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 $v w$ or $v\centerdot w$ 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

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

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

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

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

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

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

for any Lie algebra M.

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