Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Poincaré-Birkhoff-Witt theorem for quadratic algebras with group actions

Authors: Anne V. Shepler and Sarah Witherspoon
Journal: Trans. Amer. Math. Soc. 366 (2014), 6483-6506
MSC (2010): Primary 16S37, 16E40, 16S80, 16S35
Published electronically: July 21, 2014
MathSciNet review: 3267016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Braverman, Gaitsgory, Polishchuk, and Positselski gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincaré-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient of an algebra that is free over some (not necessarily semisimple) subalgebra. We show that their theorem holds under a weaker hypothesis: We require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra (semidirect product algebra) of a finite group acting on a Koszul algebra, obtaining conditions for the Poincaré-Birkhoff-Witt property over (nonsemisimple) group algebras. We prove our main results by exploiting a double complex adapted from Guccione, Guccione, and Valqui (formed from a Koszul complex and a resolution of the group), giving a practical way to analyze Hochschild cohomology and deformations of skew group algebras in positive characteristic. We apply these conditions to graded Hecke algebras and Drinfeld orbifold algebras (including rational Cherednik algebras and symplectic reflection algebras) in arbitrary characteristic, with special interest in the case when the characteristic of the underlying field divides the order of the acting group.

References [Enhancements On Off] (What's this?)

  • [1] Martina Balagović and Harrison Chen, Representations of rational Cherednik algebras in positive characteristic, J. Pure Appl. Algebra 217 (2013), no. 4, 716-740. MR 2983846,
  • [2] Yuri Bazlov and Arkady Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), no. 5, 1466-1530. MR 2493618 (2010e:16046),
  • [3] George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178-218. MR 506890 (81b:16001),
  • [4] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010),
  • [5] Alexander Braverman and Dennis Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996), no. 2, 315-328. MR 1383469 (96m:16012),
  • [6] José Bueso, José Gómez-Torrecillas, and Alain Verschoren, Algorithmic methods in non-commutative algebra, Mathematical Modelling: Theory and Applications, vol. 17, Kluwer Academic Publishers, Dordrecht, 2003. Applications to quantum groups. MR 2006329 (2005c:16069)
  • [7] V. G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69-70 (Russian). MR 831053 (87m:22044)
  • [8] Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243-348. MR 1881922 (2003b:16021),
  • [9] Marco Farinati, Hochschild duality, localization, and smash products, J. Algebra 284 (2005), no. 1, 415-434. MR 2115022 (2005j:16009),
  • [10] Murray Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103. MR 0171807 (30 #2034)
  • [11] M. Gerstenhaber and S. D. Schack, On the deformation of algebra morphisms and diagrams, Trans. Amer. Math. Soc. 279 (1983), no. 1, 1-50. MR 704600 (85d:16021),
  • [12] Murray Gerstenhaber and Samuel D. Schack, Algebraic cohomology and deformation theory, Deformation theory of algebras and structures and applications (Il Ciocco, 1986), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, pp. 11-264. MR 981619 (90c:16016)
  • [13] Edward L. Green, Idun Reiten, and Øyvind Solberg, Dualities on generalized Koszul algebras, Mem. Amer. Math. Soc. 159 (2002), no. 754, xvi+67. MR 1921583 (2004b:16042),
  • [14] Stephen Griffeth, Towards a combinatorial representation theory for the rational Cherednik algebra of type $ G(r,p,n)$, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 2, 419-445. MR 2653242 (2011f:16038),
  • [15] Jorge A. Guccione, Juan J. Guccione, and Christian Valqui, Universal deformation formulas and braided module algebras, J. Algebra 330 (2011), 263-297. MR 2774629 (2012c:16089),
  • [16] Gilles Halbout, Jean-Michel Oudom, and Xiang Tang, Deformations of orbifolds with noncommutative linear Poisson structures, Int. Math. Res. Not. IMRN 1 (2011), 1-39. MR 2755481 (2012d:53284),
  • [17] U. Krähmer, ``Notes on Koszul algebras,'' available at http: $ \slash \slash $
    $ \sim $ukraehmer/$ \char93 $research.
  • [18] Huishi Li, Gröbner bases in ring theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. MR 2894019 (2012m:16056)
  • [19] L. Li, ``A generalized Koszul theory and its application,'' Trans. Amer. Math. Soc. 366 (2014), no. 2, 931-977. MR 3130322
  • [20] George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145-202. MR 972345 (90e:22029)
  • [21] George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599-635. MR 991016 (90e:16049),
  • [22] L. E. Positselskiĭ, Nonhomogeneous quadratic duality and curvature, Funktsional. Anal. i Prilozhen. 27 (1993), no. 3, 57-66, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 27 (1993), no. 3, 197-204. MR 1250981 (95h:16041),
  • [23] Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. MR 2177131 (2006f:16043)
  • [24] Arun Ram and Anne V. Shepler, Classification of graded Hecke algebras for complex reflection groups, Comment. Math. Helv. 78 (2003), no. 2, 308-334. MR 1988199 (2004d:20007),
  • [25] Katsunori Sanada, On the Hochschild cohomology of crossed products, Comm. Algebra 21 (1993), no. 8, 2727-2748. MR 1222741 (94k:16016),
  • [26] Anne V. Shepler and Sarah Witherspoon, Quantum differentiation and chain maps of bimodule complexes, Algebra Number Theory 5 (2011), no. 3, 339-360. MR 2833794 (2012k:16026),
  • [27] Anne V. Shepler and Sarah Witherspoon, Drinfeld orbifold algebras, Pacific J. Math. 259 (2012), no. 1, 161-193. MR 2988488,
  • [28] Dragoş Ştefan, Hochschild cohomology on Hopf Galois extensions, J. Pure Appl. Algebra 103 (1995), no. 2, 221-233. MR 1358765 (96h:16013),
  • [29] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)
  • [30] D. Woodcock, Cohen-Macaulay complexes and Koszul rings, J. London Math. Soc. (2) 57 (1998), no. 2, 398-410. MR 1644229 (99g:13025),

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 16S37, 16E40, 16S80, 16S35

Retrieve articles in all journals with MSC (2010): 16S37, 16E40, 16S80, 16S35

Additional Information

Anne V. Shepler
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203

Sarah Witherspoon
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Koszul algebras, skew group algebras, Hochschild cohomology, Drinfeld orbifold algebras
Received by editor(s): September 25, 2012
Received by editor(s) in revised form: January 17, 2013
Published electronically: July 21, 2014
Additional Notes: The first author was partially supported by NSF grants #DMS-0800951 and #DMS-1101177
The second author was partially supported by NSF grants #DMS-0800832 and #DMS-1101399
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society