Symmetries Unveiled: A Journey Through Representation Theory

by Oliver Vella
Published by the author, no publication date
reviewed by T. Nelson

his book tells you why you should never let ChatGPT try to write a book on some advanced topic in mathmatics for you. Not only can't it do any mathematical reasoning, it can't even generate math formulas. Math is just not ChatGPT's forte.

It's hard to prove that a chatbot is the author, of course, but there are clues. One clue is when it simply tells you over and over how wonderful the subject is and how there are ‘profound connections’ between various things, but says nothing about what those things actually are.

Another clue is that the title is seemingly stolen from an authentic graduate-level math textbook on the subject: A Journey Through Representation Theory: From Finite Groups to Quivers via Algebras by Caroline Gruson and Vera Serganova (Springer, 2018).

What is representation theory? In Representations of Compact Lie Groups by Theodor Bröcker and Tammo tom Dieck (Springer, 2010), which I'm reading now, the authors describe it this way:

A representation of the Lie group G on the (finite-dimensional complex) vector space V is a continuous action ρ:G × V → V on V such that for each g ∈ G the translation l_g: v ↦ ρ(g,v) is a linear map. [p.65, in the chapter ‘Elementary Representation Theory’]

That might sound like pure gibberish to the average person, but at least it's a definition. Here's what Vella's book says:

As we venture into the realm of quantum groups, the landscape becomes an intriguing tapestry where the classical notions of symmetry are elegantly retold with a quantum twist. . . . As we delve into the specifics of their symmetries, we'll encounter how these beautifully complex structures not only emulate but transcend the limitations of their classical counterparts, offering foundational insights that enrich our understanding of the quantum realm.

That's from Chapter 8, “Quantum groups and their symmetries.” And indeed, the chapter titles sound okay: ‘Introduction to Lie Algebras’, ‘Kac-Moody Algebras’, ‘Symmetry Breaking and Its Consequences.’

But in fact no ‘delving’ or ‘venturing’ ever occurs: the rest of the chapter, like the rest of the book, just continues with more meaningless upbeat generalities. While it occas­ion­ally mentions technical terms like non-commutative algebras and universal enveloping algebra, it never explains them.

Who is the intended audience? Probably some poor defense­less kid who's forced to write a report on some math topic but was told not to use a chatbot.

No page numbers, no index, no diagrams, no equations; large size text. Pages are non-reflective paper. Makes excellent kindling.

apr 03, 2025