Cohomological dimension of braided Hopf algebras

Let $(H, \sigma)$ be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define $H_\sigma$, a sub-Hopf algebra of $H^0$, the finite dual of $H$. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we associate to $H$ a braided Hopf algebra structure in the category of Yetter-Drinfeld modules over $H_\sigma^{\rm cop}$. Specializing to $H={\rm SL}_q(N)$, we obtain explicit formulas which endow ${\rm SL}_q(N)$ with a braided Hopf algebra structure within the category of left Yetter-Drinfeld modules over $U_q^{\rm ext}({\rm sl}_N)^{\rm cop}$.

We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible *-structures, encompassing the quasitriangular case.

A known fundamental Theorem for braided pointed Hopf algebras states that for each coideal subalgebra, that fulfills a few properties, there is an associated quotient coalgebra right module such that the braided Hopf algebra can be decomposed into a tensor product of these two. Often one considers braided Hopf algebras in a Yetter-Drinfeld category of an ordinary Hopf algebra. In this case the braided Hopf algebra and many interesting coideal subalgebras are in particular comodules. We extend the mentioned Theorem by proving that the decomposition is compatible with this comodule structure if the underlying ordinary Hopf algebra is cosemisimple.

Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right H-comodules), which generalizes Majid’s transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber–Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

A smash coproduct in braided monoidal category C is constructed and some conditions making the smash coproduct a Hopf algebra or braided Hopf algebra are given. It is shown that the smash coproductB ×H inHM is equivalent to the transmutation of Hopf algebra. Thus a method for transmutation theory is provided. Let σ be 2-co-cycle andH a commutation Hopf algebra. A Hopf algebraHσ is constructed.Hσ≅Hσ whereHσ is a transmutation ofHσ. The braided groups from some solutions of quantum Yang-Baxter equation are obtained.

Quantum lines over non-cocommutative cosemisimple Hopf algebras

The main goal of this chapter is a detailed construction of the free braided Hopf algebra \(\mathbf{k}\langle V \rangle\) and the shuffle braided Hopf algebra Sh τ (V ) on the tensor space of a given braided space V. Then we define a Nichols algebra \(\mathcal{B}(V )\) as a subalgebra generated by V in Sh τ (V ) and provide some characterizations of it. Finally we adopt the Radford biproduct and the Majid bozonization to character Hopf algebras. All calculations are done in the braid monoid (not in the braid group), therefore in the constructions there is no need to assume that the braiding is invertible.KeywordsHopf AlgebraCommutation RuleDrinfeld ModuleBraided Monoidal CategoryNichols AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given an abelian k-linear rigid monoidal category V, where k is a perfect field, we define squared coalgebras as objects of cocompleted V tensor V (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If V is the category of k-vector spaces, squared (co)algebras coincide with conventional ones. If V is braided, a braided Hopf algebra can be obtained from a squared one. Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras in V and corresponding fibre functors to V (which is not the case with other definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.

Braided autoequivalences and the equivariant Brauer group of a quasitriangular Hopf algebra

We give the necessary and sufficient conditions for a general crossed product algebra equipped with the usual tensor product coalgebra structure to be a Hopf algebra. Furthermore, we obtain the necessary and sufficient conditions for the general crossed product Hopf algebra to be a braided Hopf algebra which generalizes some known results.

Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of Orderp3

We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic. We prove also that the category of finite dimensional left Yetter-Drinfeld modules is rigid, and then we compute explicitly the canonical isomorphisms in . Finally, we show that certain duals of H 0, the braided Hopf algebra (introduced in Bulacu and Nauwelaerts, 2002; Bulacu et al., 2000) are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra. Communicated by M. Takeuchi.

In the previous chapter we showed that braided Hopf algebras provided solutions of the Yang-Baxter equation. The problem is now to find enough such Hopf algebras. Drinfeld [Dri87] devised an ingenious method, the “quantum double construction”, which builds a braided Hopf algebra out of any finite-dimensional Hopf algebra with invertible antipode. It is the goal of this chapter to describe this construction in full detail, and to show how to apply it to the finite-dimensional quotient of the Hopf algebra Uq(sl(2)) considered in VI.5. We also give a characterization of the modules over the quantum double in Section 5.

Let H be a Hopf algebra with bijective antipode. In this paper we prove that, if A is a Hopf brace with some projection on H, then there exists a compatible braided Hopf brace R such that A is isomorphic to as Hopf braces, where is some Radford’s biproduct Hopf algebra. This should be viewed as the brace version of well-known Radford’s theorem about Hopf algebras with a projection. This provides a new method to construct Hopf braces by some braided Hopf algebras.