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general linear Lie algebra

Definition

For $V$ a vector space, the general linear Lie algebra or endomrphism Lie algebra $\mathrm{𝔤𝔩}(V)$ 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

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

the

• orthogonal Lie algebra

• and special orthogonal Lie algebra.

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

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

the

• unitary Lie algebra

• and special unitary Lie algebra.