# algebra (module)

Given a commutative ring $R$, an algebra over $R$ is a module $M$ over $R$, endowed with a law of composition

$$f:M\times M\to M$$ |

which is $R$-bilinear^{}.

Most of the important algebras^{} in mathematics belong to
one or the other of two classes: the unital associative
algebras, and the Lie algebras^{}.

## 1 Unital associative algebras

In these cases, the “product” (as it is called) of two elements $v$ and $w$ of the module, is denoted simply by $vw$ or $v\bullet w$ or the like.

Any unital associative algebra is an algebra in the sense of djao (a
sense which is also used by Lang in his book *Algebra*
(Springer-Verlag)).

Examples of unital associative algebras:

– tensor algebras and quotients of them

– Cayley algebras^{}, such as the ring of quaternions

– the ring of endomorphisms of a vector space^{}, in which the bilinear
product of two mappings is simply the composite mapping.

## 2 Lie algebras

In these cases the bilinear product is denoted by $[v,w]$, and satisfies

$$[v,v]=0\text{for all}v\in M$$ |

$$[v,[w,x]]+[w,[x,v]]+[x,[v,w]]=0\text{for all}v,w,x\in M$$ |

The second of these formulas is called the Jacobi identity^{}. One proves
easily

$$[v,w]+[w,v]=0\text{for all}v,w\in M$$ |

for any Lie algebra M.

Lie algebras arise naturally from Lie groups, q.v.

Title | algebra (module) |
---|---|

Canonical name | Algebramodule |

Date of creation | 2013-03-22 13:20:50 |

Last modified on | 2013-03-22 13:20:50 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 13B99 |

Classification | msc 20C99 |

Classification | msc 16S99 |

Defines | Jacobi identity |