Journal of East China Normal University(Natural Sc ›› 2013, Vol. 2013 ›› Issue (6): 1-13.

• Article •     Next Articles

Krull--Schmidt decomposition of the tensor  products of certain simpleUq(gln)-modules at a root of unity

GU Hai-xia, WANG Jian-pan   

  1. Department of Mathematics, East China Normal University, Shanghai  200241, China
  • Received:2013-08-01 Revised:2013-10-01 Online:2013-11-25 Published:2014-01-13

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.

Key words: quantum group, restricted quantum symmetric algebra, quantum exterior algebra, Krull--Schmidt decomposition

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