Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-01T20:18:38.264Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum Drinfeld Hecke Algebras

Published online by Cambridge University Press:  20 November 2018

Viktor Levandovskyy  and
Anne V. Shepler
Viktor Levandovskyy
Affiliation:
Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany. e-mail: levandov@math.rwth-aachen.de
Anne V. Shepler
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203, USA. e-mail: ashepler@unt.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

Keywords

16S3616S3516S8016W2016Z0516E40skew polynomial ringsnoncommutative Gröbner basesgraded Hecke algebrassymplectic reflection algebrasHochschild cohomology
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 4 , 01 August 2014 , pp. 874 - 901
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

*

Work of the second author was partially supported by National Science Foundation research grants #DMS-0800951 and #DMS-1101177 and a research fellowship from the Alexander von Humboldt Foundation.

References

[1] Alev, J. and Chamarie, M., Dérivations et automorphismes de quelques algèbres quantiques. Comm. Algebra 20(1992), no. 6, 1787–1802. http://dx.doi.org/10.1080/00927879208824431 CrossRefGoogle Scholar
[2] Artamonov, V. A. and Wisbauer, R., Homological properties of quantum polynomials. Algebr. Represent. Theory 4(2001), no. 3, 219–247. http://dx.doi.org/10.1023/A:1011458821831 CrossRefGoogle Scholar
[3] Backelin, J.et al., The Gröbner basis calculator BERGMAN. 2006. http://servus.math.su.se/bergman Google Scholar
[4] Bazlov, Y. and Berenstein, A., Noncommutative Dunkl operators and braided Cherednik algebras. Selecta Math. 14(2009), no. 3–4, 325–372. http://dx.doi.org/10.1007/s00029-009-0525-x CrossRefGoogle Scholar
[5] Bergman, G., The diamond lemma for ring theory. Adv. in Math. 29(1978), no. 2, 178–218. http://dx.doi.org/10.1016/0001-8708(78)90010-5 CrossRefGoogle Scholar
[6] Bosma, W., Cannon, J., and Playoust, C., The Magma algebra system. I: The user language. Computational algebra and number theory (London, 1993). J. Symbolic Comput. 24(1997), no. 3–4, 235–265. http://dx.doi.org/10.1006/jsco.1996.0125 CrossRefGoogle Scholar
[7] Braverman, A. and Gaitsgory, D., Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type. J. Algebra 181(1996), no. 2, 315–328. http://dx.doi.org/10.1006/jabr.1996.0122CrossRefGoogle Scholar
[8] Buchberger, B., Basic features and development of the critical-pair/completion procedure. In: Rewriting Techniques and Applications, Lecture Notes in Computer Science, 202, Springer, Berlin, 1985, pp. 1–45.CrossRefGoogle Scholar
[9] Bueso, J., Gómez–Torrecillas, J., and Verschoren, A., Algorithmic methods in non-commutative algebra. Applications to quantum groups. Mathematical Modelling: Theory and Applications, 17. Kluwer Academic Publishers, Dordrecht, 2003.Google Scholar
[10] Cohen, A. M. and Gijsbers, D. A. H., GBNP, a Non–commutative Gröbner Bases Package for GAP 4. 2007. http://www.win.tue.nl/_amc/pub/grobner Google Scholar
[11] Decker, W., Gruel, G.-M., Pfister, G., and Schönemann, H., SINGULAR 3-1-6—A computer algebra system for polynomial computations. 2012, http://www.singular.uni-kl.de. Google Scholar
[12] Drinfeld, V. G., Degenerate affine Hecke algebras and Yangians. Funct. Anal. Appl. Funktsional. Anal. i Prilozhen. 20(1986), no. 1, 69–70.Google Scholar
[13] Etingof, P.and Ginzburg, V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism Invent. Math. 147(2002), no. 2, 243–348. http://dx.doi.org/10.1007/s002220100171 CrossRefGoogle Scholar
[14] Gordon, I., On the quotient ring by diagonal invariants. Invent. Math. 153(2003), no. 3, 503–518. http://dx.doi.org/10.1007/s00222-003-0296-5 CrossRefGoogle Scholar
