[[Lie theory MOC]]
# Lie algebra representation

A **representation** $\pi : \mathfrak{g} \to \mathfrak{gl}(V)$ of a [[Lie algebra]] $\mathfrak{g}$ is a [[Lie algebra homomorphism]] from $\mathfrak{g}$ to the [[commutator]] algebra of the [[Endomorphism ring]] of $V$. #m/def/lie
Thus for any $X,Y \in \mathfrak{g}$,
$$
\begin{align*}
\pi([X,Y]) = \pi(X)\pi(Y) - \pi(Y)\pi(X)
\end{align*}
$$
A space $V$ carrying a representation of $\mathfrak{g}$ is sometimes called a $\mathfrak{g}$-[[Module over a Lie algebra|module]],[^1988] though this is an abuse of terminology.

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], p. 5

## Properties

- Every Lie algebra has a natural representation on itself, the [[adjoint Lie algebra representation]].
- Every representation $\Gamma$ of a [[Lie group]] induces an [[Infinitesimal representation]] $d\Gamma$ of the corresponding Lie algebra.
- [[Sum of commuting Lie algebra representations]]
- [[Tensor product of Lie algebra representations]]

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#state/tidy | #lang/en | #SemBr