# associator

Let $A$ be a non-associative algebra over a field. The *associator ^{}* of $A$, denoted by $[,,]$, is a trilinear (http://planetmath.org/multilinear) map from $A\times A\times A$ to $A$ given by:

$$[a,b,c]=(ab)c-a(bc).$$ |

Just as the commutator^{} measures how close an algebra^{} is to being commutative^{}, the associator measures how close it is to being associative. $[,,]=0$ identically iff $A$ is associative.

## References

- 1 R. D. Schafer, An Introduction on Nonassociative Algebras, Dover, New York (1995).

Title | associator |
---|---|

Canonical name | Associator |

Date of creation | 2013-03-22 14:43:21 |

Last modified on | 2013-03-22 14:43:21 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 10 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 17A01 |

Related topic | AlternativeAlgebra |

Related topic | PowerAssociativeAlgebra |

Related topic | FlexibleAlgebra |

Related topic | Commutator |

Defines | anti-associative |