# Lie algebra representation

A representation of a Lie algebra $\mathfrak{g}$ is a Lie algebra homomorphism

 $\rho:\mathfrak{g}\rightarrow\mathop{\mathrm{End}}\nolimits V,$

where $\mathop{\mathrm{End}}\nolimits V$ is the commutator Lie algebra of some vector space $V$. In other words, $\rho$ is a linear mapping that satisfies

 $\rho([a,b])=\rho(a)\rho(b)-\rho(b)\rho(a),\quad a,b\in\mathfrak{g}$

Alternatively, one calls $V$ a $\mathfrak{g}$-module, and calls $\rho(a),\,a\in\mathfrak{g}$ the action of $a$ on $V$.

We call the representation faithful if $\rho$ is injective.

A invariant subspace or sub-module $W\subset V$ is a subspace of $V$ satisfying $\rho(a)(W)\subset W$ for all $a\in\mathfrak{g}$. A representation is called irreducible or simple if its only invariant subspaces are $\{0\}$ and the whole representation.

The dimension of $V$ is called the dimension of the representation. If $V$ is infinite-dimensional, then one speaks of an infinite-dimensional representation.

Given a pair of representations, we can define a new representation, called the direct sum of the two given representations:

If $\rho:\mathfrak{g}\to\mathop{\mathrm{End}}\nolimits(V)$ and $\sigma:\mathfrak{g}\to\mathop{\mathrm{End}}\nolimits(W)$ are representations, then $V\oplus W$ has the obvious Lie algebra action, by the embedding $\mathop{\mathrm{End}}\nolimits(V)\times\mathop{\mathrm{End}}\nolimits(W)% \hookrightarrow\mathop{\mathrm{End}}\nolimits(V\oplus W)$.

 Title Lie algebra representation Canonical name LieAlgebraRepresentation Date of creation 2013-03-22 12:41:13 Last modified on 2013-03-22 12:41:13 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 16 Author mathcam (2727) Entry type Definition Classification msc 17B10 Synonym representation Related topic Dimension3 Defines irreducible Defines module Defines dimension Defines finite dimensional Defines finite-dimensional Defines infinite dimensional Defines infinite-dimensional Defines faithful Defines direct sum of representations