nLab endomorphism Lie algebra

Definition

Given a vector space $V$, then its endomorphism Lie algebra $\mathfrak{end}(V)$ is the Lie algebra whose elements are the linear maps $\phi \colon V \to V$ and whose Lie bracket is their commutator

$$[\phi_1, \phi_2] = \phi_1 \circ \phi_2 - \phi_2 \circ \phi_1 \,.$$

A Lie algebra homomorphisms $\mathfrak{g} \to \mathfrak{end}(V)$ is a Lie action or Lie algebra representation of $\mathfrak{g}$ on $V$.

Related concepts

• general linear Lie algebra