Krull--Schmidt decomposition of the tensor products of certain simpleUq(gln)-modules at a root of unity
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摘要: 设~$\mathscr{F}$~是特征~0~的域, $q\in\mathscr F$~是个单位根. 以~$\mathscr{F}$~为基域、以~$q$~为量子参数, 令~$\mathsf{s}_q(n)$~为秩~$n$~的限制量子对称代数, $\Wedge_q(n)$~为秩~$n$~的量子外代数. 据~[6], $\mathsf{s}_q(n)$~与~$\Wedge_q(n)$~的齐次分量都是单的~$U_q(\mathfrak{gl}_n)$-模. 本文将把~$\mathsf{s}_q(n)$~的齐次分量与~$\Wedge_q(n)$~的齐次分量的张量积分解成不可分解模的直和.
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关键词:
- 量子群 /
- 限制量子对称代数 /
- 量子外代数 /
- Krull--Schmidt\,分解
Abstract: Assume $\mathscr{F}$ to be a field of characteristic zero, and $q\in \mathscr{F}$ to be a root of unity. With $\mathscr F$ as the ground field and $q$ as the quantum parameter, let $\mathsf{s}_q(n)$ be the restricted quantum symmetric algebra of rank $n$, and $\Wedge_q(n)$ be the quantum exterior algebra of rank $n$. By [6], the homogenous components of both $\mathsf{s}_q(n)$ and $\Wedge_q(n)$ are simple $U_q(\mathfrak{gl}_n)$-modules. In this paper, we decompose the tensor product of any homogenous component of $\mathsf{s}_q(n)$ with any homogenous component of $\Wedge_q(n)$ into direct sum of indecomposable modules. -
[1] {AW} ANDERSEN H H, WEN K X. Representations of quantum algebras, the mixed case [J]. Journal f\"ur die Reine und Angewandte Mathematik, 1992, 427: 35--50.
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