Representation theory of (modified) Reflection Equation Algebra of $GL(m|n)$ type

D. I. Gurevich ; P. N. Pyatov ; P. A. Saponov

Algebra i analiz, Tome 20 (2008) no. 2, pp. 70-133.

Résumé

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.

Détail

Keywords: (Modified) reflection equation algebra, braiding, Hecke symmetry, Hilbert-Poincaré series, birank, Schur–Weyl category, (quantum) trace, (quantum) dimension, braided bialgebra.

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