The Green rings of the 2-rank Taft algebra and its two relatives twisted

Abstract

In the paper, the representation rings (or the Green rings) for a family of Hopf algebras of tame type, the 2-rank Taft algebra (at $q=-1$) and its two relatives twisted by 2-cocycles are explicitly described via a representation theoretic analysis. It turns out that the Green rings can serve to detect effectively the twist-equivalent Hopf algebras here.

arXiv commentJ. Algebra (to appear), 34 pages (revised the noncommutativity of Green ring of 2-rank Taft algebra \bar A, rewrote the proofs of Jacobson radicals of the three Green algebras, added some remarks and updated references, etc.)

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• On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras
• Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial
• Tensor product decomposition rules for weight modules over the Hopf-Ore extensions of group algebras
• The Projective Class Rings of a family of pointed Hopf algebras of Rank two
• Indecomposable decomposition of tensor products of modules over Drinfeld Doubles of Taft algebras
• Green Rings of Pointed Rank One Hopf algebras of Non-nilpotent Type
• Green Rings of Finite Dimensional Pointed Rank One Hopf algebras of Nilpotent Type
• Green rings of Drinfeld Doubles of Taft algebras
• The Green ring of a family of copointed Hopf algebras
• Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space
• Bilinear forms on Green rings of finite dimensional Hopf algebras