Quasitriangular quasi-Hopf Algebra Research Articles

Some ribbon elements for the quasi-Hopf algebra Dω(H)

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra [Formula: see text] defined in [D. Bulacu and F. Panaite, A generalization of the quasi-Hopf algebra [Formula: see text], Commun. Algebra 26 (1998) 4125–4141] and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for [Formula: see text].

Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulation of Einstein-Cartan geometry as a putative framework for a nonassociative theory of gravity.

Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

We systematically study noncommutative and nonassociative algebras A and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of A-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras A the full subcategory of symmetric A-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.

Crossed pointed categories and their equivariantizations

We propose the notion of quasi-abelian third cohomology of crossed modules, generalizing Eilenberg and MacLane's abelian cohomology and Ospel's quasi-abelian cohomology, and classify crossed pointed categories in terms of it. We apply the process of equivariantization to the latter to obtain braided fusion categories which may be viewed as generalizations of the categories of modules over twisted Drinfeld doubles of finite groups. As a consequence, we obtain a description of all braided group-theoretical categories. A criterion for these categories to be modular is given. We also describe the quasi-triangular quasi-Hopf algebras underlying these categories.

Quasisymmetric and unipotent tensor categories

We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on symmetric categories of exponential growth, and of Drinfeld on quasitriangular quasi-Hopf algebras. In particular, we classify braided categories of exponential growth which are unipotent, i.e., those whose only simple object is the unit object. We also classify fiber functors on such categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson algebraic groups, we give a classification of coconnected Hopf algebras, i.e. of unipotent categories of exponential growth with a fiber functor.

The Quantum Double for Quasitriangular Quasi-Hopf Algebras

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).

Quasitriangular and Ribbon Quasi-Hopf Algebras

Following (Drinfeld, V. G. (1990a). Quasi-Hopf algebras.Leningrad Math. J. 1:1419–1457) a quasi-Hopf algebra has, by definition, its antipode bijective. In this note, we will prove that for a quasitriangular quasi-Hopf algebra with an R-matrix R, this condition is unnecessary and also the condition of invertibility of R. Finally, we will give a characterization for a ribbon quasi-Hopf algebra. This characterization has already been given in Altschuler and Coste (Altschuler, D., Coste, A. (1992). Quasi-quantum groups, knots, three-manifolds and topological field theory. Comm. Math. Phys. 150:83–107.), but with an additional condition. We will prove that this condition is unnecessary.

DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS

A two-sided coaction [Formula: see text] of a Hopf algebra [Formula: see text] on an associative algebra ℳ is an algebra map of the form [Formula: see text] , where (λ,ρ) is a commuting pair of left and right [Formula: see text] -coactions on ℳ, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra [Formula: see text] on ℳ by ◃ and ▹, respectively, we define the diagonal crossed product[Formula: see text] to be the algebra generated by ℳ and [Formula: see text] with relations given by [Formula: see text] We give a natural generalization of this construction to the case where [Formula: see text] is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct Δ is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on [Formula: see text] extending [Formula: see text], even though the analogue of an ordinary crossed product [Formula: see text] in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case [Formula: see text] and λ=ρ=Δ we obtain an explicit definition of the quantum double [Formula: see text] for quasi-Hopf algebras [Formula: see text] , which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that [Formula: see text] is itself a (weak) quasi-bialgebra and that any diagonal crossed product [Formula: see text] naturally admits a two-sided [Formula: see text] -coaction. In particular, the above-mentioned lattice models always admit the quantum double [Formula: see text] as a localized cosymmetry, generalizing results of Nill and Szlachányi [42]. A complete proof that [Formula: see text] is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].

Doubles of Quasi-Quantum Groups

In [Dr1] Drinfeld showed that any finite dimensional Hopf algebra \(\) extends to a quasitriangular Hopf algebra \(\), the quantum double of \(\). Based on the construction of a so-called diagonal crossed product developed by the authors in [HN], we generalize this result to the case of quasi-Hopf algebras \(\). As for ordinary Hopf algebras, as a vector space the “quasi-quantum double”\(\) is isomorphic to \(\), where \(\) denotes the dual of \(\). We give explicit formulas for the product, the coproduct, the R-matrix and the antipode on \(\) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi-Hopf algebra. In particular \(\) becomes an associative algebra containing \(\) as a quasi-Hopf subalgebra. On the other hand, \(\) is not a subalgebra of \(\) unless the coproduct on \(\) is strictly coassociative. It is shown that the category \(\) of finite dimensional representations of \(\) coincides with what has been called the double category of \(\)-modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi-quantum doubles in terms of a Tannaka–Krein-like reconstruction procedure. The whole construction is shown to generalize to weak quasi-Hopf algebras with \(\) now being linearly isomorphic to a subspace of \(\).

On associators and the Grothendieck-Teichmuller group, I

We present a formalism within which the relationship (discovered by Drinfel'd in [Dr1], [Dr2]) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a simplification of Drinfel'd's original work. In particular, we reprove that rational associators exist and can be constructed iteratively, though the proof itself still depends on the apriori knowledge that a not-necessarily-rational associator exists.

Star-Products and Quasi-Quantum Groups

The purpose of the present paper is to show thatDrinfeld's theories of quasiquantum groups(quasitriangular quasi-Hopf algebras) and of quasi-Liebialgebras can be developed in terms of the starproducts on a simple quasitriangular Poisson–Liegroup.

Quantum Double for Quasi-Hopf Algebras

We introduce a quantum double quasi-triangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross-product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double Dφ(G) associated to a finite group G and group 3-cocycle φ.

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.

Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

Ifq is ap th root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU q (sl 2), whose coproduct yields the truncated tensor product of physical representations ofU q (sl 2), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (“quasi-quantum group algebras”). We describe a space of “functions on the quasi quantum plane,” i.e. of polynomials in noncommuting complex coordinate functionsz a , on which multiplication operatorsZ a and the elements of can act, so thatz a will transform according to some representation τf of can be made into a quasi-associative graded algebra on which elements of act as generalized derivations. In the special case of the truncatedU q (sl 2) algebra we show that the subspaces of monomials inz a ofn th degree vanish forn≥p−1, and that carries the 2J+ 1 dimensional irreducible representation of ifn=2J, J=0,1/2, ..., 1/2(p−2). Assuming that the representation τf of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on. This implies construction of an exterior differentiation, a graded algebra of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under gauge transformations.

A CONSTRUCTIVE QUANTUM FIELD THEORETIC APPROACH TO CHERN-SIMONS THEORY

Chern-Simons theory is formulated quantum field theoretically in terms of string operators, which are localized on finite non-selfintersecting paths in the zero-time plane R2. It is shown how the Weyl algebra approach to the abelian Chern-Simons theory leads to a bundle theoretic construction of these ‘Chern-Simons string operators’. To proceed to the non-abelian case topological Chern-Simons theory is considered as a specific model of a more general theory describing the quantum kinematics of coloured framed Plek ton s inR2. This theory allows the construction of string operators as pair-creation operators, mapping n-Plekton states into (n+2)-Plekton states. Link invariants are obtained as vacuum correlation functions of string operators and obey a natural version of reflection positivity. The vacuum sector of such a purely kinematical Plekton theory may be recovered from the link invariants by Osterwalder-Schrader reconstruction. To make the theory work one needs the structures of a ribbon graph category, for which the representations of a quasi-triangular quasi-Hopf algebra may serve as a specific realization.