Introduction to the World of Algebras

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Thread starter fresh_42
Start date Jul 8, 2023
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Algebra Introduction Mathematics

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SUMMARY

The discussion centers on Richard Pierce's book about associative algebras, emphasizing the relevance of various algebras in physics beyond Galois theory. Key algebras mentioned include tensor algebras, Graßmann algebras, Banach algebras, Lie algebras, and Virasoro algebras, all of which play significant roles in quantum physics and supermanifolds. The article serves as a comprehensive guide to understanding these algebras and their applications, highlighting the contributions of notable mathematicians and physicists.

PREREQUISITES

Understanding of vector spaces and their dimensions
Familiarity with Galois theory
Knowledge of quantum physics concepts
Basic comprehension of differential forms and functions

NEXT STEPS

Explore the properties of tensor algebras in mathematical physics
Study the applications of Graßmann and Banach algebras in differential geometry
Investigate the role of Lie algebras in quantum mechanics
Learn about Virasoro algebras and their significance in string theory

USEFUL FOR

Mathematicians, physicists, and students interested in advanced algebraic structures and their applications in theoretical physics.

Jul 8, 2023

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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
$$...

Last edited: May 17, 2025