2007/05/09 by Benjamin J. Wilson, Wilson, Benjamin J. · 2 citations
Mathematics · Physics and Astronomy · #17B10 #17B65 #17B67 #17B68 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT) #math-ph #math.MP #math.RT #msc:17B10 #msc:17B65 #msc:17B67 #msc:17B68
paper · pdf · doi:10.48550/arxiv.0705.1203
42 pages. An extract from the author's PhD thesis. See also: http://www.maths.usyd.edu.au/u/benw/
arxiv created 2007/05/09 · openalex publication_date 2007/05/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.