A family of braided cosemisimple Hopf algebras of finite dimension
The author constructs a family of Hopf algebras as described in the title over an algebraically closed field of characteristic not two. This family \(\mathcal F\) has 4 types called \(A^{ij}_{NL}\), where \(i,j\in\{+1,-1\}\) and \(N\geq 1\), \(L\geq 2\). They are all quotients of a quadratic bialgebra \(B\). It is too technical to describe \(B\) here, but its construction starts with the two-by-two matrix coalgebra. Two other quadratic bialgebras \(B^+\) and \(B^-\) are constructed, which lead to braidings on a subfamily of \(\mathcal F\). \(B\), \(B^+\) and \(B^-\) are all cosemisimple, and their braidings are determined. The main results are: (1) \(A^{ij}_{NL}\) is non-cocommutative involutory cosemisimple of dimension \(4NL\). It is non-commutative if \((L,j)\neq(2,1)\), and is semisimple if \(NL\neq 0\) in the base field. (2) Any non-commutative sub-Hopf algebra of \(A^{ij}_{NL}\) generated by a simple subcoalgebra is in \(\mathcal F\). (3) All braidings of \(A^{ij}_{NL}\) are determined. (4) It is determined when two members of \(\mathcal F\) are isomorphic.
scientific article; zbMATH DE number 1383394