# A quantum subgroup depth

 Title A quantum subgroup depth Publication Type Articles in international peer reviewed journals Year of Publication 2017 Authors Hernandez A, Kadison L, Lopes SA Journal Acta Math. Hung. Volume 152 Issue 1 Pagination 166--185 Keywords Green ring; half quantum group; Hopf subalgebra; quotient module; subgroup depth Abstract 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 $d_{ev}(H,D(H)) = 6$. URL https://doi.org/10.1007/s10474-017-0694-6 DOI 10.1007/s10474-017-0694-6 [2017-1]

Algebra