We compute Hochschild homology and cohomology of a class of generalized Weyl algebras, introduced by V. V. Bavula in St. Petersbourg Math. Journal, 4 (1) (1999), 71-90. Examples of such algebras are the n-th Weyl algebras, , primitive quotients of , and subalgebras of invariants of these algebras under finite cyclic groups of automorphisms. We answer a question of Bavula–Jordan (Trans. A.M.S., 353 (2) (2001), 769-794) concerning the generators of the group of automorphisms of a generalized Weyl algebra. We also explain previous results on the invariants of Weyl algebras and of primitive quotients
Nous calculons l'homologie et la cohomologie de Hochschild des algèbres de Weyl généraliseés introduites par V. Bavula. Nous répondons à une question de Bavula-Jordan concernant les générateurs du groupe d'automorphismes d'une telle algèbre. Ce calcul explique aussi des résultats connus sur les invariants des algèbres de Weyl et des quotients primitifs.
Keywords: Hochschild cohomology, generalized Weyl algebras, automorphism group
Mot clés : cohomologie de Hochschild, algèbres de Weyl généralisées, groupe d'automorphismes
Farinati, Marco A. 1; Solotar, Andrea L. 1; Suárez-Álvarez, Mariano 1
@article{AIF_2003__53_2_465_0, author = {Farinati, Marco A. and Solotar, Andrea L. and Su\'arez-\'Alvarez, Mariano}, title = {Hochschild homology and cohomology of generalized {Weyl} algebras}, journal = {Annales de l'Institut Fourier}, pages = {465--488}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {2}, year = {2003}, doi = {10.5802/aif.1950}, zbl = {1100.16008}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1950/} }
TY - JOUR AU - Farinati, Marco A. AU - Solotar, Andrea L. AU - Suárez-Álvarez, Mariano TI - Hochschild homology and cohomology of generalized Weyl algebras JO - Annales de l'Institut Fourier PY - 2003 SP - 465 EP - 488 VL - 53 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1950/ DO - 10.5802/aif.1950 LA - en ID - AIF_2003__53_2_465_0 ER -
%0 Journal Article %A Farinati, Marco A. %A Solotar, Andrea L. %A Suárez-Álvarez, Mariano %T Hochschild homology and cohomology of generalized Weyl algebras %J Annales de l'Institut Fourier %D 2003 %P 465-488 %V 53 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1950/ %R 10.5802/aif.1950 %G en %F AIF_2003__53_2_465_0
Farinati, Marco A.; Solotar, Andrea L.; Suárez-Álvarez, Mariano. Hochschild homology and cohomology of generalized Weyl algebras. Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 465-488. doi : 10.5802/aif.1950. https://aif.centre-mersenne.org/articles/10.5802/aif.1950/
[1] Homologie des invariants d'une algèbre de Weyl sous l'action d'un groupe fini, J. of Alg., Volume Vol. 232 (2000) no. 2, pp. 564-577 | DOI | MR | Zbl
[2] Comparaison de l'homologie de Hochschild et de l'homologie de Poisson pour une déformation des surfaces de Klein, Algebra and Operator Theory, Proceedings of the Colloquium, Tashkent (1998), pp. 25-38 | Zbl
[3] Homologie des invariants d'une algèbre de Weyl, K-Theory, Volume 18 (1999), pp. 401-411 | DOI | MR | Zbl
[4] Generalized Weyl algebras and their representations, St. Petersbourg Math. J, Volume 4 (1990) no. 1, pp. 71-90 | MR | Zbl
[5] Isomorphism problems and groups automorphisms for generalized Weyl algebras, Trans. Am. Math. Soc, Volume 353 (2001) no. 2, pp. 769-794 | DOI | MR | Zbl
[6] Krull dimension of generalized Weyl algebras with noncommutative coefficients, J. of Alg, Volume 235 (2001) no. 1, pp. 315-358 | DOI | MR | Zbl
[7] Cyclic homology of commutative algebras, I, Springer Lecture Notes in Math, Volume 1318 (1988), pp. 51-72 | DOI | MR | Zbl
[8] Homological algebra, Princeton Univ. Press, Princeton, 1956 | MR | Zbl
[9] Quotients simples de l'algèbre enveloppante de , J. of Alg., Volume 24 (1973), pp. 551-574 | MR | Zbl
[10] Symplectic reflection algebras, Calogero-Moser spaces, and deformed Harish-Chandra homomorphism (e-print, arXiv:math.AG/0011114 v5) | MR | Zbl
[11] Automorphismes d'algèbres enveloppantes classiques et quantifiées : sous-groupes finis et invariants (1997) (Thèse Université de Reims, Champagne-Ardenne)
[12] Sur les invariants de sous l'action de sous-groupes finis d'automorphismes: conjecture de Gelfand - Kirillov et homologie de Hochschild, Comm. in Alg, Volume 29 (2001) no. 8, pp. 3535-3557 | DOI | MR | Zbl
[13] Noncommutative deformations of type-A Kleinian singularities, J. Algebra, Volume 161 (1993) no. 2, pp. 271-290 | DOI | MR | Zbl
[14] L'homologie cyclique des algèbres enveloppantes, Invent. Math, Volume 91 (1988) no. 2, pp. 221-251 | DOI | EuDML | MR | Zbl
[15] Homologie des quotients primitifs de l'algèbre enveloppante de , Math. Ann, Volume 294 (1992) no. 3, pp. 483-502 | EuDML | MR | Zbl
[16] A class of algebras similar to the enveloping algebra of , Trans. Am. Math. Soc, Volume 322 (1990) no. 1, pp. 285-314 | MR | Zbl
[17] Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, Berlin-Heidelberg-New York, 1977 | MR | Zbl
[18] Filtered algebras and representations of Lie algebras, Trans. Am. Math. Soc, Volume 100 (1961), pp. 530-550 | DOI | MR | Zbl
[19] Hochschild cohomology on Hopf-Galois extensions, J. Pure Appl. Alg, Volume 103 (1995) no. 2, pp. 221-233 | DOI | MR | Zbl
[20] Multiplicative structure of Hochschild cohomology of the ring of invariants of a Weyl algebra under finite groups (To appear in J. of Alg) | MR | Zbl
[21] Hochschild cohomology of primitive quotients of and their rings of invariants (Preprint, http://www.math.jussieu.fr/\mariano)
[22] A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Am. Math. Soc, Volume 126 (1998) no. 5, pp. 1345-1348 | DOI | MR | Zbl
[23] A relation between Hochschild homology and cohomology for Gorenstein rings (Erratum) (to appear) | MR | Zbl
Cited by Sources: