Examples of tensor categories

Let be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐ modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient . For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Quantum group and quantum symmetry

We apply quantum group methods to quantum computing, starting with the notion of interacting Frobenius Hopf algebras for ZX calculus with noncommutative algebra and noncocommutative coalgebra. We introduce the notion of *-structures in ZX calculus at this algebraic level and construct examples based on the quantum group u q (sl 2) at a root of unity. We provide an abstract formulation of the Hadamard gate related to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX calculus to take place in a braided tensor category. In the ribbon case, the Hadamard gate coming from braided self-duality obeys a modular identity. We give the example of b q (sl 2), the self-dual braided version of u q (sl 2).

Let p be an odd prime number and K a number field having a primitive p th root of unity \zeta_p . We prove that Nikshych's non group-theoretical Hopf algebra H_p , which is defined over \mathbb Q(\zeta_p) , admits a Hopf order over the ring of integers \mathcal O_K if and only if there is an ideal I of \mathcal O_K such that I^{2(p-1)} = (p) . This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over \mathcal O_K exists, it is unique and we describe it explicitly.

Introduction. In [Jon83], Jones constructs, from any III subfactor pair M,N with finite index [M, N] = q + q-I + 2, a canonical hyperfinite subfactor pair of the same index. The pair is constructed from the completion of the TemperleyLieb algebra by a Markov trace. By showing that such a completion yields a II, subfactor pair exactly when q is a positive real or of the form e27ri/k, he obtained his index of subfactors result. The fact that the set of admissible values of q divides into a discrete piece and a continuous piece is puzzling and is indicative of qualitative differences between the subfactors themselves. For one thing, at roots of unity the subfactors are of finite depth. That is, after a certain point in the Bratteli diagram the centers of the finite-dimensional approximating algebras are periodically isomorphic. On the contrary, in the positive real case each finite-dimensional algebra in the Bratteli diagram has a strictly larger center than the previous one. Furthermore, the relative commutant of the smaller algebra in the larger algebra consists only of multiples of the identity in the root of unity case. For positive real q, not only is there a relative commutant, but in fact the subfactor is locally trivial. That is, the subfactor pair, restricted to a minimal projection in the relative commutant, is the trivial subfactor pair. In addition to its direct relevance to the study of subfactors, Jones' work had indirect but important ramifications. The Temperley-Lieb algebra in the construction forms a representation of the braid group, and the Markov trace as a functional on the braid group is, with proper normalization, a polynomial link invariant. This discovery triggered an explosion of mathematics which eventually came to see the algebra, trace, and invariant as a special case of a general construction which associates, via quantum groups, an example of all three to each irreducible representation of each. simple Lie algebra. The original Jones polynomial comes from the fundamental representation of s12. The quantum groups construction can be done entirely without reference to subfactors, or even analysis. Nevertheless, there are echos of Jones' work, and in particular both the roots of unity and the positive real q continue to play a distinct and special role. For example, when q is a root of unity, the quantum group

It has become clear over the years that quantum groups (i.e., quasitriangular Hopf algebras, see [D]) and their semiclassical counterpart, Poisson Lie groups, are an essential algebraic structure underlying three related subjects: integrable models of statistical mechanics, conformal field theory, and integrable models of quantum field theory in 1+1 dimensions. Still, some points remain obscure from the point of view of Hopf algebras. In particular, integrable models associated with elliptic curves are still poorly understood. We propose her an elliptic version of quantum groups, based on the relation to conformal field theory, which hopefully will be helpful to complete the picture.

Given a dynamical twist for a finite-dimensional Hopf algebra, we construct two weak Hopf algebras, using methods of P. Xu and of P. Etingof and A. Varchenko, and we show that they are dual to each other. We generalize the theory of dynamical quantum groups to the case when the quantum parameter q is a root of unity. These objects turn out to be self-dual—which is a fundamentally new property, not satisfied by the usual Drinfeld-Jimbo quantum groups.

The two major approaches to chiral conformal field theory - one based on operator algebras and one based on vertex algebras - both lead to representation categories which are tensor categories and, in the case of rational chiral conformal field theories, more specifically modular tensor cat- egories. In this Arbeitsgemeinschaft, we have studied algebraic structures related to tensor categories arising in conformal field theory. The notion of a module category over this tensor category is central in the construction of a full local conformal field theory in various frameworks.

Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory

We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided monoidal category of Yetter–Drinfeld modules over a Hopf algebra. This gives rise to the notion of ribbon Yetter–Drinfeld modules over a Hopf algebra, which form ribbon categories. This gives an invariant of tangles.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess–Zumino–Novikov–Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet -algebras as an example of the applications of our general construction of the braided tensor category structure.

Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of Lie groups and algebras quantum groups, algebras and their duality non-commutative spaces and quantum groups invariant differential calculi elements of quantum groups representations theory tensor products of representations q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator different forms of commutation relations representations q real and roots of unity cases algebraic maps from non-deformed to deformed oscillators path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: relativistic space-time symmetries Poincare group as a classical deformation of Galilei group its representations multiparametric q-deformation of linear groups, twisted groups and algebras q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields unified models of Higgs fields in the frame of non-commutative geometry and others.

Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map τis a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H)=H ⋈ r H *cop in C. In this paper, we first show that D(H) is a quasitriangular Hopf algebra in C. Next, let H be an ordinary triangular finite-dimensional Hopf algebra. Then one can form quasitriangular Hopf algebras B(H,H) and B(H,D(H)) (in a rigid braided monoidal category) by Majid’s method associated to the ordinary Hopf algebra maps H →H and iH H → D(H), where D(H) is the Drin-fePd quantum double. We show that D (B(H,H)) and B(H,D(H)) are isomorphic Hopf algebras in the braided monoidal category.