A quantum subgroup depth

2016/10/04 by Alberto Hernández, Hernandez, Alberto, Lars Kadison +3
Mathematics · #16D20 #16D90 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1610.00923

openalex publication_date 2016/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^{⊗ (n)} has the same indecomposable constituents as Q^{⊗ (n+1)} is n = 2, since Q^{⊗ (2)} contains all H-module indecomposables, which determines the minimum even depth d_ev(H,D(H)) = 6.

Related

• Algebraic quotient modules and subgroup depth
• Simple modules over finite quantum groups and their Drinfel'd doubles
• The projective class ring of half-quantum groups at roots of unity
• Subgroup depth and twisted coefficients
• Minimum depth of double cross product extensions