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Research Papers
Representation theory of (modified) Reflection Equation Algebra of $GL(m|n)$ type
D. I. Gurevicha,
P. N. Pyatovb,
P. A. Saponovc a USTV, Université de Valenciennes, Valenciennes, France
b Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna
c Institute for High Energy Physics, Russian Academy of Scienses
Abstract:
Let
$R\colon V^{\otimes 2}\to V^{\otimes 2}$ 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
$\mathrm{SW}(V_{(m|n)})$ 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
$\mathrm{SW}(V_{(m|n)})$ 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.
Keywords:
(Modified) reflection equation algebra, braiding, Hecke symmetry, Hilbert-Poincaré series, birank, Schur–Weyl category, (quantum) trace, (quantum) dimension, braided bialgebra.
MSC: 81R50 Received: 13.07.2007