04 Apr 2014 No Comments

# Lie Groups: A Reading Guide

The present web document is essentially all the notes I have made to myself and proofs I have constructed for myself to help me learn and understand the theory of Lie groups.

These notes will therefore work for those multicellular organisms who think like I do. Naturally, this is not everybody. This document is a product of the web era, and there is a great deal of knowledge on the Internet which you can use to help you if my explanations are not clear. Google the terms you find unwonted or thorny and read far and wide. I should be very interested to hear from anybody seeking clarification of my explanations for, in corresponding with you:

- I may learn something new about my subject matter, Lie theory;
- I will certainly learn something about technical writing in pondering what it is you have difficulty in reading.

Although I was learning and applying Lie theory to engineering and physics problems in my day job from about 1993 onwards, for me, the central reference I used to teach myself the theory of Lie groups, or at least the one that got me off to a sound beginning:

I see this book as an example of excellent technical writing. Here are links to two reviews, by Anthony Knapp and Peter E. Trapa. Prof. Knapp gives the book by Andrew Baker a bit of a beating in his review: I have no experience of the book and so are neither in agreement nor disagreement with the comments about it.

Naturally, when I heard tell of a publication of a book by my former teacher at Monash University:

John Stillwell, “*Naïve Lie Theory*“, Springer Science + Business Media, New York, 2008

well that certainly got my attention and I learnt several new ways of thinking about Lie groups from this book. As with all things Stillwell, I would recommend that anyone seeking to learn a topic thoroughly from scratch should read a Stillwell publication if there is one on the topic. Not only will you learn sound mathematics, you will also learn a great deal of history. I took four of Prof. Stillwell’s courses in all in the mid 1990s, in general group and Galois theory as well two topics (topology and Riemann Surfaces) which were very much subtopics of his books “*Classical Topology and Combinatorial Group Theory**“* (Springer-Verlag Graduate Texts in Mathematics) and “*Geometry of Surfaces*“, (Springer Verlag, New York, 1992). I hope he wouldn’t mind my saying that his gift for explanation did not appear magically: sheer hard work was evident in his lecture notes and he gave me the impression of someone never happy with an explanation as it was, he was always striving for a simpler and cleaner one for everything he lectured. Perhaps a mathematical analogue of Richard Feynman as a teacher. In his Galois theory lectures I and a few other students were lucky enough to join him as fellow learners: he was still getting his lectures straight and, in his honest way, warned us that this would be the case. So we “read” Emil Artin’s “Galois Theory” toge ther. Thus I got to see first hand the staggering amount of work he put into building his explanations.

One last book which has become wonted to me over the last few months is

Harriet Pollatsek, “*A Problem Oriented Introduction via Matrix Groups*“

and the reason I am keen on this book can be seen in its title: a Problem Oriented Approach. Needless to say, the book is true to this title. Essential to the learning of any subject involves *play* and Prof. Pollatsek’s is what I call a *playful* book. I do not mean the word *play* to bear disparaging connotations: far from it. Throughout the time I have built up my own notes on Lie theory, I have been the primary carer of my daughter and son, now eight and four years old. Over that time I have witnessed the absolute necessity of play to learning. Furthermore, academic education research bears out more and more the idea that the capacity for abstract mathematical thought later in life is wrought early – the kindergarten and early primary years – in a child’s life, through play-oriented learning and certainly not through mindless drilling in arithmetic rules that characterised early mathematics education forty years ago. I have spent a great deal of time not only watching my own children, but those of other (mostly mothers) parents and have witnessed that one of the instincts most manifest in the human child is the learning instinct. In my observation, play seems to split into four broad categories: bonding, allegorical, rehearsal and exploration. The middle two are stereotyped by doll play, but they are not only this. The first three build what I call the child’s first great literacy: the connexion with fellow social animals and readiness to be a social animal with worthwhile personal relationships in one’s own right. The last – exploration – builds what I call the second great literacy: the child’s grasping of the physics and logic of the World about her or him. My own definition of mathematics is as the *science that describes the relationships between categories, objects and processes in the World*. Exploratory play is pretty much wholly aimed at the building of the child’s wherewithal for that description, and inferences one can make from it. Think of the little boy or girl learning to build a block stack, or fitting shapes into a shape sorter: it’s pure exploration, unbidden, spontaneous and simply “for the sheer fun of it”. My partner Merrindal described our little daughter Nakira thus: one of her first thoughts on finding something new seemed to be the question “Does this come off? How does it fit with other things?”. We joked that she was the little topologist and nicknamed her *Epsilon Alexandrof* (one of our many nicknames for her which brought us great joy and all emphasised the wonder of this little being exploring her World). So, in this light, the outcome of the latest research is unsurprising. Problem solving, with a text like Prod Pollatsek’s, is the analogue of building block stacks and utterly indispensible. Child’s play is never, ever to be sneered at.

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