AlgebraModule 2002-12-31 03:54:32 2005-04-14 19:32:09 Definition algebra Jacobi identity % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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. \section{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 $v\centerdot w$ 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 \emph{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 bilinear product of two mappings is simply the composite mapping. \section{Lie algebras} 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.