A quantum subgroup depth

TitleA quantum subgroup depth
Publication TypeArticles in international peer reviewed journals
Year of Publication2017
AuthorsHernandez A, Kadison L, Lopes SA
JournalActa Math. Hung.
KeywordsGreen ring; half quantum group; Hopf subalgebra; quotient module; subgroup depth

The Green ring of the half quantum group $H=U_n(q)$ is computed in [Chen, Van
Oystaeyen, Zhang]. The tensor product formulas between indecomposables may be
used for a generalized subgroup depth computation in the setting of quantum
groups -- to compute depth of the Hopf subalgebra $H$ in its Drinfeld double
$D(H)$. In this paper the Hopf subalgebra quotient module $Q$ (a generalization
of the permutation module of cosets for a group extension) is computed and, as
$H$-modules, $Q$ and its second tensor power are decomposed into a direct sum
of indecomposables. We note that the least power $n$, referred to as depth, for
which $Q^{\otimes (n)}$ has the same indecomposable constituents as $Q^{\otimes
(n+1)}$ is $n = 2$, since $ Q^{\otimes (2)}$ contains all $H$-module
indecomposables, which determines the minimum even depth $d_{ev}(H,D(H)) = 6$.


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