) = D 0 (x k |1 -1|x k ) = D(x k )|1 -1|D(x k ) because x k (x r k -x r 1 ) is D(x k ).
Proposition 5.2. For every p ≥ 0 we have that

D 1 (η p 1 ) ≡ pr(x r 3 + x r 2 -x r 1 )η p 1 + rx r-1 1 x 2 η p 2 + rx r-1 1 x 3 η p 3 mod F p-1 X 1 mod Im d 0 D 1 (η p 2 ) ≡ ((1 -p)x r 1 + (p -r -1)x r 2 + px r 3 ) η p 2 mod F p-1 X 1 , D 1 (η p 3 ) ≡ ((1 -p)x r 1 + px r 2 + (p -r -1)x r 3 ) η p 3 mod F p-1 X 1 .
Proof. Recall from Corollary 4.4 the cocycle η p l = u p l xl for each l ∈ 3 that is such that u l commutes with every x j with j = l. The commuting relations (9) in L give

[D, u 3 ] = [D, C] = r(x r 3 + x r 2 -x r 1 )C = r(x r 3 + x r 2 -x r 1 )u 3 and [D, u 2 ] = [D, C -(x r 3 -x r 2 )D] = r(x r 3 + x r 2 -x r 1 )C -(rx r 3 (x r 3 -x r 1 ) -rx r 2 (x r 2 -x r 1 )) D = r(x r 3 + x r 2 -x r 1 ) (C -(x r 3 -x r 2 )D) = r(x r 3 + x r 2 -x r 1 )u 2 . It follows that if l = 2, 3 then [D, u p l ] ≡ p(x r 3 + x r 2 -x r 1 )u p l modulo F p-1 U . We can now compute D 1 (η p l )(1|x 1 |1) = [D, η p l (1|x 1 |1)] -η p l • D 1 (1|x 1 |1) = [D, 0] -η p l (0) = 0; D 1 (η p l )(1|x l |1) = [D, η p l (1|x l |1)] -η p l • D 1 (1|x l |1) = [D, u p l ] -η p l s+t=r x s l |x l |x t l - s+t=r-1 x s l |x 1 |x t 1 x l -x r 1 |x l |1 ≡ p(x r 3 + x r 2 -x r 1 )u p l -((r + 1)x r l -x r 1 ) u p l mod F p-1 U and for m = 1, l D 1 (η p l )(1|x m |1) = [D, η p l (1|x m |1)] -η p l • D 1 (1|x m |1) = [D, 0] -η p l s+t=r x s m |x m |x t m - s+t=r-1 x s 1 |x 1 |x t 1 x m -x r 1 |x m |1 = 0.
(η p 1 ) is cohomologous modulo F p-1 X 1 to pr(x r 3 + x r 2 -x r 1 )η p 1 + rx r-1 1 x 2 η p 2 + rx r-1 1 x 3 η p 3 .
5.3. Invariants of H 1 (S, U ) by the action of L. We already have explicit descriptions of H 1 (S, U ), in Section 4, and of the action of L thereon, in Subsection 5.1 above: the next step is to calculate the intersection of the kernels of the actions of E, D and C on H 1 (S, U ).

Proposition 5.3. H 0 S (L, H 1 (S, U )) = 0.

Proof. Recall that the polynomial grading on S induces a grading on S, on U and on the cohomology H • (S, U ). Since the derivation E induces the linear endomorphism ∇ 1 E of H 1 (S, U ) that sends the class of an homogeneous element a of degree |a| to the class of |a|a it follows that ker ∇ 1 E = H 1 (S, U ) 0 , where H 1 (S, U ) 0 is the subspace of H 1 (S, U ) formed by elements of degree zero. Remember that if k ∈ 3 then |u k | = |α 3 | = 2r, and therefore |η k | = 2r -1. In view of our calculation in Corollary 4.6 this means that ker ∇

