# weight (Lie algebras)

Let $\U0001d525$ be an abelian Lie algebra, $V$ a vector space^{} and
$\rho :\U0001d525\to \mathrm{End}V$ a representation. Then the representation
is said to be *diagonalisable*, if $V$ can be written as a direct
sum

$$V=\underset{\lambda \in {\U0001d525}^{*}}{\oplus}{V}_{\lambda}$$ |

where ${\U0001d525}^{*}$ is the dual space^{} of $\U0001d525$ and

$${V}_{\lambda}=\{v\in V\mid \rho (h)v=\lambda (h)v\text{for all}h\in \U0001d525\}.$$ |

Now let $\U0001d524$ be a semi-simple Lie algebra. Fix a Cartan subalgebra^{}
$\U0001d525$, then $\U0001d525$ is abelian. Let $\rho :\U0001d524\to \mathrm{End}V$ be a representation whose restriction to $\U0001d525$ is
diagonalisable. Then for any $\lambda \in {\U0001d525}^{*}$, the space
${V}_{\lambda}$ is the *weight space* of $\lambda $ with respect to
$\rho $. The *multiplicity* of
$\lambda $ with respect to $\rho $ is the dimension^{} of ${V}_{\lambda}$:

$${\mathrm{mult}}_{\rho}(\lambda ):=dim{V}_{\lambda}.$$ |

If the multiplicity of $\lambda $ is greater than zero, then $\lambda $
is called a *weight* of the representation $\rho $.

A representation of a semi-simple Lie algebra is determined by the multiplicities of its weights.

Title | weight (Lie algebras) |
---|---|

Canonical name | WeightLieAlgebras |

Date of creation | 2013-03-22 13:11:42 |

Last modified on | 2013-03-22 13:11:42 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 7 |

Author | GrafZahl (9234) |

Entry type | Definition |

Classification | msc 17B20 |

Synonym | weight |

Defines | diagonalisable |

Defines | diagonalizable |

Defines | multiplicity |

Defines | weight space |