Given a commutative ring , an algebra over is a module over , endowed with a law of composition
which is -bilinear.
1 Unital associative algebras
In these cases, the “product” (as it is called) of two elements and of the module, is denoted simply by or 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
2 Lie algebras
In these cases the bilinear product is denoted by , and satisfies
The second of these formulas is called the Jacobi identity. One proves easily
for any Lie algebra M.
Lie algebras arise naturally from Lie groups, q.v.
|Date of creation||2013-03-22 13:20:50|
|Last modified on||2013-03-22 13:20:50|
|Last modified by||mathcam (2727)|