Braided Morita equivalence for finite-dimensional semisimple and cosemisimple Hopf algebras

TL;DR

This paper develops refined braided Morita invariants for finite-dimensional semisimple and cosemisimple Hopf algebras, computes these invariants for specific examples, and classifies their equivalence classes, advancing understanding of their algebraic structures.

Contribution

It introduces new refined invariants for braided Morita equivalence and applies them to classify certain finite-dimensional Hopf algebras, including the 8-dimensional Kac-Paljutkin algebra.

Findings

Computed invariants for duals of Suzuki's braided Hopf algebras

Determined braided Morita equivalence classes over the 8-dimensional Kac-Paljutkin algebra

Refined the understanding of coribbon elements in these algebras

Abstract

Braided Morita invariants of finite-dimensional semisimple and cosemisimple Hopf algebras with braidings are constructed by refining the polynomial invariants introduced by the author. The invariants are computed for the duals of Suzuki's braided Hopf algebras, and as an application of that, the braided Morita equivalence classes over the $8$-dimensional Kac-Paljutkin algebra are determined. This paper also includes the modified results and proofs on determination of the coribbon elements of Suzuki's braided Hopf algebras, that are discussed and given in my previous paper "The coribbon structures of some finite dimensional braided Hopf algebras generated by $2\times 2$-matrix coalgebras" which is published in Banach Center Publication.