∞-Lie theory

# Contents

## Definition

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

This is also the endomorphism L-∞ algebra of $V$

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

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

the

If $V$ is a complex vector space with an inner product there is

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

the

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