Minoru Wakimoto

Fock representations of the affine Lie algebraA 1 (1)

The aim of this note is to show that the affine Lie algebraA1(1) has a natural family πμ, υ,v of Fock representations on the spaceC[xi,yj;i ∈ ℤ andj ∈ ℕ], parametrized by (μ,v) ∈C2. By corresponding the highest weightΛμ, υ of πμ, υ to each (μ,ν), the parameter spaceC2 forms a double cover of the weight spaceCΛ0⊕C −1 with singularities at linear forms of level −2; this number is (−1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA1(1)-modules for generic (μ,v).

Modular and conformal invariance constraints in representation theory of affine algebras

On etudie les contraintes de linvariance modulaire et conforme dans la theorie de la representation des algebres de Kac-Moody affines. On considere les regles de ramification dune representation integrable de poids le plus eleve dune algebre affine g par rapport a une sous-algebre affine p. On etudie certaines fonctions generatrices pour ces coefficients de ramification

Integrable Highest Weight Modules over Affine Superalgebras and Number Theory

The problem of representing an integer as a sum of squares of integers has had a long history. One of the first after antiquity was A. Girard who in 1632 conjectured that an odd prime p can be represented as a sum of two squares iff p ≡ 1 mod 4, and P. Fermat in 1641 gave an “irrefutable proof” of this conjecture. The subsequent work on this problem culminated in papers by A.M. Legendre (1798) and C.F. Gauss (1801) who found explicit formulas for the number of representations of an integer as a sum of two squares. C.G. Bachet in 1621 conjectured that any positive integer can be represented as a sum of four squares of integers, and it took efforts of many mathematicians for about 150 years before J.-L. Lagrange gave a proof of this conjecture in 1770.

Characters and fusion rules for

Using the cohomological approach toW-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld-Sokolov reduction.

W

Using the cohomological approach toW-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld-Sokolov reduction.

-algebras via quantized Drinfel'd-Sokolov reduction

0.1. One of the basic problems of representation theory is to find a decomposition of an irreducible representation of a group with respect to a subgroup. Namely, suppose that we have a representation π of a group G in a vector space V and suppose that with respect to a subgroup S this representation decomposes into a direct sum of irreducible representations: n n

Branching functions for winding subalgebras and tensor products

pi = { oplus_i}{pi_i},;V = { oplus_i}{V_i}

On classification of poisson vertex algebras

n n(3.1) n nGiven an irreducible representation σ of S one denotes by [π: σ] the number of representations of S equivalent to σ in this decomposition, and calls this number a branching coefficient. An important problem is to compute the branching coefficients.