Quasifinite representations of classical Lie subalgebras of W_1+infty

1998/01/29 by Victor G. Kac, Weiqiang Wang, Catherine H. Yan
Mathematics · Physics and Astronomy · #math.QA #hep-th #math.RT

paper · pdf

published as Adv. in Math. 39 (1998) 56--140 · Latex, 77 pages

arxiv created 1998/01/29 · arxiv updated 2009/11/30

Abstract

We show that there are precisely two, up to conjugation, anti-involutions sigma_± of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D^± of the Lie subalgebra of this algebra fixed by - sigma_±, and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the truncated polynomial algebra C[u] / (u^m+1) and its classical Lie subalgebras of B, C and D types. Character formulas for positive primitive representations of D^± (including all the unitary ones) are obtained. We also realize a class of primitive representations of D^± in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum module V_c of D⁺ carries a vertex algebra structure and establish a relationship between V_c for half-integral central charge c and W-algebras.

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