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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Representation theory of (modified) reflection equation algebra of $ GL(m\vert n)$ type

Authors: D. Gurevich, P. Pyatov and P. Saponov
Translated by: the authors
Original publication: Algebra i Analiz, tom 20 (2008), nomer 2.
Journal: St. Petersburg Math. J. 20 (2009), 213-253
MSC (2000): Primary 81R50
Published electronically: January 30, 2009
MathSciNet review: 2423997
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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 $ {\rm SW}(V_{(m\vert 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 $ {\rm SW}(V_{(m\vert 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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Additional Information

D. Gurevich
Affiliation: ISTV, Université de Valenciennes, Valenciennes 59304, France

P. Pyatov
Affiliation: Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia

P. Saponov
Affiliation: Division of Theoretical Physics, IHEP, Protvino, Moscow Region 142281, Russia

Keywords: (Modified) reflection equation algebra, braiding, Hecke symmetry, Hilbert--Poincar\'e series, birank, Schur--Weyl category, (quantum) trace, (quantum) dimension, braided bialgebra
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.
Article copyright: © Copyright 2009 American Mathematical Society