Self-extensions of Verma Modules and Differential Forms on Opers

Verma Modules of Critical Level and Differential Forms on Opers

Let g be a simple finite-dimensional Lie algebra and ĝκ, where κ is an invariant inner product on g, the corresponding affine Kac-Moody algebra. Consider the vacuum module Vκ over ĝκ (see Section 2 for the precise definitions). According to the results of [FF, Fr], the algebra of endomorphisms of Vκ is trivial, i.e., isomorphic to C, unless κ = κc, the critical value. The algebra Endĝκc Vκc is ...

Extensions of Baer and quasi-Baer modules

Modules of the toroidal Lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$

‎Highest weight modules of the double affine Lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎Singular vectors of Verma modules are‎ ‎determined using a similar condition with horizontal affine Lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

Lectures on Wakimoto Modules, Opers and the Center at the Critical Level

Introduction 1 1. Finite-dimensional case 4 2. The case of affine algebras 8 3. Comparison of cohomology classes 19 4. Wakimoto modules of critical level 25 5. Deforming to other levels 33 6. Semi-infinite parabolic induction 40 7. Wakimoto modules over ŝl2 46 8. Intertwining operators for an arbitrary g 51 9. Description of the center of Vκc(g) 57 10. Opers and Miura opers 66 11. Identificatio...

New Generalized Verma Modules and Multilinear Intertwining Differential Operators

The present paper contains two interrelated developments. First are proposed new generalized Verma modules. They are called k Verma modules, k ∈ IN , and coincide with the usual Verma modules for k = 1. As a vector space a k Verma module is isomorphic to the symmetric tensor product of k copies of the universal enveloping algebra U(G), where G is the subalgebra of lowering generators in the sta...