[15] Green, E. L., Noncommutative Groebner bases, and projective resolutions. In: Computational methods for representations of groups and algebras (Essen, 1997), Prog. Math., 173, Birkhäuser, Basel, 1999, pp. 29–60.CrossRefGoogle Scholar
[16] Green, E. L., An introduction to noncommutative Gröbner bases. In: Computational algebra, Lecture Notes in Pure and Appl. Math., 151, Dekker, New York, 1994, pp. 167–190.Google Scholar
[17] Griffeth, S., 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. http://dx.doi.org/10.1017/S0013091508000904 CrossRefGoogle Scholar
[18] Greuel, G.-M. and Pfister, G., A SINGULAR introduction to commutative algebra. Second ed., Springer, 2008.Google Scholar
[19] J.W. Helton and M. Stankus, NCGB 4.0, a Noncommutative Gröbner Basis Package for Mathematica. 2012. http://www.math.ucsd.edu/_ncalg/ Google Scholar
[20] Kirkman, E., Kuzmanovich, J., and Zhang, J. J., Shephard-Todd-Chevalley theorem for skew polynomial rings. Algebr. Represent. Theory 13(2010), no. 2, 127–158. http://dx.doi.org/10.1007/s10468-008-9109-2 CrossRefGoogle Scholar
[21] La Scala, R. and Levandovskyy, V., Letterplace ideals and non-commutative Gröbner bases. J. Symbollic Comput. 44(2009), no. 10, 1374–1393. http://dx.doi.org/10.1016/j.jsc.2009.03.002 CrossRefGoogle Scholar
[22] La Scala, R., Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra. J. Symbolic Comput. 48(2013), no. 1, 110–131. http://dx.doi.org/10.1016/j.jsc.2012.05.003 CrossRefGoogle Scholar
[23] Levandovskyy, V., PBW bases, non-degeneracy conditions and applications. In: Representation of algebras and related topics. Fields Inst. Commun., 45, American Mathematical Society, Providence, RI, 2005, pp. 229–246.Google Scholar
[24] Li, H., Gröbner bases in ring theory. World Scientific Publishing, 2012.Google Scholar
[25] Lusztig, G., Cuspidal local systems and graded Hecke algebras. I. Inst. Haute Études Sci. Publ. Math. 67(1988), 145–202.CrossRefGoogle Scholar
[26] Lusztig, G., Affine Hecke algebras and their graded version. J. Amer. Math. Soc. 2(1989), no. 3, 599–635. http://dx.doi.org/10.1090/S0894-0347-1989-0991016-9 CrossRefGoogle Scholar
[27] McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings. Graduate Studies in Mathematics, 30, American Mathematical Society, Providence, RI. 2001.Google Scholar
[28] Mora, T., An introduction to commutative and non-commutative Gröbner bases. Second International Colloquium on Words, Languages and Combinatorics (Kyoto, 1992). Theor. Comput. Sci. 134(1994), no. 1, 131–173. http://dx.doi.org/10.1016/0304-3975(94)90283-6 CrossRefGoogle Scholar
[29] Naidu, D. and Witherspoon, S., Hochschild cohohomology and quantum Drinfeld Hecke algebras. 2011, arxiv:1111.5243v1.Google Scholar
[30] Passman, D. S., Infinite crossed products. Pure and Applied Mathematics, 135, Academic Press, Boston, MA, 1989.Google Scholar
[31] Ram, A. and Shepler, A. V., Classification of graded Hecke algebras for complex reflection groups. Comment. Math. Helv. 78(2003), no. 2, 308–334. http://dx.doi.org/10.1007/s000140300013 CrossRefGoogle Scholar
[32] Shepler, A. V. and Witherspoon, S., Group actions on algebras and the graded Lie structure of Hochschild cohomology. J. Algebra 351(2012), 350–381. http://dx.doi.org/10.1016/j.jalgebra.2011.10.038 CrossRefGoogle Scholar
[33] Shepler, A. V., Hochschild cohomology and graded Hecke algebras. Trans. Amer. Math. Soc. 360(2008), no. 8, 3975–4005. http://dx.doi.org/10.1090/S0002-9947-08-04396-1 CrossRefGoogle Scholar
[34] Shepler, A. V., A Poincaré-Birkhoff-Witt Theorem for quadratic algebras with group actions. Trans. Amer. Math. Soc., to appear.Google Scholar
[35] Ufnarovski, V., Introduction to noncommutative Gröbner bases theory. In: Gröbner bases and applications, Cambridge University Press, 1998, pp. 259–280.CrossRefGoogle Scholar