Insights Introduction to the World of Algebras

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Abstract

Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois theory. Whereas Galois theory might not really be on the agenda of physicists, many algebras are: from tensor algebras as the gown for infinitesimal coordinates over Graßmann and Banach algebras for the theory of differential forms and functions up to Lie and Virasoro algebras in quantum physics and supermanifolds. This article is meant to provide a guide and a presentation of the main parts of this zoo of algebras. And we will meet many famous mathematicians and physicists on the way.
Definitions and Distinctions
Algebras
An algebra $\mathcal{A}$ is in the first place a vector space. This provides already two significant distinguishing features: the dimension of $\mathcal{A}$, i.e. whether it is an $n$- or infinite-dimensional vector space, and the characteristic of the field, i.e. the number $p$ such that
$$...

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