On the structure of (co-Frobenius) Hopf algebras

Abstract

Abstract. We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).

On finite dimensional Nichols algebras of diagonal type

“…Nichols algebras are basic invariants of Hopf algebras that are not generated by its coradical [11,18]; see the discussion in Sect. 1.7.…”

“…Clearly, A 0 is a subalgebra iff A 0 = A [0] ; in this case the method outlined below was introduced in [18,20], see also [21], the extension being proposed in [11]. For simplicity we address the problem of classifying finite-dimensional Hopf algebras, but this could be adjusted to finite Gelfand-Kirillov dimension.…”

“…Formally, this means: Question 1.11 Classify all finite-dimensional Hopf algebras generated as algebras by their coradicals. This seems to be out of reach presently; see the discussion in [11]. Notice that there are plenty of finite-dimensional Hopf algebras generated by the coradical; pick one of them, say L. Recall that L L YD is the category of its Yetter-Drinfeld modules.…”

Hopf algebras having a dense big cell

“…We first note that the result generalizes as follows. If H is a Hopf algebra, we denote by H [0] the Hopf subalgebra generated by the simple subcoalgebras of H, see [1]. …”

“…where (·) (1) denotes the Frobenius twist. Hence to describe tensor products of simple O(SL q (2))-comodules it is sufficient to describe tensor products L(m) ⊗ L(m ′ ) where m, m ′ < N and L(n) (1) ⊗ L(n ′ ) (1) .…”