Extensions of “thickened” Verma modules of the Virasoro algebra

✍ Scribed by Karen S. Brown

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 350 KB
• Volume: 269
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00452-6

✦ Synopsis

Let g denote the Virasoro Lie algebra, h its Cartan subalgebra, and S(h) the symmetric algebra on h. In this paper we consider "thickened" Verma modules M(λ) which are (U (g), S(h))-bimodules satisfying M(λ) ⊗ S(h) C ∼ = M(λ) where M(λ) is the usual Verma module with highest weight λ ∈ h * . We determine Ext 1 ( M(µ), M(λ)) to be S(h)/φ µ,λ S(h) where φ µ,λ is, up to a C-algebra automorphism of S(h), a product of irreducible factors of the determinant of the Shapovalov matrix. This result provides a conceptual explanation of the factorization of the Shapovalov determinant and implies that the inverse of the Shapovalov matrix has only simple poles.