A Home Page for Physics 467, Spring, 2011
A Study in Lie Algebra and Lie Group Theory

I plan to begin by introducing some of the jargon used in group theory, in the context of very simple groups such as the rotations of a crystal or the permutations of a few objects, but then following through the material in Robert N. Cahn's book, Semi-Simple Lie Algebras and Their Representations, which is available from Dover books for about $11, although electronic versions are actually free online from the right places. We will go to other texts and readings as well, although will not use any of those other places sufficiently often to refer to them as texts. Therefore, at this link I have put a list of books that seem suitable to me, along with some notes concerning some of them.

Along the way we will talk a little bit about the rotation group and some others as examples, and about the basic language of group theory, as there is a language all its own, which only pulls in a bit from other areas. With respect to understanding Lie algebras, we will introduce the idea of the Cartan subalgebra and the root system associated with it which allows us to understand Cartan's derivation of all possible, finite-dimensional, complex, semi-simple Lie algebras. We will then introduce (highest) weight systems and Young tableaux to discuss irreducible representations of the algebras and how to decompose products of such representations again into sums of irreducible ones.
If, by now, I've used too many words with meanings of which you are unsure, rest assured that we will work hard at explaining all this jargon. In particular notice below just the titles of the handouts that will be available from this website, to try to help with this effort.

A Lie group is a group with continuously-many elements, varying in a continuous way away from the identity transformation, with an underlying structure that allows for arbitrarily-many derivatives of its elements and their expression via (convergent) infinite series. As such an n-dimensional Lie group has local, invertible mappings into ordinary n-dimensional, flat space, so that it can be viewed as a differentiable manifold. One parameter groups of transformations, such as a rotation about some fixed direction but with varying angle, may be thought of as paths, or curves, on the group, and therefore have tangent vectors along these curves associated with them them.
At any particular element in the group the set of all of these possible tangent vectors forms a vector space. The particular tangent space at the identity element, i.e., the set of tangent vectors to all possible curves originating at the identity element, is referred to as the Lie algebra of the Lie group. Physically it corresponds to the set of all "infinitesimal generators" of the group of transformations.
When a transformation is made on some physical system, the parts of that system are usually described by elements in some space, either a vector space or a space of functions. Therefore to describe the transformation of the description of that system we need a representation of the group of transformations in the space where its description lives. This is how the entire notion of looking for representations of groups comes about.

Homework Assignments, Solutions, and related Things: as pdf-files (Acrobat)

Some handouts are also available, listed below:

1. Some basic definitions for various algebraic objects whose names may come up in our discussions. Just a general overview. (4 pages)
2. Some more particular definitions needed for general group theory: Some useful definitions and notes (3 pages).
3. Lie Algebras: A basic, abstract introduction and various useful definitions and notes, with descriptions of Lie groups and their associated Lie algebras, in a general way, headed toward the simple Lie algebras. (8 pages)
4. Root Systems for Simple Lie Algebras: Works through a description of the properties of all simple, finite-dimensional Lie algebras, with good descriptions of their basic properties. (32 pages)
5. Simple Lie Algebras:  More details for algebras of Ranks 1 and 2, except for G2, which has its own handout. Also has simple lists of all Ranks of Lie algebras.(15 pages).
6. The root system for G₂ and its lowest-dimensional representation, in 7 dimensions. (11pages)
7. A useful summary of background on the rotation group in 3 dimensions as a Lie group and its Lie algebra, and some representations. (17 pages)
8. A brief summary of details about the Lorentz and Poincaré Lie algebras. (15 pages)
9. This is a basic handout on manifolds and tangent vectors as well as other things related to them. Although it is not rigorous, it is truly more detailed than will be needed for this class, so you should not worry with this version too much unless you feel compelled. (33 pages)
10. Brief Introduction to the use of Weights to Label (Irreducible) Representations. (4 pages)
11. We need some decent notes on Young tableaux. My first efforts were computer-lost. Therefore, I can now only present the following:
    • My notes on uses of Young Tableaux for su(n), and also
    • Some notes on Young Tableaux and their use for Irreps of su(n) , written by Yufei Zhao
12. Some specific notes on the uses of su(3):
    • Descriptions of the standard particle descriptions for su(3).
    • Discussion of the mass formula for su(3) published by Gell-Mann and Okubo
13. Some handouts relevant to discussions of Kac-Moody algebras, and the use of formal series to discuss them.
    • Notes on A₁⁽¹⁾, the smallest Kac-Moody algebra;
    • Notes on the general ideas of loop algebras and their relationships to untwisted Kac-Moody algebras;
    • Notes on formal series and their combinatoric uses.
14. Detailed descriptions of Spinors to Describe Representations for SO(3,1).

1. General Contragredient Algebras:   some very brief notes on more general, infinite-dimensional algebras, but of finite growth (8 pages).

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