Related papers: Algebraic quantum groupoids - An example
We prove a number of structural and representation-theoretic results on linearly reductive quantum groups, i.e. objects dual to that of cosemisimple Hopf algebras: (a) a closed normal quantum subgroup is automatically linearly reductive if…
We show that the algebra of the bicovariant differential calculus on a quantum group can be understood as a projection of the cross product between a braided Hopf algebra and the quantum double of the quantum group. The resulting super-Hopf…
Let $H$ be a weak Hopf algebra with a bijective antipode and $A$ an $H$-comodule Poisson algebra. In this paper, we mainly generalize the fundamental theorem of Poisson Hopf modules to the case of weak Hopf algebras. Besides we will deduce…
Family of doublings of Hopf algeras based on the product of algebra and its dual are constructed and studied. Special cases of these construction may be considered as natural quantum analogs of rings of differential operators on groups.…
For a given finite dimensional Hopf algebra $H$ we describe the set of all equivalence classes of cocycle deformations of $H$ as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the…
We present two classes of examples of Hopf algebroids associated with noncommutative principal bundles. The first comes from deforming the principal bundle while leaving unchanged the structure Hopf algebra. The second is related to…
Let $L$ be a finite dimensional Lie algebra over a field of characteristic $0$. Then by the original Levi theorem, $L = B \oplus R$ where $R$ is the solvable radical and $B$ is some maximal semisimple subalgebra. We prove that if $L$ is an…
Using the fact that the algebra M := M_N(C) of NxN complex matrices can be considered as a reduced quantum plane, and that it is a module algebra for a finite dimensional Hopf algebra quotient H of U_q(sl(2)) when q is a root of unity, we…
Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic zero, where p, q are odd primes with p < q < 4p+12. We prove that H is semisimple and thus isomorphic to a group algebra, or the dual of a group…
The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum $n$-space. A kind of braided category $\Cal {GB}$ of $\La$-graded $\th$-commutative associative algebras over a field $k$ is…
The ``local'' structure of a quantum group G_q is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping…
We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff \'{e}tale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse…
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…
A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a…
The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring $B$ can be embedded as a right coideal subalgebra into a Hopf algebra $A$ such that $A$ is faithfully flat as a $B$-module. In…
Let $(A,\Delta)$ be a multiplier Hopf algebra. In general, the underlying algebra $A$ need not have an identity and the coproduct $\Delta$ does not map $A$ into $A\otimes A$ but rather into its multiplier algebra $M(A\otimes A)$. In this…
In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a…
In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra H. We show that the second cohomology group classifies completely the weak crossed products, having a common preunit, of H with a commutative left…
For any open, connected and bounded set $\Omega \subseteq \mathbb C^m$, let $\mathcal A$ be a natural function algebra consisting of functions holomorphic on $\Omega$. Let $\mathcal M$ be a Hilbert module over the algebra $\mathcal A$ and…
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…