1 E = H 1 (S, U ) 0 ∼ =    S1⊗ x1,x2,x3 k(x1 x+x2 x2+x3 x3) ⊕ η 1 , η 2 , η 3 if r = 1; S1⊗ x1,x2,x3 k(x1 x+x2 x2+x3 x3) if r ≥ 2.
We begin by supposing that r ≥ 2. We observe that if

f 1 , f 2 , f 3 ∈ S 1 then D 1 f i xi = D(f i )x i + f i D 1 (x i ) = D(f i )x i + f 1 r x r-1 1 x 2 x2 + x r-1 1 x 3 x3 + f 2 (x r 1 -(r + 1)x r 2 )x 2 + f 3 (x r 1 -(r + 1)x r 3 )x 3 = D(f 1 )x 1 + D(f 2 ) + f 1 rx r-1 1 x 2 + f 2 (x r 1 -(r + 1)x r 2 ) x2 + D(f 3 ) + f 1 rx r-1 1 x 3 + f 3 (x r 1 -(r + 1)x r 3 ) x3
belongs to the homogeneous component of degree r of F 0 H 1 (S, U ), which is precisely

S ⊗ x1 , x2 , x3 Sd 0 (E) + Sd 0 (D) + Sd 0 (C) r = S r+1 ⊗ x1 , x2 , x3 S r d 0 (E) + kd 0 (D) . It follows that if ∇ 1 D ([ f i xi ]) = D 1 ( f i xi ) is zero in cohomology there must exist g ∈ S r and µ ∈ k such that D 1 f i xi = g(x 1 x1 + x 2 x2 + x 3 x3 ) + µ (x 2 (x r 2 -x r 1 )x 2 -x 3 (x r 3 -x r 1 )x 3 ) . ( 10
)
Let us write f i = f i,1 x 1 + f i,2 x 2 + f i,3 x 3 with f i,j ∈ k for i, j ∈ 3 . Up to the addition of coboundary that is a scalar multiple of d 0 (E) = x 1 x1 + x 2 x2 + x 3 x3 we may suppose that f 1,1 = 0. In x1 we have D(f 1 ) = gx 1 , or, in other words,

f 1,2 (x r+1 2 -x 1 x r 2 ) + f 1,3 (x r+1 3 -x 1 x r 3 ) = gx 1 .
The components in x r+1 2 and in x r+1 3 of this equality read f 1,2 = 0 and f 1,3 = 0: this implies that g = 0, and of course that f 1 = 0. Next, equation (10) in x2 yields the equality in S r+1 D(f 2 ) + f 2 (x r 1 -(r + 1)x r 2 ) = µ(x 2 (x r 2 -x r 1 )).

In x r+1 1 and x r+1 3 we have f 2,1 = 0 and f 2,3 = 0, and what remains is -rf 2,2 x r+1 2 = µ(x 2 (x r 2x r 1 ). It follows that µ = 0 and therefore f 2 = 0; analogously, f 3 = 0. We conclude that ker ∇ 1 D | H 1 (S,U )0 = 0 when r ≥ 2.

Let us now suppose that r = 1 and compute the kernel of the restriction of ∇ 1 D to H 1 (S, U ) 0 . Let then f 1 , f 2 , f 3 ∈ S and λ 1 , λ 2 , λ 3 be such that ∇ 1 D ([ f i xi + λ i η i ]) is zero in cohomology. Since

H 1 (S, U ) 1 ∼ = S 2 ⊗ x1 , x2 , x3 S 1 (x 1 x + x 2 x2 + x 3 x3 ) + k(x 2 (x 2 -x 1 )x 2 + x 3 (x 3 -x 1 )x 3 ) ⊕ S 1 η 1 , η 2 , η 3 k(x 1 η 1 + x 2 η 2 + x 3 η 3 ) ⊕ η 2 1 , η 2 2 , η 2 3
there exist µ 1 , µ 2 ∈ k and g ∈ S 1 such that

D 1 f i xi + λ i η i = g(x 1 x + x 2 x2 + x 3 x3 ) + µ 2 (x 2 (x 2 -x 1 )x 2 + x 3 (x 3 -x 1 )x 3 ) + µ 1 (x 1 η 1 + x 2 η 2 + x 3 η 3 ) (11)
We know from Proposition 5.2 that modulo S D 1 (η 1 ) ≡ (x 3 + x 2 -x 1 )η 1 + x 2 η 2 + x 3 η 3 , D 1 (η 2 ) ≡ (-x 2 + x 3 )η 2 and D 1 (η 3 ) ≡ (x 2 -x 3 )η 3 and since D 1 ( S xi ) ⊂ S the equality (11) implies that λ 1 (x 3 + x 2 -x 1 )η 1 + (λ 1 x 2 + λ 2 (-x 2 + x 3 )) η 2 + (λ 1 x 3 + λ 3 (x 2 -x 3 )) η 3 = µ 1 (x 1 η 1 + x 2 η 2 + x 3 η 3 )

