PlanetMath: Lie algebra representation

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Lie algebra representation

(Definition)

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

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

where $\mathop{\mathrm{End}}\nolimits V$ is the commutator Lie algebra of some vector space

. In other words, $\rho$ is a linear mapping that satisfies

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

Alternatively, one calls

a $\mathfrak{g}$ -module, and calls $\rho(a),\, a\in \mathfrak{g}$ the action of

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

A invariant subspace or sub-module $W\subset V$ is a subspace of 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 is called the dimension of the representation. If 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 $\mathop{\mathrm{End}}\nolimits (V)\times\mathop{\mathrm{End}}\nolimits (W)\hookrightarrow\mathop{\mathrm{End}}\nolimits (V\oplus W)$ .

"Lie algebra representation" is owned by mathcam. [ full author list (3) | owner history (2) ]

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