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# 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 $\mathop{\mathrm{End}}\nolimits(V)\times\mathop{\mathrm{End}}\nolimits(W)% \hookrightarrow\mathop{\mathrm{End}}\nolimits(V\oplus W)$.

## Mathematics Subject Classification

17B10*no label found*

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## Corrections

faithful, tensor product, etc. by bwebste ✓

Tensor product? by archibal ✓

subsspace by mps ✓

Tensor product by lars_h ✓