This is an equality in 3 i=1 S 1 η i . In S 1 η 3 we have λ 3 x 2 + (λ 1 -λ 3 )x 3 = µ 1 x 3 , so λ 3 = 0 and λ 1 = µ 1 . In S 1 η 2 an analogous argument shows that λ 2 = 0 and λ 1 = µ 1 again, and finally in S 1 η 1 we have

λ 1 (x 3 + x 2 -x 1 ) = λ 1 x 1 .
It follows that λ 1 = µ 1 = 0. Consider now what is left of (11): it is precisely (10) replacing r by 1. The same argument, therefore, allows us to see that ker ∇ 1 D | H 1 (S,U )0 = 0 when r = 1.

We conclude that H 0 S (L, H 0 (S, L)) ⊂ ker ∇ 1 D : H 1 (S, U ) 0 → H 1 (S, U ) 1 = 0, from which H 0 S (L, H 1 (S, L)) = 0 independently of r ≥ 1.

Corollary 5.4. Let r ≥ 1 and A r = A(C r S 3 ). If (S, L) is its associated Lie-Rinehart algebra and U its enveloping algebra then HH 1 (U ) ∼ = H 1 S (L, S). In particular, the dimension of HH 1 (U ) is 3r + 3, the number of hyperplanes of A r .

Proof. Thanks to Theorem 1.27 HH 1 (U ) ∼ = H 1 S (L, H 0 (S, U )) ⊕ H 0 S (L, H 1 (S, U )); Proposition 3.3 tells us that H 0 (S, U ) = S and Proposition 5.3 above that the second summand is zero.

Let f ∈ S 1 be a linear form whose kernel is one of the hyperplanes in A 3 . It is a direct verification that there is a unique derivation ∂ f : U → U such that

∂ f (g) = 0 if g ∈ S; ∂ f (θ) = θ(f )/f if θ ∈ Der A r .
Fix as well k = C and factorize the defining polynomial Q(A r ) = x 1 x 2 x 3 1≤i<j≤3 (x r j -x r i ) as

Q(A r ) = x 1 x 2 x 3
r-1 j=0 (x 2 -e 2jπi/r x 1 )(x 3 -e 2jπi/r x 1 )(x 3 -e 2jπi/r x 2 ) (12)

Corollary 5.5. The Lie algebra of outer derivations of Diff A r together with the commutator is an abelian Lie algebra of dimension 3r + 3 generated by the classes of the derivations ∂ f with f in a linear factor of (12).

Proof. We claim that the classes of ∂ f , with f one of the linear factors in (12), are linearly independent in OutDer(U ). Indeed, let u ∈ U and λ f ∈ k be such that

λ f ∂ f (v) = [u, v] for every v ∈ U . ( 13
)
Evaluating (13) on each v = g ∈ S we obtain that the left side vanishes and therefore u ∈ H 0 (S, U ), which is equal to S in view of Proposition 3.3. Write u = j≥0 u j with u j ∈ S j . Evaluating now (13) on E we obtain that f ∈A λ f = -j≥0 ju j . In each homogeneous component S j with j = 0 we have ju j = 0 and therefore u ∈ S 0 = k and, when j = 0, f λ f = 0. Evaluating the left hand side of (13) on C gives f λ f ∂ f (C). Now, if ∂ f (C) = C(f )/f = ∂ 3 (f )C(x 3 )/f is nonzero then ∂ 3 (f ) = 0 and thus f is a factor of C(x 3 ): let us, then, factor

Proposition 3. 2 .

 2 If p > 0 and u ∈ F p U are such that d 0 (u) ≡ 0 modulo F p-2 X 1 then u ∈ F p-1 U . Proof. Let {f I : I ∈ N n , |I| = p} ⊂ S be such that u ≡ |I|=p f I α I modulo F p-1 U : thanks to Proposition 2.4 we have that d 0 (u) ≡ |J=(jn,...,j1)|=p-1

Corollary 3. 5 .

 5 Let r, n ≥ 1. The centers of Diff (A(C r S n )) and of Diff B n are k. Proof. The algebras considered have been shown to satisfy the hypotheses of Theorem 3.4 in Examples 1.13 and 1.8.

