Seminar of Algebra

A Lie–Rinehart pair (S, L) 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 asin [1]. The universal enveloping algebra U of such a pair is an associative algebra that resembles the enveloping alegebra of L but takes into account is the action of S. We are interested in computing the Hochschild cohomology HH(U). Our main result, available at [4], is the construction of a spectral sequence converging to it that is based on ideas by Th. Lambre and P. Le Meur in [2]. In this talk, we will go over its construction, we will see some immediate consequences and we will illustrate the fact that the spectral sequence makes it concretely easier to compute the Hochschild cohomology of U in some examples. In particular, since for a free hyperplane arrangement A the enveloping algebra of the pair (S, Der A) is isomorphic to the algebra of differential operators tangent to A, this spectral sequence gives us an alternative way to re-obtain and extend our results in [3], where we computed the Hochschild cohomology of this algebra for a very special kind of arrangements.

[1] G. S. Rinehart, Differential Forms on General Commutative Algebras, Transactions of the American Mathematical Society 108 (1963), no. 2, 195-222. ↑35, 36, 37, 93, 94, 95

[2] Lambre, Thierry & Le Meur, Patrick. (2017). Duality for Differential Operators of Lie-Rinehart Algebras. Pacific Journal of Mathematics. 297. 10.2140/pjm.2018.297.405.

[3] F. Kordon and M. Suárez-Álvarez, Hochschild cohomology of algebras of differential operators tangent to a central arrangement of lines (2018). Accepted for publication by Documenta Mathematica, available at arXiv:1807.10372.

[4] F. Kordon, The Hochschild cohomology of the enveloping algebra of a Lie-Rinehart pair (2018). Available at arXiv:1810.02901.