Insights Introduction to the World of Algebras

fresh_42
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
2024 Award
Messages
20,626
Reaction score
27,753
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
$$...

Continue reading...
 
Last edited:
  • Like
  • Informative
Likes bhobba, pinball1970, DeBangis21 and 6 others
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...

Similar threads

Back
Top