## Exercise Your Brain

As many may know or have already heard, Dr. Mike Miller, a retired mathematician from RAND and long-time math professor at UCLA, is offering a course on Introduction to Lie Groups and Lie Algebras this fall through UCLA Extension. Whether you’re a professional mathematician, engineer, physicist, physician, or even a hobbyist interested in mathematics you’ll be sure to get something interesting out of this course, not to mention the camaraderie of 20-30 other “regulars” with widely varying backgrounds (actors to surgeons and evolutionary theorists to engineers) who’ve been taking almost everything Mike has offered over the years (and yes, he’s THAT good — we’re sure you’ll be addicted too.)

## “Beginners” Welcome!

Even if it’s been years since you last took Calculus or Linear Algebra, Mike (and the rest of the class) will help you get quickly back up to speed to delve into what is often otherwise a very deep subject. If you’re interested in advanced physics, quantum mechanics, quantum information or string theory, this is one of the topics that is de rigueur for delving in deeply and being able to understand them better. The topic is also one near and dear to the hearts of those in robotics, graphics, 3-D modelling, gaming, and areas utilizing multi-dimensional rotations. And naturally, it’s simply a beautiful and elegant subject for those who have no need to apply it to anything, but who just want to meander their way through higher mathematics for the fun of it (this will comprise the largest majority of the class by the way.)

Whether you’ve been away from serious math for decades or use it every day or even if you’ve never gone past Calculus or Linear Algebra, this is bound to be the most entertaining thing you can do with your Tuesday nights in the fall. If you’re not sure what you’re getting into (or are scared a bit by the course description), I highly encourage to come and join us for at least the first class before you pass up on the opportunity. I’ll mention that the greater majority of new students to Mike’s classes join the ever-growing group of regulars who take almost everything he teaches subsequently. (For the reticent, I’ll mention that one of the first courses I took from Mike was Algebraic Topology which generally requires a few semesters of Abstract Algebra and a semester of Topology as prerequisites. I’d taken neither of these prerequisites, but due to Mike’s excellent lecture style and desire to make everything comprehensible, I was able to do exceedingly well in the course.) I’m happy to chat with those who may be reticent. Also keep in mind that you can register to take the class for a grade, pass/fail, or even no grade at all to suit your needs/lifestyle.

As a group, some of us have a collection of a few dozen texts in the area which we’re happy to loan out as well. In addition to the one recommended text (Mike always gives such comprehensive notes that any text for his classes is purely supplemental at best), several of us have also found some good similar texts:

- Stillwell, John. Naïve Lie Theory. (Springer, 2008) ISBN: 9780387782157
- Baker, Andrew. Matrix Groups: An Introduction to Lie Group Theory (Springer, 2002) ISBN: 9781447101833
- Tapp, Kristopher. Matrix Groups for Undergraduates (AMS, 2005) ISBN: 0821837850

Given the breadth and diversity of the backgrounds of students in the class, I’m sure Mike will spend some reasonable time at the beginning [or later in the class, as necessary] doing a quick overview of some linear algebra and calculus related topics that will be needed later in the quarter(s).

Further information on the class and a link to register can be found below. If you know of others who might be interested in this, please feel free to forward it along – the more the merrier.

I hope to see you all soon.

## Introduction to Lie Groups and Lie Algebras

MATH X 450.6 / 3.00 units / Reg. # 249254W

Professor: Michael Miller, Ph.D.

Start Date: 9/30/2014

Location UCLA: 5137 Math Sciences Building

Tuesday, 7-10pm

September 30 – December 16, 2014

11 meetings total *(no mtg 11/11)*

Register here: https://www.uclaextension.edu/Pages/Course.aspx?reg=249254

### Course Description

A *Lie group* is a differentiable manifold that is also a group for which the product and inverse maps are differentiable. A *Lie algebra* is a vector space endowed with a binary operation that is bilinear, alternating, and satisfies the so-called Jacobi identity. This course, the first in a 2-quarter sequence, is an introductory survey of Lie groups, their associated Lie algebras, and their representations. This first quarter will focus on the special case of matrix Lie groups–including general linear, special linear, orthogonal, unitary, and symplectic. The second quarter will generalize the theory developed to the case of arbitrary Lie groups. Topics to be discussed include compactness and connectedness, homomorphisms and isomorphisms, exponential mappings, the Baker-Campbell-Hausdorff formula, covering groups, and the Weyl group. This is an advanced course, requiring a solid understanding of linear algebra and basic analysis.

### Recommended Textbook

Hall, Brian. *Lie Groups, Lie Algebras, & Representations* (Springer, 2004) ISBN: 9781441923134

@JenLucPiquant I heard you were an Angeleno & thought you might like to join us for a local math class: http://boffosocko.com/2014/09/02/introduction-to-lie-groups-and-lie-algebras/ #LieGroups, #LieAlgebra, #Mathematics

Manny wrote: “I see Jacobson was an expert on Lie algebras… yes, that might certainly be useful! Thanks…”Many/most graduate math texts on Lie Groups/Algebras utilize manifold theory as their basis, which can make the field more daunting for physicists. As a result, I recommend looking at books that take a more linear algebraic bent to the subject, which can make some of the quantum mechanics related areas more transparent. I’ve used [book:Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|1051566], [book:Matrix Groups for Undergraduates.|1162374], and [book:Lie Groups, Lie Algebras, and Some of Their Applications|424103] for these types of applications and viewpoints.

About a year ago, I took a two quarter sequence at UCLA on Lie Groups from an (“easier”) matrix group perspective which I imagine you may find somewhat useful in your reading on quantum mechanics. Several engineers, programmers, amateur mathematicians, physicists, and quantum mechanics enthusiasts had spent several years coaxing the professor into teaching it from this perspective. It was geared toward the advanced undergraduate level, and based on your comments here and your reading of Shankar, should be relatively easily followable.

We loosely followed Hall’s textbook and portions of [book:Matrix Analysis|647523], which will give you some of the advanced linear algebra you could possibly be missing depending on your background. If it helps, here’s a link to a downloadable pdf copy of the notes for the first class with the audio of the lecture embedded (using Livescribe.com digital pen technology which should let you click on the notes and jump to the audio portion related to where you’ve clicked): http://bit.ly/1KMxj0R. If it’s useful, let me know and I can give you links for others. You may need to open it up in a more recent version of Acrobat Reader to be able to access the audio portion of the lecture, which will go a long way to assisting the clarity of the notes.)

Having this background may make Weyl and Woit’s developing textbook more easily manageable. My guess is that Woit is doing a more thorough job of developing the math than typical physics-oriented texts like [book:Geometry, Topology and Physics|439357] which do a lot of hand-waving at the math in an effort to get to the physics more quickly, but at the detriment of understanding what is happening mathematically.

I’ve dipped into some of Weyl’s work in the past, but also keep in mind that some of his notation and definitions can be dated in relation to more modern presentations.