We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain func...

The theory of Doi-Hopf modules [7, 10] is generalized to Weak Hopf Algebras [1, 12, 2].

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s automorphisms of quantized Kac-Moody algebras to the nilpotent part. As a direct application we co...

In this paper, we mainly investigate a revised induction funtor to study Hom-Doi-Hopf modules and the coinvariant for Hom-Hopf comodule algebras.

Let k be an algebraically closed field. The main goal of this paper is to classify the finite-dimensional pointed Hopf algebras over k of finite corepresentation type. To do so, we give a necessary and sufficient condition for a basic Hopf algebra over k to be of finite representation type firstly. Explicitly, we prove that a basic Hopf algebra over k is of finite representation type if and onl...

We develop a non-compact version of the Hopf maps based on the split-algebras. The split-algebras consist of three species: split-complex numbers, split-quaternions, and split-octonions. They correspond to three non-compact Hopf maps that represent topological mapping between higher dimensional hyperboloids in different dimensions with hyperboloid-bundles. We realize such non-compact Hopf maps ...

In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows us to give a new construction of the universal enveloping algebras of Malcev algebras. Our work relies on the approach of Grishkov and ...

Actually to any spin model one can associate a vertex model (this is clear from V. Jones’ initial interpretation – in terms of statistical mechanics – of these objects) and the construction of Hopf algebras from complex Hadamard matrices is a particular case of the construction of Hopf algebras from biunitary matrices. This is done in section 5 in [3]. The construction of Hopf algebras from biu...

Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero [Mo and references therein]. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky’s...