Related papers: Hopf-algebraic techniques applied to super Lie gro…
Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some…
In this note we prove a generalization of the Frobenius-Schur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic $p > 2$ if the…
Definitions of the parastatistics algebras and known results on their Lie (super)algebraic structure are reviewed. The notion of super-Hopf algebra is discussed. The bosonisation technique for switching a Hopf algebra in a braided category…
The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough $\mathcal E$-projective objects with respect to the class…
Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…
Let G be an exceptional Lie group with a maximal torus T. Based on Schubert calculus on the flag manifold G/T we have described the integral cohomology ring H*(G) by explicitely constructed generators in [DZ2], and determined the structure…
One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to…
In this paper, we provide a uniform method to thoroughly classify all Harish-Chandra modules over some Lie algebras related to the Virasoro algebras. We first classify such modules over the Lie algebra $W(\varrho)[s]$ for $s=0,\frac12$.…
We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that…
Given a crossed module $\chi$, we introduce Hopf $\chi$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of…
We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld. In the course of studying…
A Hom-Lie algebra $(L, \alpha_L)$ is said to be capable if there exists a Hom-Lie algebra $(H, \alpha_H)$ such that $L \cong H/Z(H)$. We obtain a characterisation of capable Hom-Lie algebras involving its epicentre and we use this theory to…
We studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general $(\alpha,\beta)$-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are…
This text collects useful results concerning the quasi-Hopf algebra $\D $. We give a review of issues related to its use in conformal theories and physical mathematics. Existence of such algebras based on 3-cocycles with values in $ {R} /…
We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf…
We define the fundamental group of a Hopf algebra over a field. For this purpose we first consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf…
We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by Andr\'e and Neto. Also, we make further comments…
A connection between the Galois-theoretic approach to semi-abelian homology and the homological closure operators is established. In particular, a generalised Hopf formula for homology is obtained, allowing the choice of a new kind of…
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for…
Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop $\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this paper, all the finite-dimensional commutative Hopf algebras over the sub coalgebras…