Remarks on the structure constants of the Verlinde algebra associated to sl3

Abstract

The Verlinde fusion algebra is an associative commutative algebra associated to a Wess-Zumino-Witten model of conformal field theory [V,F,GW,K,S]. Such a model is labelled by a simple Lie algebra g and a natural number k called level. The Verlinde algebra A(g, k) is a finitely generated algebra with generators V λ enumerated by irreducible g-modules admissible for the model. The structure constants N ν λ,µ of the multiplication V λ · V µ = ν N ν λ,µ V ν are non-negative integers important for applications. ( We use the formula in [K, Sec.13.35] as a definition of the structure constants.) Example 1. The algebra A(sl 2 , k) has k+1 generators V 0 , ..., V k . For fixed λ, ν and varying µ, the structure constants N µ+ν λ,µ are either zero or form the characteristic function of an interval with respect to µ. Namely, N µ+ν λ,µ = 0, if λ − ν is odd or if |ν| > λ. If λ−ν is even and |ν| < λ, then N µ+ν λ,µ = 1 for µ ∈ [(λ−ν)/2, k −(λ+ν)/2] and N µ+ν λ,µ = 0 otherwise. It is interesting that after an affine change of the variable the function N µ+ν λ,µ of µ becomes the weight function of the irreducible sl 2 -module with highest weight k − λ.In this paper we give a similar formula for the structure constants of the Verlinde algebra associated to sl 3 . Weight Functions.Let P = Z 3 /Z · (1, 1, 1) be the two dimensional weight lattice of sl 3 . LetFor a natural number k introduce coordinates on P:y 1 (µ) := (α 1 , µ), y 2 (µ) := (α 2 , µ), y 3 (µ) := k + (α 3 , µ) = k − y 1 − y 2 ,where (x, y) = x 1 y 1 + x 2 y 2 + x 3 y 3 .

A geometric description of tensor product decompositions in su(3)

“…The problem of finding this direct sum decomposition is important in many parts of physics. 15 These tensor products are also strongly related to affine sû͑3͒ k fusions, 5,13 which originate from conformal field theory. 14,16 This decomposition could also be used to find decompositions in the asymptotic limit, 10,11 which are used in the study of the noncompact rigid rotor algebra.…”