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"Introduction to representation theory", Pavel Etingof et al., 2009

Reviewed September 4, 2023

Citation: Etingof, Pavel, et al. "Introduction to representation theory." arXiv preprint arXiv:0901.0827 (2009).

Web: https://math.mit.edu/~etingof/repb.pdf

Tags: Mathematical, Expository

This representation theory textbook has a fun premise. It was born out of a course given by Etingof to six high school students, who are now the coauthors for this book. Clearly they understood the subject quite well, and took it upon themselves to write up the notes as a textbook to push their understanding to the limit.

Another nice premise of this book: it is a proper introduction to representation theory, with special focus towards the things that Etingof cares about. Namely, the things which are applicable to quantum field theory! Hence, this is a great textbook to point to as an "introduction to representation theory for people who want to learn TQC".

For instance, it contains theorems like Theorem 9.6.4: Every finite abelian category over a field k is equivalent to the category of modules over a finite dimensional k-algebra. Seeing as fusion categories are finite abelian categories over fields, this immediately implies that we are only looking at finite dimensional k algebras. In fact, this theorem shows that we are only looking at finite dimensional k-algebras earlier than I would have expected.