Abstract
This paper improves several previously known results. First, the results describing the R-skewsymmetric algebra and the quadratic dual of the R-symmetric algebra as Frobenius algebras are shown to be true with any restriction on the parameter q of the Hecke relation being removed. An even Hecke symmetry gives rise to a pair of graded Frobenius algebras. We describe interrelation between the Nakayama automorphisms of the two algebras. As an illustration of general technique we give full details of the verification that Artin–Schelter regular algebras of global dimension 3 and elliptic type A are not associated with any quantum GL(3).
Similar content being viewed by others
References
Abella, A., Andruskiewitsch, N.: Compact quantum groups arising from the FRT-construction. Bol. Acad. Nac. Cienc. Córdoba 63, 15–44 (1999)
Andruskiewitsch, N.: An introduction to Nichols algebras, Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Springer, pp. 135–195 (2017)
Artin, M., Schelter, W.F.: Graded algebras of global dimension \(3\). Adv. Math. 66, 171–216 (1987)
Artin, M., Tate, J., Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Volume I. Birkhäuser, pp. 33–85 (1990)
Artin, M., Tate, J., Van den Bergh, M.: Modules over regular algebras of dimension \(3\). Invent. Math 106, 335–388 (1991)
Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Springer, Berlin (2005)
Bondal, A.I., Polishchuk, A.E.: Homological properties of associative algebras: The method of helices(in Russian) Izv. Ross. Akad. Nauk Ser. Mat. 57 (2) 3–50,: Russian Acad. Sci. Izv. Math. 42(1994), 219–260 (1993)
Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhäuser, Basel (1986)
Chirvasitu, A., Walton, C., Wang, X.: On quantum groups associated to a pair of preregular forms. J. Noncommut. Geom. 13, 115–159 (2019)
Dubois-Violette, M.: Poincaré duality for Koszul algebras. In: Algebra, Geometry and Mathematical Physics, pp. 3–26. Springer (2014)
Ewen, H., Ogievetsky, O.: Classification of the \(GL(3)\) quantum matrix groups. arxiv:9412009
Fischman, D., Montgomery, S., Schneider, H.-J.: Frobenius extensions of subalgebras of Hopf algebras. Trans. Am. Math. Soc. 349, 4857–4895 (1997)
Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras. Clarendon Press, Oxford (2000)
Gelfand, I.M., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Basel (1994)
Gurevich, D.I.: Algebraic aspects of the quantum Yang-Baxter equation(in Russian), Algebra i Analiz 2 4 119–148(1990). Leningrad Math. J. 2, 801–828 (1991)
Hai, P.H.: Poincaré series of quantum spaces associated to Hecke operators. Acta Math. Vietnam 24, 235–246 (1999)
Heckenberger, I., Schneider, H.-J.: Hopf algebras and root systems Amer. Math, Soc (2020)
Lyubashenko, V.V.: Vectorsymmetries, Reports Dept. Math. Univ. Stockholm , No. 19 (1987)
Manin, Yu.: Quantum Groups and Non-Commutative Geometry, 2nd edn. Springer, Berlin (2018)
Ohn, C.: Quantum \(SL(3,{\mathbb{C} })\)’s with classical representation theory. J. Algebra 213, 721–756 (1999)
Ohn, C.: Quantum \(SL(3,{C })\)’s: the missing case Hopf Algebras in Noncommutative Geometry and Physics Marcel Dekker, pp. 245–255 (2005)
Polishchuk, A., Positselski, L.: Quadratic Algebras. Am. Math, Soc (2005)
Rosso, M.: Quantum groups and quantum shuffles. Invent. Math. 133, 399–416 (1998)
Skryabin, S.: On the graded algebras associated with Hecke symmetries. J. Noncommut. Geom. 14, 937–986 (2020)
Smith, S.P.: Some finite dimensional algebras related to elliptic curves. In: Representation Theory of Algebras and Related Topics. Am. Math. Soc., pp. 315–348 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Skryabin, S. Hecke symmetries: an overview of Frobenius properties. Sel. Math. New Ser. 29, 35 (2023). https://doi.org/10.1007/s00029-023-00843-0
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-023-00843-0