A quantum subgroup depth

TL;DR

This paper computes the depth of a quantum subgroup within its Drinfeld double by analyzing the tensor powers of a specific quotient module, revealing the minimal depth as 6.

Contribution

It introduces a method to determine the depth of a Hopf subalgebra in its Drinfeld double using module decomposition and tensor power analysis.

Findings

The quotient module Q and its second tensor power decompose into indecomposables.

The minimal depth n for which Q^n and Q^{n+1} share indecomposables is 2.

The minimum even depth of the subalgebra in its Drinfeld double is 6.

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 $d_{ev}(H,D(H)) = 6$.