# (special) unitary Lie algebra

Let $V$ be a vector space^{} over a field $K$ admitting an involution^{} $\sigma :K\to K$, and let $B:V\times V\to \mathbb{F}$ be a http://planetmath.org/node/SesquilinearFormsOverGeneralFieldshermitian form^{} relative to $\sigma $. Then the *unitary Lie algebra* $\U0001d532(V,B)$, or just $\U0001d532(V)$, consists of the linear transformations $T$ satisfying

$$B(Tx,y)+B(x,Ty)=0,$$ |

for all $x,y\in V$. This is a Lie algebra^{} over $k=\{\alpha \in K\mid {\alpha}^{\sigma}=\alpha \}$, but not over $K$ in the case that $K\ne k$ (because $B$ is linear in the first, but not in the second variable).

The *special unitary Lie algebra* $\U0001d530u(V,B)$, or just $\U0001d530u(V)$, consists of those linear transformations in $\U0001d532(V,B)$ with trace zero.

Title | (special) unitary Lie algebra |
---|---|

Canonical name | specialUnitaryLieAlgebra |

Date of creation | 2013-03-22 18:45:21 |

Last modified on | 2013-03-22 18:45:21 |

Owner | pan (17366) |

Last modified by | pan (17366) |

Numerical id | 6 |

Author | pan (17366) |

Entry type | Definition |

Classification | msc 17B99 |

Synonym | unitary algebra |

Synonym | special unitary algebra |

Defines | unitary algebra |

Defines | special unitary algebra |