algebra (module)

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


which is R-bilinearPlanetmathPlanetmath.

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

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 vw 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 algebrasMathworldPlanetmathPlanetmath, such as the ring of quaternions

– the ring of endomorphisms of a vector spaceMathworldPlanetmath, 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 for all vM
[v,[w,x]]+[w,[x,v]]+[x,[v,w]]=0 for all v,w,xM

The second of these formulas is called the Jacobi identityMathworldPlanetmath. One proves easily

[v,w]+[w,v]=0 for all v,wM

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