nLab
general linear Lie algebra

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∞-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

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∞-Lie groupoids

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∞-Lie algebras

Contents

Definition

For V a vector space, the general linear Lie algebra or endomrphism Lie algebra 𝔤𝔩(V) of V is the Lie algebra whose elements are linear endomorphisms V→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

𝔰𝔬(V)↪𝔬(V)↪𝔤𝔩(V)\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

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)
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