×

zbMATH — the first resource for mathematics

PBW deformations of braided products. (English) Zbl 1420.16021
Summary: We present new examples of deformations of smash product algebras that arise from Hopf algebra actions on pairs of module algebras. These examples involve module algebras that are Koszul, in which case a PBW theorem we established previously applies. Our construction generalizes several ‘double’ constructions appearing in the literature, including Weyl algebras and some types of Cherednik algebras, and it complements the braided double construction of Y. Bazlov and A. Berenstein [Sel. Math., New Ser. 14, No. 3–4, 325–372 (2009; Zbl 1220.16027)]. Many suggestions of further directions are provided at the end of the work.

MSC:
16T05 Hopf algebras and their applications
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Baez, J. C., Hochschild homology in a braided tensor category, Trans. Amer. Math. Soc., 344, 2, 885-906, (1994) · Zbl 0814.18003
[2] Bakalov, B.; Kirillov, A., Lectures on tensor categories and modular functors, AMS University Lecture Series, vol. 21, (2001) · Zbl 0965.18002
[3] Bazlov, Y.; Berenstein, A., Braided doubles and rational Cherednik algebras, Adv. Math., 220, 5, 1466-1530, (2009) · Zbl 1167.16017
[4] Bazlov, Y.; Berenstein, A., Noncommutative Dunkl operators and braided Cherednik algebras, Selecta Math., 14, 3-4, 325-372, (2009) · Zbl 1220.16027
[5] Bueso, J. L.; Gómez-Torrecillas, J.; Verschoren, A., Algorithmic methods in non-commutative algebra: applications to quantum groups, vol. 17, (2003), Kluwer · Zbl 1063.16054
[6] Čap, A.; Schichl, H.; Vanžura, J., On twisted tensor products of algebras, Comm. Algebra, 23, 12, 4701-4735, (1995) · Zbl 0842.16005
[7] Cherednik, I., Double affine Hecke algebras, vol. 319, (2005), Cambridge University Press
[8] Drinfel’d, V., Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl., 20, 1, 58-60, (1986) · Zbl 0599.20049
[9] Etingof, P., Cherednik and Hecke algebras of varieties with a finite group action, preprint available at · Zbl 07069275
[10] Etingof, P.; Gan, W. L.; Ginzburg, V., Continuous Hecke algebras, Transform. Groups, 10, 3-4, 423-447, (2005) · Zbl 1115.20005
[11] Etingof, P.; Ginzburg, V., Symplectic reflection algebras, Calogero-Moser space, and deformed harish-chandra homomorphism, Invent. Math., 147, 2, 243-348, (2002) · Zbl 1061.16032
[12] Jantzen, J. C., Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, (1996), American Mathematical Society · Zbl 0842.17012
[13] Jara Martinez, P.; López Peña, J.; Ştefan, D., Koszul pairs: applications, preprint available at · Zbl 1381.16028
[14] Kassel, C., Quantum groups, Graduate Texts in Mathematics, vol. 155, (1995), Springer-Verlag New York · Zbl 0808.17003
[15] Krähmer, U., Notes on Koszul algebras, available at
[16] Losev, I.; Tsymbaliuk, A., Infinitesimal Cherednik algebras as W-algebras, Transform. Groups, 19, 2, 495-526, (2014) · Zbl 1336.17008
[17] Majid, S., Algebras and Hopf algebras in braided categories, (Advances in Hopf Algebras, Chicago, IL, 1992, Lecture Notes in Pure and Appl. Math., vol. 158, (1994), Dekker New York), 55-105 · Zbl 0812.18004
[18] Majid, S., Foundations of quantum group theory, (1995), Cambridge University Press · Zbl 0857.17009
[19] Manin, Yu. I., Quantum groups and noncommutative geometry, (1988), Université de Montréal, Centre de Recherches Mathématiques · Zbl 0724.17006
[20] Polishchuk, A.; Positselski, L., Quadratic algebras, University Lecture Series, vol. 37, (2005), American Mathematical Society Providence, RI · Zbl 1145.16009
[21] Radford, D. E., Minimal quasitriangular Hopf algebras, J. Algebra, 157, 2, 285-315, (1993) · Zbl 0787.16028
[22] Rosso, M., Quantum groups at a root of 1 and tangle invariants, (Topological and Geometrical Methods in Field Theory, Turku, 1991, (1992), World Sci. Publ. River Edge, NJ), 347-358 · Zbl 1103.17303
[23] Rouquier, R., Representations of rational Cherednik algebras, Contemp. Math., 392, 103-131, (2005) · Zbl 1171.20303
[24] Sevostyanov, A., Conjugacy classes in Weyl groups and q-W algebras, Adv. Math., 228, 3, 1315-1376, (2011) · Zbl 1229.17017
[25] Smith, S. P., Some finite dimensional algebras related to elliptic curves, (Representation Theory of Algebras and Related Topics, Mexico City, 1994, CMS Conf. Proc., vol. 19, (1996)), 315-348 · Zbl 0856.16009
[26] Suárez-Alvarez, M.; Vivas, Q., Automorphisms and isomorphisms of quantum generalized Weyl algebras, J. Algebra, 424, 540-552, (2015) · Zbl 1312.16021
[27] Walton, C., Representation theory of three-dimensional Sklyanin algebras, Nuclear Phys. B, 860, 1, 167-185, (2012) · Zbl 1246.81156
[28] Walton, C.; Witherspoon, S., Poincaré-Birkhoff-Witt deformations of smash product algebras from Hopf actions on Koszul algebras, Algebra Number Theory, 8, 7, 1701-1731, (2014) · Zbl 1334.16032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.