1 x 2 η p 2 -rx r-1 1 x 3 η p 3

 13 has component zero in C p x1 , C p x2 and C p x3 , and then Proposition 4.2 tells us that D 1

Lie-Rinehart algebras. Definition 1.10. Let

	S and (L, [-, -]) be, respectively, a commutative and a Lie algebra endowed
	with a morphism of Lie algebras L → Der(S) that we write α → α S and a left S-module structure
	on L which we simply denote by juxtaposition. We say that the pair (S, L) is a Lie-Rinehart
	algebra, or that L is a Lie-Rinehart algebra over S, if the equalities
	(sα) S (t) = sα S (t),	[α, sβ] = s[α, β] + α S (s)β
	hold whenever s, t ∈ S and α, β ∈ L.

  Example 1.13. For any n ≥ 2 the Lie-Rinehart algebra Der B n is triangular and satisfies the Bézout condition with the basis {θ 1 , . . . , θ n } given in Example 1.5. The same goes to Der Bn with the basis given in Example 1.6.

	Example 1.14. Let r, n ≥ 1. The Lie-Rinehart algebra associated to A(C r S n ) is triangularizable,
	as follows immediately from Example 1.8.
	Example 1.15. Let r ≥ 1. The arrangement A r = A(C r S 3 ) from Example 1.9 is triangularizable
	thanks to Example 1.8. Moreover, it satisfies the Bézout condition: indeed, α 2

Differential operators associated to an arrangement. Remember

  from J. C. McConnell and J. C. Robson's [MR01, §15] that the algebra Diff S of differential operators on S = k[x 1 , . . . , x n ] is the subalgebra of End S generated by Der S and the set of maps given by left multiplication by elements of S. Recall as well from [MR01, §5] that if R is an algebra and I ⊂ R is a right ideal, the largest subalgebra I R (I) of R that contains I as an ideal -the idealizer of I in Ris {r ∈ R : rI ⊂ I}.

	1.3. Definition 1.16. Let A be a central arrangement of hyperplanes with defining polynomial Q.
	The algebra of differential operators tangent to the arrangement A is

  Proof. This is a consequence of applying the Künneth's formula for Hochschild cohomology as in H. Cartan and S. Eilenberg's [CE56, XI.3.I] to the isomorphism Diff B n+1 ∼ = A 1 ⊗ Diff Bn in Proposition 1.21 and the observation that HH 0 (A 1 ) ∼ = k and HH

	E p,q 2 = H p S (L, H

• S L, N ) with Chevalley-Eilenberg differentials computes H • S (L, N ).

Theorem 1.27

[START_REF] Kordon | Lie-Rinehart and Hochschild cohomology for algebras of differential operators[END_REF]

). Let (S, L) be a Lie-Rinehart algebra with enveloping algebra U , and suppose that L is an S-projective module. There exist a U -module structure on H • (S, U ) and a first-quadrant spectral sequence E • converging to HH • (U ) with second page q (S, U )).

Proposition 1.28. There are isomorphisms HH • (Diff B n+1 ) ∼ = HH • (Diff Bn ) for any n ≥ 1.

With this information at hand we are able to see that

Let us now consider the action of D on η p 1 . To begin with, we have

and we observe that

From these computations we see that the cocycle

C(x 3 ) by x 3 and f l,j = x 3 -e 2jπi/r x l for l = 1, 2 and j ∈ 0, r -1 , and in this way reformulate the evaluation of (13) at C as the nullity of

Fix now l ∈ 2 and j ∈ 0, r -1 and apply the morphism of algebras ε l,j : S → k[x 1 , x 2 ] that sends x 3 to e 2kπi/r x l : since ε l,j ((x r 3 -x r l )/f j ,l ) = 0 whenever l = l and j = j we obtain that ε l,j :

As the expression at which we evaluated ε l,j was zero, it follows that λ f l,j = 0 and, immediately, that also λ x3 = 0. We observe that the indexes that survive in the sum λ f ∂ f are x 1 , x 2 and f j = x 2 -e 2jπi/r x 1 with j ∈ 0, r -1 ; evaluating at D we obtain

Reasoning as above we get that λ x2 = λ fj = 0 for every j. Recalling now that f ∈A λ f = 0 we see that λ x1 = 0 as well.

The classes of ∂ f with f a linear factor in (12) span OutDer U because the dimension of OutDer U ∼ = HH 1 (U ) is, thanks to Corollary 5.4, precisely |A|. The composition ∂ f • ∂ g : U → U is evidently equal to zero for any f, g ∈ A, as a straightforward calculation shows, and therefore the Lie algebra structure in OutDer U vanishes.