Finite-dimensional simple-pointed Hopf algebras
Quantized universal enveloping algebras are generated by pairs of elements \(a\) and \(x\), where \(a\neq 0\) is group-like, i.e., \(\Delta a=a\otimes a\), \(x\) is \((1,a)\)-primitive, i.e., \(\Delta x=1\otimes x+x\otimes a\) and \(xa=qax\). They are pointed as coalgebras, i.e., every simple subcoalgebra is one-dimensional, spanned by a group-like element. Let \(A\) be a pointed Hopf algebra over a field \(k\). The author calls \(A\) simple-pointed if \(A\) is not cocommutative, and every proper Hopf subalgebra of \(A\) is contained in the coradical \(A_0\) of \(A\), where \(A_0\) is the span of the group-like elements of \(A\). In this paper, the author describes the structure of simple-pointed Hopf algebras generated by pairs \(a\) and \(x\) as above, when \(k\) is algebraically closed. In the finite-dimensional characteristic zero case, he characterizes the coalgebra structure of the duals of these simple-pointed Hopf algebras. He also determines the Hopf algebra projections of such a finite-dimensional simple-pointed Hopf algebra onto proper sub-Hopf algebras. This enables one to determine whether or not such a simple-pointed Hopf algebra is a non-trivial biproduct. The techniques in the paper involve Hopf modules and \(q\)-binomial coefficients.
scientific article; zbMATH DE number 1271344