Representation theory of (modified) reflection equation algebra of $GL(m|n)$ type
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D. Gurevich, P. Pyatov and P. Saponov
Translated by: the authors - St. Petersburg Math. J. 20 (2009), 213-253
- DOI: https://doi.org/10.1090/S1061-0022-09-01045-0
- Published electronically: January 30, 2009
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Abstract:
Let $R:V^{\otimes 2}\to V^{\otimes 2}$ be a Hecke type solution of the quantum Yang–Baxter equation (a Hecke symmetry). Then, the Hilbert–Poincaré series of the associated $R$-exterior algebra of the space $V$ is the ratio of two polynomials of degrees $m$ (numerator) and $n$ (denominator).
Under the assumption that $R$ is skew-invertible, a rigid quasitensor category $\textrm {SW}(V_{(m|n)})$ of vector spaces is defined, generated by the space $V$ and its dual $V^*$, and certain numerical characteristics of its objects are computed. Moreover, a braided bialgebra structure is introduced in the modified reflection equation algebra associated with $R$, and the objects of the category $\textrm {SW}(V_{(m|n)})$ are equipped with an action of this algebra. In the case related to the quantum group $U_q(sl(m))$, the Poisson counterpart of the modified reflection equation algebra is considered and the semiclassical term of the pairing defined via the categorical (or quantum) trace is computed.
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Bibliographic Information
- D. Gurevich
- Affiliation: ISTV, Université de Valenciennes, Valenciennes 59304, France
- Email: gurevich@univ-valenciennes.fr
- P. Pyatov
- Affiliation: Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia
- Email: pyatov@thsun1.jinr.ru
- P. Saponov
- Affiliation: Division of Theoretical Physics, IHEP, Protvino, Moscow Region 142281, Russia
- Email: Pavel.Saponov@ihep.ru
- Received by editor(s): July 13, 2007
- Published electronically: January 30, 2009
- Additional Notes: The work of D.G. was partially supported by the grant ANR-05-BLAN-0029-01; the work of P.P. and P.S. was partially supported by the RFBR grant 05-01-01086.
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 213-253
- MSC (2000): Primary 81R50
- DOI: https://doi.org/10.1090/S1061-0022-09-01045-0
- MathSciNet review: 2423997