Representations of ${\bar{U}}_{q}s\ell (2|1)$ at even roots of unity

Abstract

We construct all projective modules of the restricted quantum group ${\bar{U}}_{q}s\ell (2|1)$ at an even, $2p$th, root of unity. This $64p^4$-dimensional Hopf algebra is a common double bosonization, $B(X^{\ast })\otimes B(X)\otimes H$, of two rank-2 Nichols algebras $B(X)$ with fermionic generator(s), with $H=Z_{2p}\otimes Z_{2p}$. The category of ${\bar{U}}_{q}s\ell (2|1)$-modules is equivalent to the category of Yetter--Drinfeld $B(X)$-modules in $C_{\rho }={}_H^{H}YD$, where coaction is defined by a universal $R$-matrix $\rho$. As an application of the projective module construction, we find the associative algebra structure and the dimension, $5p^2-p+4$, of the ${\bar{U}}_{q}s\ell (2|1)$ center.

55 pp, amsart + xy + mathabx + t-angles +

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Cited by in corpus (2)

• Quasi-lisse extension of affine ${\mathfrak{sl}}_2$ à la Feigin--Tipunin
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra