Extensions of 'thickened' Verma modules of the Virasoro algebra.

摘要

The intent of this dissertation is to calculate Ext1c&parl0; M&d4;&parl0;m&parr0;,M&d4; &parl0;m&parr0;&parr0; for any two "thickened" Verma modules M&d4;m and M&d4;l of the Virasoro Lie algebra g in a suitable category C . Consequences of this result are also discussed.; Initially, a category C consisting of h* -graded Ug, Sh -bimodules is defined, where Ug is the enveloping algebra of g , h is a Cartan subalgebra of g and Sh is the symmetric algebra generated by the abelian Lie subalgebra h . The category includes as objects highest weight modules M&d4;l for each l ∈ h* which will be called "thickened" Verma modules. A family of symmetric, contravariant bilinear forms, known as the Shapovalov forms, is introduced. Using the Jantzen filtration on Verma modules Ml of g , projective objects in the category C and a description of singular vectors of Ml , it is proved that as a right Sh -module, Ext1c&parl0; M&d4;&parl0;m&parr0;,M&d4; &parl0;m&parr0;&parr0; is isomorphic to a quotient of Sh by an ideal generated by a product of irreducible factors of the determinant of the Shapovalov form. The analogous result for the Neveu-Schwarz Lie superalgebra is also proved.; The theorem provides a conceptual explanation for the factorization of the Shapovalov determinant and the presence of only simple poles in the inverse of the Shapovalov matrix. It also makes it possible to describe the block structure of category O in terms of the nonzero localizations of Ext1c&parl0; M&d4;&parl0;m&parr0;,M&d4; &parl0;m&parr0;&parr0; at the prime ideal of Sh consisting of all elements with zero constant term. A conjectural description of the blocks of O is also given.