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- W3094863812 abstract "For a semisimple Lie algebra $mathfrak{g}$ of rank $n$, let $overline{U}_zeta(mathfrak{g})$ be the restricted quantum group of $mathfrak{g}$ at a primitive fourth root of unity. This quantum group admits a natural Borel-induced representation $V({boldsymbol{t}})$, with ${boldsymbol{t}}in(mathbb{C}^times)^n$ determined by a character on the Cartan subalgebra. Ohtsuki showed that for $mathfrak{g}=mathfrak{sl}_2$, the braid group representation determined by tensor powers of $V({boldsymbol{t}})$ is the exterior algebra of the Burau representation. In this paper, we generalize the tensor decomposition of $V({boldsymbol{t}})otimes V({boldsymbol{s}})$ used in Ohtsuki's proof to any semisimple $mathfrak{g}$. Upon specializing to the $mathfrak{sl}_3$ case, we describe all projective covers of $V({boldsymbol{t}})$ in terms of induced representations. The above decomposition formula for $V({boldsymbol{t}})otimes V({boldsymbol{s}})$ is then extended to more general $boldsymbol{t}$ and $boldsymbol{s}$ where these projective covers occur as indecomposable summands. We also define a stratification of $(mathbb{C}^times)^{4}$ whose points $({boldsymbol{t}},{boldsymbol{s}})$ in the lower strata are associated with representations $V({boldsymbol{t}})otimes V({boldsymbol{s}})$ that do not have a homogeneous cyclic generator. With this information, we characterize under what conditions the isomorphism ${V({boldsymbol{t}})otimes V({boldsymbol{s}})cong V({boldsymbol{lambda t}})otimes V({boldsymbol{lambda^{-1} s}})}$ holds." @default.
- W3094863812 created "2020-11-09" @default.
- W3094863812 creator A5040251872 @default.
- W3094863812 date "2019-11-02" @default.
- W3094863812 modified "2023-09-27" @default.
- W3094863812 title "Verma Modules for Restricted Quantum Groups at a Fourth Root of Unity." @default.
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