Fusion Rules for Affine Kac-Moody Algebras

We review some contributions on fusion rules that were inspired by the work of Sharp, in particular, the generating-function method for tensor-product coefficients that he developed with Patera. We also review the Kac-Walton formula, the concepts of threshold level, fusion elementary couplings, fusion generating functions and fusion bases. We try to keep the presentation elementary and exemplify each concept with the simple $\su(2)_k$ case.

Fusion product originates in the algebraisation of the operator product expansion in conformal field theory. Read and Saleur (2007) introduced an analogue of fusion for modules over associative algebras, for example those appearing in the description of 2d lattice models. The article extends their definition for modules over the affine Temperley-Lieb algebra $\atl n$. Since the regular Temperley-Lieb algebra $\tl n$ is a subalgebra of the affine $\atl n$, there is a natural...

These are informal lecture notes for a three-hour minicourse on Kac-Moody groups, given at the workshop "Kac-Moody geometry" in July 2023 in Kiel. They provide a concise overview of the book "An introduction to Kac-Moody groups over fields", EMS Textbooks in Mathematics (2018). They assume a previous familiarity with the (very) basics of Kac-Moody algebras. For readers unfamiliar with the latter topic, short "Prerequisites" notes (referenced within the text) are also freely a...

An irreducible weight module of an affine Kac-Moody algebra $\mathfrak{g}$ is called dense if its support is equal to a coset in $\mathfrak{h}^{*}/Q$. Following a conjecture of V. Futorny about affine Kac-Moody algebras $\mathfrak{g}$, an irreducible weight $\mathfrak{g}$-module is dense if and only if it is cuspidal (i.e. not a quotient of an induced module). The conjecture is confirmed for $\mathfrak{g}=A_{2}^{\left(1\right)}$, $A_{3}^{\left(1\right)}$ and$A_{4}^{\left(1\ri...

In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\widetilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\widetilde{\mathfrak{g}}$-modules on which the action of each root vector in $\widetilde{\mathfrak{n}}_+$ is locally finite, where $\widetilde{\mathfrak{n}}_+$ is the locally nilpotent subalgebra ...

Affine Kac-Moody algebras give rise to interesting systems of differential equations, so-called Knizhnik-Zamolodchikov equations. The monodromy properties of their solutions can be encoded in the structure of a modular tensor category on (a subcategory of) the representation category of the affine Lie algebra. We discuss the relation between these solutions and physical correlation functions in two-dimensional conformal field theory. In particular we report on a proof for the...

Invited lecture at the International Congress of Mathematicians, Zuerich, August 3-11, 1994 (extended version), reviews free field realizations of affine Kac-Moody and W-algebras and their applications.

In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak g$. Let $P\_+$ be the set of dominant integral weights. For $\lambda\in P\_+$ , $L(\lambda)$ denotes the irreducible, integrable, highest weight representation of g with highest weight $\lambda$. Let $P\_{+,\mathbb Q}$ be the rational convex cone generated by $P\_+$. Consider the tensor cone $\Gamma(\mathfrak g) := \{(\lambda\_...

The total multiplicity in the decomposition into irreducibles of the tensor product i x j of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them sum_k N_{i j}^{k}= sum_k N_{ibar j}^{k}. This also applies to the fusion multiplicities of affine algebras in conformal WZW theories. In that context, the statement is equivalent to a property of the modular S matrix, Sigma(k)= sum_j S_{j k}=0 if k is a complex representation. Curious...

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues known as Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side: More precisely, we introduce OSAKAs, the algebraic str...