Lie–Rinehart and Hochschild cohomology for algebras of differential operators
Introduction
The goal of this article is to apply homological algebra techniques for Lie–Rinehart algebras to a problem of algebras of differential operators. We begin by describing a spectral sequence that converges to the Hochschild cohomology of the enveloping algebra of a Lie–Rinehart algebra. After that, we focus on the algebra of differential operators associated to a central arrangement of three lines. This is a graded associative algebra that is at the same time the enveloping algebra of a Lie–Rinehart algebra: an explicit calculation with the spectral sequence allows us to compute the Hilbert series of its Hochschild cohomology. We conclude by giving two other examples of algebras in which the spectral sequence proves useful.
Let be a field of characteristic zero and let be a central hyperplane arrangement in a finite dimensional -vector space V. Let S be the algebra of coordinates on V and let be a defining polynomial for . The arrangement is free if the Lie algebra of derivations of S tangent to is a free S-module. It is not known what makes an arrangement free, but this condition is nevertheless satisfied in many important examples; for instance, it is a theorem by H. Terao in [18] that reflection arrangements over are free. We refer to P. Orlik and H. Terao's book [15] for a general reference of hyperplane arrangements.
The algebra of differential operators tangent to an arrangement , first considered by F. J. Calderón-Moreno in [5], is the algebra of differential operators on S which preserve the ideal QS of S and all its powers. We are interested in the Hochschild cohomology of when is free.
The first and simplest example of a free arrangement is that of a central line arrangement, that is, when . Let l be the number of lines of such an arrangement: for , the Hochschild cohomology of has been obtained as a Gerstenhaber algebra by the first author and M. Suárez-Álvarez in [9] starting from a projective resolution of as a bimodule over itself by means of explicit calculations that exploit a graded algebra structure on , but the calculations performed in this situation seem impossible to emulate when or . In this paper we are able to extend, in Corollary 5.9, Corollary 5.10, some of these results to the most complicated case, which is when :
Theorem A Let be a central arrangement of three lines. The Hilbert series of is . The first cohomology space is an abelian Lie algebra of dimension three.
It is to prove Theorem A that Lie–Rinehart algebras come to into play: the pair is a Lie-Rinehart algebra. Recall that a Lie–Rinehart algebra consists of a commutative algebra S and a Lie algebra L with an S-module structure that acts on S by derivations and which satisfies certain compatibility conditions analogous to those satisfied by the pair . The universal enveloping algebra U of a Lie–Rinehart algebra and the Lie–Rinehart cohomology are an associative algebra and a cohomology theory that generalize the usual enveloping algebra and the Lie algebra cohomology of the Lie algebra L by taking into account its interaction with S — see the original paper [16] by G. Rinehart or the more modern exposition [8] by J. Huebschmann.
If is free, as remarked by L. Narváez Macarro in [12, Theorem 1.3.1], the enveloping algebra of is isomorphic to . To compute the Hochschild cohomology in Theorem A above we employ a strategy that gives rise to a general method to approach this kind of computations: we construct, in Corollary 3.3, a spectral sequence converging to the Hochschild cohomology of the enveloping algebra U with values on an U-bimodule M. For this sequence we need an U-module structure on , the Hochschild cohomology of S with values on M. This U-module structure is constructed using an injective resolution of M by U-bimodules and we see in Theorem 2.8 that it can be computed explicitly from a projective resolution of S by S-bimodules. Moreover, the action of each on , computed using projectives, by the endomorphism given in Remark 2.5 turns out suitable for computations.
Theorem B Let be a Lie–Rinehart pair such that L is an S-projective module and let M be an U-bimodule. There exist a U-module structure on and a first-quadrant spectral sequence converging to with second page
We give two other applications of Theorem B. First, in Subsection 6.1 we compute the Hochschild cohomology of a family of subalgebras of the Weyl algebra over a field of characteristic zero, that is, the algebras generated by elements x and y satisfying the relation for a given . These algebras have been studied by G. Benkart, S. Lopes and M. Ondrus in the series of articles that start with [2] for a field of arbitrary characteristic and, more recently, S. Lopes and A. Solotar in [11] have described their Hochschild cohomology, with special emphasis on the Lie module structure of the second cohomology space over the first one, also in arbitrary characteristic. Some of the expressions we provide were nevertheless not found before and might be of interest. Second, in Subsection 6.2 we recover in a more direct and clear way a result by the second author and P. Le Meur in [10] that states that the enveloping algebra U of a Lie–Rinehart algebra has Van den Bergh duality in dimension if S has Van den Bergh duality in dimension n and L is finitely generated and projective with constant rank d.
