Quasifinite Representations of Classical Lie Subalgebras of W_1+∞

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• 作者：Kac ; Victor G. ; Wang ; Weiqiang ; Yan ; Catherine H.
• 刊名：Advances in Mathematics
• 出版年：1998
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：October 15, 1998
• 卷：139
• 期：1
• 页码：56-140
• 文件大小：998 K

摘要

We show that there are precisely two, up to conjugation, anti-involutionsσ_±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 −σ_±, 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]/(u^m+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 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 moduleV_cof D^±carries a vertex algebra structure and establish a relationship betweenV_cforcZ and W-algebras.