Related papers: Duality properties for quantum groups

Let $(A,\Delta)$ be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. The main difference with multiplier Hopf algebras is…

We classify equivalence classes of Hopf algebra quotient pairs $(D,\theta)$ of the Drinfeld double $D(G)$ of a finite group scheme $G$ over an algebraically closed field $\mathbf{k}$ of characteristic $p\ge 0$, in terms of group…

We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincar\'e duality for the Lie algebra cohomology of the infinite-dimensional gauge transformation group. We show that…

In the present work, we study Yetter-Drinfeld algebras over a pairing of multiplier Hopf algebras. Our main motivation is the construction of a self-dual theory of (C*-)algebraic quantum transformation groupoids. Instead of the standard…

The Hopf algebra generated by the l-functionals on the quantum double C_q[G] \bowtie C_q[G] is considered, where C_q[G] is the coordinate algebra of a standard quantum group and q is not a root of unity. It is shown to be isomorphic to…

We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras ${\cal H}^c$ that we recently introduced. We show that if the standard Hopf (super)algebras ${\cal H}_q$ that are the building blocks of…

Let $G$ be any group and let $K(G)$ denote the multiplier Hopf algebra of complex functions with finite support in $G$. The product in $K(G)$ is pointwise. The comultiplication on $K(G)$ is defined with values in the multiplier algebra…

The quantum double construction of a $q$-deformed boson algebra possessing a Hopf algebra structure is carried out explicitly. The $R$-matrix thus obtained is compared with the existing literature.

Let $H$ be a finite Hopf algebra with $C_{H,H} = C_{H,H}^{-1}.$ The duality theorem is shown for $H$, i.e., $$ (R # H)# H^{\hat *} \cong R \otimes (H \bar \otimes H^{\hat *}) \hbox {as algebras in} {\cal C}.$$ Also, it is proved that the…

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as…

Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra $H=kM\cobicross k(G)$ is itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where $Y=G\times…

Let G be a topological group such that its homology H(G) with coefficients in a principal ideal domain R is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between G-spaces and…

A two-parameter quantum deformation of the affine Lie super algebra $osp(2|2)^{(2)}$ is introduced and studied in some detail. This algebra is the first example associated with nonsimply-laced and twisted root systems of a quantum current…

If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have families of…

We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the `quantum groups approach'. We apply the author's previous classification theorem, implying that calculi in the factorisable case…

Let $H$ be a Hopf algebra over a field $K$ of characteristic $0$ and let $A$ be a bialgebra or Hopf algebra such that $H$ is isomorphic to a sub-Hopf algebra of $A$ and there is an $H$-bilinear coalgebra projection $\pi$ from $A$ to $H$…

Let A be a commutative unital algebra over an algebraically closed field k of characteristic not equal to 2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let Q be a Hopf algebra…

Twist deformation U_F(g) is equivalent to the quantum group Fun_d(G#) and has two preferred bases: the one originating from U(g) and that of the coordinate functions on the dual Lie group G#. The costructure of the Hopf algebra U_F(g) is…

We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If $K$ is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra $K^n$:…

Let $ L/K $ be a finite separable extension of fields whose Galois closure $ E/K $ has group $ G $. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on $ L/K $ has the form $ E[N]^{G}…