Let us outline the organization of this article. In Section 1 we recall the definition of Lie–Rinehart pairs, their universal enveloping algebras and their cohomology theory. In Sections 2 and 3 we describe the module structure on and present the spectral sequence. After proving some useful lemmas regarding eulerian modules in Section 4 we devote Section 5 to the computation of the Hochschild cohomology of the algebra of differential operators of a central arrangement of three lines. Finally, in Section 6 we provide the two other applications described above.
We will denote the tensor product over the base field simply by ⊗ or, sometimes, by |. Unless it is otherwise specified, all vector spaces and algebras will be over . Given an associative algebra A, the enveloping algebra is the vector space endowed with the product ⋅ defined by , so that the category of -modules is equivalent to that of A-bimodules. The Hochschild cohomology of A with values on an -module M is defined as and will be denoted by or, if , by . The book [22] by C. Weibel may serve as general reference on this subject.
The first author heartfully thanks his PhD advisor M. Suárez-Álvarez for his collaboration, fruitful suggestions and overall help. 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 as well as for the support with the projet émergence “homologies et symétrie quantiques” (2019). Part of this work was done during the time the first author was supported by a full doctoral grant by CONICET and by the projects PIP-CONICET 12-20150100483 (CONICET), PICT 2015-0366 (AGENCIA) and UBACyT 20020170100613BA.
Section snippets
Lie–Rinehart algebras
We begin by recalling some basic facts about Lie-Rinehart algebras available in [16] and in [8]. Until Section 3 we assume to be a field of arbitrary characteristic.
Definition 1.1 Let S and be a commutative and a Lie algebra endowed with a morphism of Lie algebras that we write and a left S-module structure on L which we simply denote by juxtaposition. The pair is a Lie–Rinehart algebra if the equalities hold whenever and .
Definition 1.2 Let
The U-module structure on
Let be a Lie–Rinehart algebra such that L is a projective S-module. Let U be its enveloping algebra and M be an -module. Since the inclusion of S in U is a morphism of algebras we can regard M as an -module and consider the Hochschild cohomology of S with values on M, denoted as before by . In this section we first construct an U-module structure on from an -injective resolution of M; afterwards, we construct S- and L-module structures on from an -projective
The spectral sequence
Let be a Lie–Rinehart algebra, let U be its enveloping algebra and let M be an -module. In this section we construct a spectral sequence which converges to the Hochschild cohomology of U with values on M and whose second page involves the Lie–Rinehart cohomology of and the Hochschild cohomology of S with values on M.
Recall that in (3) we considered a functor defined on objects as . We now consider the functor where we give to
Eulerian modules
We assume from now on that is a field of characteristic zero. In this section we pay attention to a particular but rather frequent situation in which some calculations to attain the second page of the spectral sequence in Corollary 3.3 can be significantly shortened. Let . The usual graded algebra structure on S, such that if , induces a grading on the Lie algebra Der S that makes each partial derivative have degree −1. Let L be a Lie subalgebra of Der S that is
The algebra of differential operators tangent to a central arrangement of three lines
In this section we describe the example that motivated us to construct the spectral sequence of Corollary 3.3: it is the algebra of differential operators tangent to a central arrangement of lines , whose Hochschild cohomology was studied by the first author and M. Suárez-Álvarez in [9]. We will regard as the enveloping algebra of a Lie–Rinehart algebra and compute the second page of the spectral sequence of Corollary 3.3 for a central line arrangement of three
The Hochschild cohomology of a family of subalgebras of the Weyl algebra
Let be a field of characteristic zero, fix a nonzero and consider the algebra with presentation Setting the algebra is the Weyl algebra that already appeared in Example 1.4, when it is the universal enveloping algebra of the two-dimensional non-abelian Lie algebra and if , it is the Jordan plane studied in [1].
We let and consider the Lie algebra L freely generated by as an S-submodule of Der S. It is straightforward to see that
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