Coquasitriangular Hopf Algebra Research Articles

Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R(H), which we call its remnant and a co-quasitriangular structure on H induces a YBE solution on R(H), which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, H, this secondary group structure appears as the projection of the transmutation of H. Finally, for any YBE solution, we obtain a SupLat-FRT Hopf algebra in SupLat, whose remnant recovers the universal skew brace of the solution.

The gauge group of a noncommutative principal bundle and twist deformations

We study noncommutative principal bundles (Hopf–Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps.We study Drinfeld twist (2-cocycle) deformations of Hopf–Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular, noncommutative principal bundles arising via twist deformation of commutative principal bundles have classical gauge group. We illustrate the theory with a few examples.

Co-double bosonisation and dual bases of cq[SL2] and cq[SL3

We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra B in the category of comodules of a coquasitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra ▪. As an application we find new generators for cq[SL2] reduced at q a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld–Jimbo quantum enveloping algebra uq(sl2). Our methods apply in principle for general cq[G] as we demonstrate for cq[SL3] at certain odd roots of unity.

Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ1 over a Hopf algebra A which are quotients of the augmentation ideal A+ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation provides a bicovariant differential graded algebra on A. We introduce Λmax providing the maximal prolongation, while the canonical braided-exterior algebra Λmin = B−(Λ1) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ♯ by super-braided Fourier transform on B−(Λ1) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S3] with its 3D calculus and obeying the q-Hecke relation ♯2 = 1 + (q − q−1)♯ in middle degree on kq[SL2] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra mathcal {L}subseteq A defines a sub-braided Lie algebra and {Lambda }^{1}subseteq mathcal {L}^{*} provides the required data A+ → Λ1.

The group of braided autoequivalences of the category of comodules over a coquasi-triangular Hopf algebra

Let H be a coquasi-triangular Hopf algebra. We first show that the group of braided autoequivalences of the category of H -comodules is isomorphic to the group of braided-commutative bi-Galois objects. Next, by investigating the latter, we obtain that the group of braided autoequivalences of the representation category of Lusztig’s quantum group u q ( s l ( 2 ) ) ′ is isomorphic to the projective special linear group P S L ( 2 ) , where q is a root of unity of odd order N > 1 .

On the Drinfeld Center of the Category of Comodules over a Co-quasitriangular Hopf Algebra

Let $H$ be a co-quasitriangular Hopf algebra with bijective antipode. We prove that the Drinfeld center of the category of $H$-comodules is equivalent to the category of modules over some braided group. In particular, the equivalence holds not only for a finite dimensional $H$, but also for an infinite dimensional one.

Crossed Product of Hopf Algebras

The main properties of the crossed product in the category of Hopf algebras are investigated. Let A and H be two Hopf algebras connected by two morphism of coalgebras ▹: H ⊗ A → A, f: H ⊗ H → A. The crossed product is a new Hopf algebra containing A as a normal Hopf subalgebra. Furthermore, a Hopf algebra E is isomorphic as a Hopf algebra to a crossed product of Hopf algebras if and only if E factorizes through a normal Hopf subalgebra A and a subcoalgebra H such that 1 E ∈ H. The universality of the construction, the existence of integrals, commutativity or involutivity of the crossed product are studied. Looking at the quantum side of the construction we shall give necessary and sufficient conditions for a crossed product to be a coquasitriangular Hopf algebra. In particular, all braided structures on the monoidal category of -comodules are explicitly described in terms of their components. As an example, the braidings on a crossed product between H 4 and k[C 3] are described in detail.

COQUASITRIANGULAR STRUCTURES FOR EXTENSIONS OF HOPF ALGEBRAS. APPLICATIONS

AbstractLet A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

Transmutation theory of a coquasitriangular weak Hopf algebra

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.

On quiver-theoretic description for quasitriangularity of Hopf algebras

This paper is devoted to the study of the quasitriangularity of Hopf algebras via Hopf quiver approaches. We give a combinatorial description of the Hopf quivers whose path coalgebras give rise to coquasitriangular Hopf algebras. With a help of the quiver setting, we study general coquasitriangular pointed Hopf algebras and obtain a complete classification of the finite-dimensional ones over an algebraically closed field of characteristic 0.

Braided Hopf Algebras Obtained from Coquasitriangular 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 $U_q^{\rm ext}({\rm sl}_N)^{\rm cop}$.

Cocycle Deformations and Brauer Groups

Let (H, R) be a finite coquasitriangular Hopf algebra over a commutative ring k with unity. In this article, we show that the exact sequence for the equivariant Brauer group obtained in the article Zhang (2004), is stable under cocycle twisting for any cocycle σ:H⊗ H→ k on H.

