Highest-Weight Theory for Truncated Current Lie Algebras

Let g denote a Lie algebra over a field of characteristic zero, and let T(g) denote the tensor product of g with a ring of truncated polynomials. The Lie algebra T(g) is called a truncated current Lie algebra, or in the special case when g is finite-dimensional and semisimple, a generalized Takiff algebra. In this paper a highest-weight theory for T(g) is developed when the underlying Lie algebra g possesses a triangular decomposition. The principal result is the reducibility criterion for the Verma modules of T(g) for a wide class of Lie algebras g, including the symmetrizable Kac-Moody Lie algebras, the Heisenberg algebra, and the Virasoro algebra. This is achieved through a study of the Shapovalov form.Comment: 42 pages. An extract from the author's PhD thesis. See also: http://www.maths.usyd.edu.au/u/benw

Representations of Truncated Current Lie Algebras

Let g denote a Lie algebra, and let T(g) denote the tensor product of g with a ring of truncated polynomials. The Lie algebra T(g) is called a truncated current Lie algebra. The highest-weight representation theory of T(g) is developed, and a reducibility criterion for the Verma modules is described.Comment: 5 pages. A summary of the article 'Highest-Weight Theory for Truncated Current Lie Algebras' published on the arxi

Imaginary highest-weight representation theory and symmetric functions

Affine Lie algebras admit non-classical highest-weight theories through alternative partitions of the root system. Although significant inroads have been made, much of the classical machinery is inapplicable in this broader context, and some fundamental questions remain unanswered. In particular, the structure of the reducible objects in non-classical theories has not yet been fully understood. This question is addressed here for affine sl(2), which has a unique non-classical highest-weight theory, termed "imaginary". The reducible Verma modules in the imaginary theory possess an infinite descending series, with all factors isomorphic to a certain canonically associated module, the structure of which depends upon the highest weight. If the highest weight is non-zero, then this factor module is irreducible, and conversely. This paper examines the degeneracy of the factor module of highest-weight zero. The intricate structure of this module is understood via a realization in terms of the symmetric functions. The realization permits the description of a family of singular (critical) vectors, and the classification of the irreducible subquotients. The irreducible subquotients are characterized as those modules with an action given in terms of exponential functions, in the sense of Billig and Zhao.Comment: 23 page