algebra (module)
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algebra (module)
Given a commutative ring , an algebra over is a module over , endowed with a law of composition
which is -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 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
– 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 , 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.
Mathematics Subject Classification
13B99 None of the above, but in MSC2010 section 13Bxx20C99 None of the above, but in MSC2010 section 20Cxx
16S99 None of the above, but in MSC2010 section 16Sxx
