In [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category $\mathcal{A}$ extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) is described using the space of "stringnets with boundary conditions" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory in [CY1993].

An extension of the Poincare-Birkhoff fixed point theorem to noninvariant under area-preserving homeomorphism annuli is considered. Unlike the well-known W.-Y. Ding's theorem [7], the inner boundary component of an annulus is not assumed to be star-shaped, while the outer boundary component is star-shaped. The existence of at least two fixed points for area preserving homeomorphism satisfying some twist condition is proved.

We consider infinitely connected domain continuous mappings, satisfying the condition of interior boundary components interchange. The fixed point theorem is proved.

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.

We represent several results on the existence of fixed points of the arbitrary topological annulus maps. The celebrated boundary twist condition of the Poincaré–Birkhoff theorem is replaced by its essentially weakest analogue for two points in the annulus. We do not use area-preserving and homeomorphic maps. We consider continuous maps satisfying some modification of T. Ding’s bend condition and a special monotonicity condition. We also reject the often used 2π-periodicity angle displacement condition. Besides, we obtain the description of the fixed points set structure for continuously differentiable maps.

Let Γbe a Dynkin diagram of type A,D,E and let R denote the corresponding root system. In this paper we give a categorical construction of R from Γ. Instead of choosing an orientation of Γand studying representations of the associated quiver, we study representations of a canonical quiver associated to Γ. This construction is very closely related to the preprojective algebra of Γ. In particular, the construction gives a certain periodicity result about the preprojective algebra.

With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras.

In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between these notions.

In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between these notions.

This is an overview article on compact Lie groups and their representations, written for the Encyclopedia of Mathematical Physics to be published by Elsevier.

Let $G$ be a finite subgroup in SU(2), and $Q$ the corresponding affine Dynkin diagram. In this paper, we review the relation between the categories of $G$-equivariant sheaves on $P^1$ and $Rep Q_h$, where $h$ is an orientation of $Q$, constructing an explicit equivalence of corresponding derived categories.

: In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G , is it possible to describe in a suitable sense the “quotient” category ?/ G ? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/ G turns out to be the category of representation of Drinfeld's double D ( G ). This should be considered as the category theory analog of the topological identity { pt }/ G = BG . This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ? G , is equivalent as a tensor category to the category of representations of Drinfeld's double D ( G ). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module.

The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{\pi\i/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to sl^(2) at level k=l-2. We show that ``finite subgroups in U_q sl(2)'' are classified by Dynkin diagrams of types A_n, D_{2n}, E_6, E_8 with Coxeter number equal to $l$, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in (sl(2))_k conformal field theory.

Introduction Braided tensor categories Ribbon categories Modular tensor categories 3-dimensional topological quantum field theory Modular functor Moduli spaces and complex modular functor Wess-Zumino-Witten model Bibliography Index Index of notation.