Verma module


Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebraMathworldPlanetmath, and 𝔟 a Borel subalgebra. We work over a field F. Given a weight λ∈𝔥*, let Fλ be the 1-d dimensional 𝔟-module on which 𝔥 acts by multiplicationPlanetmathPlanetmath by λ, and the positive root spaces act trivially. Now, the Verma moduleMathworldPlanetmath Mλ of the weight λ is the 𝔤-module induced from Fλ, i.e.

Mλ=𝒰⁢(𝔤)⊗𝒰⁢(𝔟)Fλ.

Using the Poincaré-Birkhoff-Witt theorem we see that as a vector space Mλ is isomorphicPlanetmathPlanetmathPlanetmath to 𝒰⁢(𝔫¯), where 𝔫¯ is the sum of the negative weight spaces (so 𝔤=𝔟⊕𝔫¯). In particular Mλ is infinite dimensional.

We say a 𝔤-module V is a highest weight module if it has a weight μ∈𝔥* and a non-zero vector v∈Vμ with X⁢v=0 for any X in a positive root space and such that V is generated as a 𝔤-module by v. The Verma module Mλ is a highest weight module and we fix a generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 1⊗1.

The most important property of Verma modules is that they are universalPlanetmathPlanetmathPlanetmath amongst highest weight modules, in the following sense. If V is a highest weight module generated by v which has weight λ then there is a unique surjectivePlanetmathPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath Mλ→V which sends 1⊗1 to v. That is, all highest weight modules with highest weight λ are quotients of Mλ. Also, Mλ has a unique maximal submodule, so there is a unique irreducible representation with highest weight λ. If λ is dominant and integral then this module is finite dimensional.

Title Verma module
Canonical name VermaModule
Date of creation 2013-03-22 13:12:13
Last modified on 2013-03-22 13:12:13
Owner owenjonesuk (12024)
Last modified by owenjonesuk (12024)
Numerical id 8
Author owenjonesuk (12024)
Entry type Definition
Classification msc 17B20
Defines highest weight module