# Lie algebra representation

A representation^{} of a Lie algebra^{} $\U0001d524$ is a Lie algebra homomorphism^{}

$$\rho :\U0001d524\to EndV,$$ |

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

$$\rho ([a,b])=\rho (a)\rho (b)-\rho (b)\rho (a),a,b\in \U0001d524$$ |

Alternatively, one calls $V$ a $\U0001d524$-module, and calls $\rho (a),a\in \U0001d524$ 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 $a\in \U0001d524$. 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 $\rho :\U0001d524\to End(V)$ and $\sigma :\U0001d524\to End(W)$ are representations, then $V\oplus W$ has the obvious Lie algebra action, by the embedding $End(V)\times End(W)\hookrightarrow End(V\oplus W)$.

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 |