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, $B(X^*)\otimes B(X)\otimes H$, 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.