algebra (module)

Given a commutative ring $R$, an algebra over $R$ is amodule $M$ 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 toone or the other of two classes: the unital associativealgebras, 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 thelike.

Any unital associative algebra is an algebra in the sense of djao (asense 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

– polynomial rings

– the ring of endomorphisms of a vector space, in which the bilinearproduct 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 proveseasily

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

for any Lie algebra M.

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