nLab
general linear Lie algebra

Context

\infty-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

Definition

For VV a vector space, the general linear Lie algebra or endomrphism Lie algebra 𝔤𝔩(V)\mathfrak{gl}(V) of VV is the Lie algebra whose elements are linear endomorphisms VVV \to V and whose Lie bracket is given by the commutator of endomorphisms.

This is also the endomorphism L-∞ algebra of VV

If VV is a real vector space that carries an inner product there are the sub-Lie algebras

𝔰𝔬(V)𝔬(V)𝔤𝔩(V) \mathfrak{so}(V) \hookrightarrow \mathfrak{o}(V) \hookrightarrow \mathfrak{gl}(V)

the

If VV is a complex vector space with an inner product there is

𝔰𝔬(V)𝔬(V)𝔤𝔩(V) \mathfrak{so}(V) \hookrightarrow \mathfrak{o}(V) \hookrightarrow \mathfrak{gl}(V)

the

Created on April 2, 2011 10:50:00 by Urs Schreiber (89.204.153.118)