Quasifinite Representations of Classical Lie Subalgebras of W1+ Academic Article uri icon

abstract

  • We show that there are precisely two, up to conjugation, anti-involutionsof the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension Dof the Lie subalgebra of this algebra fixed by -, 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 algebra C[u]/(um+1) and its classical Lie subalgebras ofB,CandDtypes. Character formulas forpositive primitiverepresentations of D(including all the unitary ones) are obtained. We also realize a class of primitive representations of Din 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 moduleVcof Dcarries a vertex algebra structure and establish a relationship betweenVcforc12Z and W-algebras. 1998 Academic Press.

published proceedings

  • Advances in Mathematics

author list (cited authors)

  • Kac, V. G., Wang, W., & Yan, C. H.

citation count

  • 53

complete list of authors

  • Kac, Victor G||Wang, Weiqiang||Yan, Catherine H

publication date

  • January 1998