# Jacobi identity interpretations

The Jacobi identity^{} in a Lie algebra $\U0001d524$ has various interpretations^{} that are more transparent, whence easier to remember, than the usual form

$$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.$$ |

One is the fact that the adjoint representation^{}
^{1}^{1}Here, “$\U0001d524\U0001d529(\U0001d524)$” means the space o
endomorphisms^{} of $\U0001d524$, viewed as a vector space, with Lie
bracket on $\U0001d524\U0001d529(\U0001d524)$being commutator^{}.
$\mathrm{ad}:\U0001d524\to \U0001d524\U0001d529(\U0001d524)$ really is a representation^{}. Yet another way to formulate the identity^{} is

$$\mathrm{ad}(x)[y,z]=[\mathrm{ad}(x)y,z]+[y,\mathrm{ad}(x)z],$$ |

i.e., $\mathrm{ad}(x)$ is a derivation^{} on $\U0001d524$ for all $x\in \U0001d524$.

Title | Jacobi identity interpretations |
---|---|

Canonical name | JacobiIdentityInterpretations |

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

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

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 17B99 |