L-R smash products for bimodule algebras*

Abstract In this paper, we prove that the L-R smash product A ♮ H is exactly the twisted smash product A * H if H is a finite dimensional cocommutative Hopf algebra, and give a sufficient and necessary condition for L-R smash products to be bialgebras (Hopf algebras). For any finite dimensional coquasitriangular Hopf algebra (H, σ), we prove that the L-R smash product H ♮ H is semisimple Artinian if H is semisimple and H* is unimodular. In particular, the L-R smash product D(H)* ♮D(H) * semisimple Artinian if the Drinfel’d double D(H) is semisimple. * Supported by National Natural Science Foundation of China (Grant No. 10571153), the Postdoctoral Science Foundation of China (Grant No. 2005037713), and the Postdoctoral Science Foundation of Jiangsu (Grant No. 0203003403)

An exact sequence for the Brauer group of a finite quantum group

In this paper, we give a general method to compute the Brauer group of a finite quantum group, i.e., a faithfully projective coquasitriangular Hopf algebra over a commutative ring with unity. Let ( H, R) be a finite quantum group with an R-matrix R on H⊗ H. There exists a braided Hopf algebra H R in the braided monoidal category of right H-comodules [S. Majid, J. Pure Appl. Algebra 86 (1993) 187–221]. We construct a group Gal( H R) consisting of quantum commutative H ∗ R -bigalois objects and show that there is an exact sequence of group homomorphisms: 1→ Br(k)→ BC(k,H,R)→ Gal( H R), where Br( k) is the usual Brauer group of k and BC( k, H, R) is the Brauer group of ( H, R) with respect to the R-matrix R.

Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras

We show that if g Γ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first-order differential calculus over a co-quasitriangular Hopf algebra (A, r ) , then a certain extension of it is a braided Lie algebra in the category of A -comodules. This is used to show that the Woronowicz quantum universal enveloping algebra U( g Γ ) is a bialgebra in the braided category of A -comodules. We show that this algebra is quadratic when the calculus is inner. Examples with this unexpected property include finite groups and quantum groups with their standard differential calculi. We also find a quantum Lie functor for co-quasitriangular Hopf algebras, which has properties analogous to the classical one. This functor gives trivial results on standard quantum groups O q (G) , but reasonable ones on examples closer to the classical case, such as the cotriangular Jordanian deformations. In addition, we show that split braided Lie algebras define ‘generalized-Lie algebras’ in a different sense of deforming the adjoint representation. We construct these and their enveloping algebras for O q (SL(n)) , recovering the Witten algebra for n =2.

Coribbon Hopf (Face) Algebras Generated by Lattice Models

By studying “points of the underlying quantum groups” of coquasitriangular Hopf (face) algebras, we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type. Also, we give a classification of the braiding and the ribbon structure on quantized classical groups and modular tensor categories closely related to quantum SU(N)L-invariants of 3-manifolds.

DIFFERENTIAL CALCULUS ON INHOMOGENEOUS QUANTUM GROUPS

We investigate the question of covariant differential calculi on the bosonisation of a coquasitriangular Hopf algebra and an associated braided Hopf algebra. As a result we present a general way of obtaining such calculi on inhomogeneous quantum groups.

On the construction of covariant differential calculi on quantum homogeneous spaces

Let A be a coquasitriangular Hopf algebra and X the subalgebra of A generated by a row of a matrix corepresentation u or by a row of u and a row of the contragredient corepresentation u c. In the paper left-covariant first order differential calculi on the quantum group A are constructed and the corresponding induced calculi on the left quantum space X are described. The main tool for these constructions are the L-functionals associated with u. The results are applied to the quantum homogeneous space GL q ( N)/ GL q ( N − 1).

Making the Category of Doi-Hopf Modules into a Braided Monoidal Category

We investigate how the category of Doi-Hopf modules can be made into a monoidal category. It suffices that the algebra and coalgebra in question are both bialgebras with some extra compatibility relation. We study tensor identies for monoidal categories of Doi-Hopf modules. Finally, we construct braidings on a monoidal category of Doi-Hopf modules. Our construction unifies quasitriangular and coquasitriangular Hopf algebras, and Yetter-Drinfel'd modules.

Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces

A method of constructing covariant differential calculi on a quantum homogeneous space is devised. The function algebra χ of the quantum homogeneous space is assumed to be a left coideal of a coquasitriangular Hopf algebra ℋ and to contain the coefficients of any matrix over ℋ which is the two-sided inverse of one with entries in χ. The method is based on partial derivatives. For the quantum sphere of Podleś and the quantizations of symmetric spaces due to Noumi, Dijkhuizen and Sugitani, the construction produces the subcalculi of the standard bicovariant calculus on the quantum group.