nLab
Verma module

Context

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

Given a semisimple Lie algebra 𝔤\mathfrak{g} over a field FF with Borel subalgebra? 𝔟𝔤\mathfrak{b}\subset\mathfrak{g} a Verma module is the induced module

U(𝔤) U(𝔟)F λ U(\mathfrak{g})\otimes_{U(\mathfrak{b})} F_\lambda

where F λF_\lambda is one dimensional representation of U(𝔟)U(\mathfrak{b}) corresponding to the character λ\lambda of 𝔟\mathfrak{b}.

This construction is used also for many other Lie algebras with triangular decomposition, for quantized enveloping algebras and many other generalizations.

Revised on January 21, 2015 22:56:56 by Urs Schreiber (88.100.66.95)