Simple Conformal Algebras Generated by Jordan Algebras

1 Background and Motivation We start with an example of affine Kac-Moody algebras and the Virasoro algebra. In this talk, F will be a field with characteristic 0, and all the vector spaces are assumed over F. Denote by Z the ring of integers and by N the set of nonnegative integers. Let 2 ≤ n ∈ N. Set sl(n,F) = {A ∈ Mn×n(F) | tr A = 0}, (1.1) 〈A,B〉 = tr AB for A,B ∈ sl(n,F), (1.2) where Mn×n(F) is the algebra of n×n matrices. Then sl(n,F) forms a simple Lie algebra with the Lie bracket [A,B] = AB −BA for A,B ∈ sl(n,F), (1.3) and 〈·, ·〉 forms a symmetric invariant bilinear form, that is, 〈[A,B], C〉 = 〈A, [B,C]〉 for A,B ∈ sl(n,F). (1.4) Let t be an indeterminant and set ŝl(n,F) = sl(n,F)⊗F F[t, t ]⊕ Fκ⊕ Fd, (1.5) where κ and d are symbols serving as base elements. We define a Lie bracket on ŝl(n,F) by [u⊗ t, v ⊗ t ] = [u, v]⊗ t +mδl+j,0, [d, u⊗ t ] = lu⊗ t (1.6) for u, v ∈ sl(n,F), l, j ∈ Z and [ŝl(n,F), κ] = 0. (1.7) The Lie algebra (ŝl(n,F), [·, ·]) is an affine Kac-Moody algebra. Define the generating function u(z) = ∑