Let (H,R) be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field k. In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module ...category YDHH. Let HR be the Majid's transmuted braided group of (H,R), we show that HR is cosemisimple. As a coalgebra, let HR=D1⊕⋯⊕Dr be the sum of minimal H-adjoint-stable subcoalgebras. For each i(1≤i≤r), we choose a minimal left coideal Wi of Di, and we can define the R-adjoint-stable algebra NWi of Wi. Using Ostrik's theorem on characterizing module categories over monoidal categories, we prove that V∈HHYD is irreducible if and only if there exists an i(1≤i≤r) and an irreducible right NWi-module Ui, such that V≅Ui⊗NWi(H⊗Wi).
Our structure theorem generalizes the results of Dijkgraaf-Pasquier-Roche and Gould on Yetter-Drinfeld modules over finite group algebras. If k is an algebraically closed field of characteristic 0, we stress that the R-adjoint-stable algebra NWi is an algebra over which the dimension of each irreducible right module divides its dimension.
We show that two module homomorphisms for groups and Lie algebras established by Xi （2012） can be generalized to the setting of quasi-triangular Hopf algebras. These module homomorphisms played a key ...role in his proof of a conjecture of Yau （1998）. They will also be useful in the problem of decomposition of tensor products of modules. Additionally, we give another generalization of result of Xi （2012） in terms of Chevalley-Eilenberg complex.
The main problem with current approaches to quantum computing is the difficulty of establishing and maintaining entanglement. A Topological Quantum Computer (TQC) aims to overcome this by using ...different physical processes that are topological in nature and which are less susceptible to disturbance by the environment. In a (2+1)-dimensional system, pseudoparticles called anyons have statistics that fall somewhere between bosons and fermions. The exchange of two anyons, an effect called braiding from knot theory, can occur in two different ways. The quantum states corresponding to the two elementary braids constitute a two-state system allowing the definition of a computational basis. Quantum gates can be built up from patterns of braids and for quantum computing it is essential that the operator describing the braiding—the R-matrix—be described by a unitary operator. The physics of anyonic systems is governed by quantum groups, in particular the quasi-triangular Hopf algebras obtained from finite groups by the application of the Drinfeld quantum double construction. Their representation theory has been described in detail by Gould and Tsohantjis, and in this review article we relate the work of Gould to TQC schemes, particularly that of Kauffman.
We construct the group of H-Galois objects for a flat and cocommutative Hopf algebra in a braided monoidal category with equalizers provided that a certain assumption on the braiding is fulfilled. We ...show that it is a subgroup of the group of BiGalois objects of Schauenburg, and prove that the latter group is isomorphic to the semidirect product of the group of Hopf automorphisms of H and the group of H-Galois objects. Dropping the assumption on the braiding, we construct the group of H-Galois objects with normal basis. For H cocommutative we construct Sweedler cohomology and prove that the second cohomology group is isomorphic to the group of H-Galois objects with normal basis. We construct the Picard group of invertible H-comodules for a flat and cocommutative Hopf algebra H. We show that every H-Galois object is an invertible H-comodule, yielding a group morphism from the group of H-Galois objects to the Picard group of H. A short exact sequence is constructed relating the second cohomology group and the two latter groups, under the above mentioned assumption on the braiding. We show how our constructions generalize some results for modules over commutative rings, and some other known for symmetric monoidal categories. Examples of Hopf algebras are discussed for which we compute the second cohomology group and the group of Galois objects.
The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang–Baxter equation. Use of the two-dimensional representations recovers the six-vertex model solution. ...Solutions in arbitrary dimensions, which are viewed as descendants of the six-vertex model case, are then obtained using tensor product graph methods which were originally formulated for quantum algebras. Connections with the Fateev–Zamolodchikov model are discussed.
A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted ...factor occurring in it. For a Hopf algebra
B
in a braided monoidal category
, and under certain assumptions on the braiding (fulfilled if
is symmetric), we construct a sequence for the Brauer group
of
B
-module algebras, generalizing Beattie’s one. It allows one to prove that
, where
is the Brauer group of
and
the group of
B
-Galois objects. We also show that
contains a subgroup isomorphic to
where
is the second Sweedler cohomology group of
B
with values in the unit object
I
of
. These results are applied to the Brauer group
of a quasi-triangular Hopf algebra that is a Radford biproduct
B
×
H
, where
H
is a usual Hopf algebra over a field
K
, the Hopf subalgebra generated by the quasi-triangular structure
is contained in
H
and
B
is a Hopf algebra in the category
of left
H
-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that
is a subgroup of
, confirming the suspicion that a certain cohomology group of
B
×
H
(second lazy cohomology group was conjectured) embeds into it. New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.
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