Abstract
We give an interpretation of the Cremmer–Gervais r-matrices for \({\mathfrak{sl}_n}\) in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL 2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer–Gervais quantization in terms of the corresponding double affine Hecke algebra.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Belavin A.A., Drinfel’d V.G.: Solutions of the classical Yang–Baxter equations for simple Lie algebras. Funct. Anal. Appl. 16, 159–180 (1982)
Berest Y., Etingof P., Ginzburg V.: Finite dimensional representations of rational Cherednik algebras. In. Math. Res. Notices 19, 1053–1090 (2003)
Chari V., Pressley A.: A Guide to Quantum Groups. Cambridge University Press, New York (1994)
Cherednik I.: Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald operators. Int. Math. Res. Notices 9, 171–180 (1992)
Cherednik, I.: Double Affine Hecke Algebras. London Mathematical Society Lecture Note Series (2005)
Cremmer E., Gervais J.L.: The quantum group structure associated with non-linearly extended Virasoro algebras. Comm. Math. Phys. 134, 619–632 (1990)
Endelman R., Hodges T.J.: Generalized Jordanian R-matrices of Cremmer–Gervais Type. Lett. Math. Phys. 32, 225–237 (2000)
Etingof P., Schiffman O.: Lectures on Quantum Groups. International Press, Inc., Boston (1998)
Gerstenhaber M., Giaquinto A.: Boundary Solutions of the classical Yang–Baxter equation. Lett. Math. Phys. 40, 337–353 (1997)
Gerstenhaber M., Giaquinto A.: Boundary Solutions of the quantum Yang–Baxter equation and solutions in three dimensions. Lett. Math. Phys. 44, 131–141 (1998)
Hodges T.J.: On the Cremmer–Gervais quantizations of SL(n). Int. Math. Res. Notices 10, 465–481 (1995)
Hodges T.J.: The Cremmer–Gervais solution of the Yang–Baxter equation. Proc. Am. Math. Soc. 127(6), 1819–1826 (1999)
Khoroshkin S.M., Pop I.I., Samsonov M.E., Stolin A.A., Tolstoy V.N.: On some Lie bialgebra structures on polynomial algebras and their quantization. Comm. Math. Phys. 282(3), 625–662 (2008)
Kulish P.P., Mudrov A.I.: On twisting solutions to the Yang–Baxter equation. Czechoslov. J. Phys. 50(1), 115–122 (2000)
Stolin A.: On rational solutions of Yang–Baxter equation for \({\mathfrak{sl}_n}\) . Math. Scand. 69, 57–80 (1991)
Suzuki T.: Rational and trigonometric degeneration of the double affine Hecke algebra of type A. Int. Math. Res. Not. 37, 2249–2262 (2005)
Acknowledgements
I am grateful to Milen Yakimov for many helpful discussions. The author was partially supported by NSF grant DMS-0701107.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Johnson, G. Cremmer–Gervais r-Matrices and the Cherednik Algebras of Type GL 2 . Lett Math Phys 94, 115–122 (2010). https://doi.org/10.1007/s11005-010-0421-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-010-0421-5