Solution Set 4 has been re-posted to Section IV of this website.
The slides from the class presentations are available for viewing in Section V of this website.
A revised version of the class handout entitled The Cartan-Killing Form (which contains two new appendices) has posted to Section VI of this website.
Aaron Wangberg and Tevian Dray have provided a pedagogical introduction to root and weight diagrams of Lie algebras. A link has been provided in Section VIII of this website. Related links can be found in Section X of this website.
A link to some notes by Terence Tao on the classification of complex Lie algebras has been added to Section X of this website.
The General Information and Syllabus handout is available
in either PDF or Postscript format
[PDF | Postscript]
Some of the information in this handout is reproduced here.
General Information | ||
---|---|---|
Instructor | Howard Haber | |
Office | ISB 326 | |
Phone | 459-4228 | |
Office Hours | Mondays and Tuesdays 3--4 pm | |
haber@scipp.ucsc.edu | ||
webpage | scipp.ucsc.edu/~haber/ | |
Lectures: Tuesdays and Thursdays, 1:350-3:05 pm, ISB 231
There is no required textbook. However, I can recommend a number of textbooks that you may find useful during this course. The first two textbooks listed below come closest to covering the syllabus of this course. Apart from the first two textbooks, all other textbooks listed below served as the required textbook for this course at one time or another in years past.
A Physicist's Introduction to Algebraic Structures, by Palash
B. Pal
Continuous Groups for Physicists, by Narasimhaiengar Mukunda and
Subhash Chaturvedi
Theory of Groups and
Symmetries--Finite Groups, Lie Groups, and Lie Algebras , by Alexey P. Isaev and Valery A. Rubakov
Theory of Groups and
Symmetries--Representations of Groups and Lie Algebras, Applications , by Alexey P. Isaev and Valery A. Rubakov
Group Theory in
Physics: A Practitioner's Guide, by Rutwig Campoamor-Stursberg and
Michel Rausch de Traubenberg
Group Theory in Physics, by Wu-Ki Tung
Groups,
Representations and Physics, Second edition, by H.F. Jones
Group
Theory in Physics: An Introduction, by J.F. Cornwell
Group
Theory for Physicists, Second edition, by Zhong-Qi Ma and Xiao-Yan Gu
Lie Groups, Lie Algebras, and Some of Their Applications,
by Robert Gilmore
Lie
Algebras in Particle Physics, Second edition, by Howard Georgi
Group
Theory: A Physicist's Survey, by Pierre Ramond
Symmetries,
Lie Algebras and Representations, by Jürgen Fuchs and
Christoph Schweigert (Errata)
Group
Theory in a Nutshell for Physicists, by Anthony Zee
R. Slansky, Group theory for unified model building, Physics Reports 79 (1981) 1--128.
The basic course requirements consist of four problem sets, which will be handed out during the quarter, and a term project. (There will be no exams.) Due to the limited time in a quarter, it will be impossible to do more than sketch some of the most basic applications of group theory to modern physics. To encourage students to delve deeper, all students will be required to complete a term project based on their reading of a particular topic in group theory and its applications to physics. The project may be presented orally or in written form at the end of the term. Oral presentations are encouraged since they will benefit all members of the class. Please follow the following schedule:
The oral presentations will take place on Thursday June 1 from 11:45--1 pm and on Wednesday June 14 from 4--7 pm in ISB 231.
All projects should include a one page bibliography (containing references pertinent to the project). Copies of this bibliography should be made available to all students in the class. For those projects presented orally, a xerox of transparencies and/or a readable set of notes should be made available to the class. If an oral presentation is not possible (not the preferred option), a full written version of the project is an acceptable substitute.
Problem sets and exams are available in either PDF or Postscript formats.
The problem set solutions are available in either PDF or postscript formats.
Students were required to give half hour presentations on a project
involving an application of group theory to physics. These
presentations have been collected below.
1. Robert Gilmore provides a very nice introduction to the the basic building blocks of algebra---groups, fields, vector spaces and linear algebras. I have scanned the first ten pages of his book, Lie Groups, Lie Algebras, and Some of Their Applications, and provide it here. [PDF]
2. John Sullivan provided a nice handout with a table of groups of order 15 or less. You can find the table in PDF format. Note that in this table, V stands for the Klein group (Viergruppe in German), also called the Klein 4-group. It is the smallest non-cyclic group and is isomorphic to the dihedral group D_{2}. The group T (denoted as Q_{6} by Ramond), also called the dicyclic group Dic_{3}, can be defined as the order-12 group generated by two elements a and b such that a^{6}=e (where e is the identity element) and b^{2}=a^{3}=(ab)^{2}.
3. Check out the periodic table of finite simple groups. [PDF]
4. Chapter 1 of the book by Alexey P. Isaev and Valery A. Rubakov entitled Theory of Groups and Symmetries (World Scientific, Singapore, 2018) is provided free of charge by the publisher. It can be found at this link.
5. Chapter 1 of the book by Zhong-Qi Ma and Xiao-Yan Gu entitled Group Theory for Physicists (World Scientific, Singapore, 2007) provides a brief review of linear algebra that is often used in group theory. This chapter is provided free of charge by the publisher and can be found at this link.
6. These notes, entitled Notes on Basis Changes and Matrix Diagonalization, provide a review of some important concepts in linear algebra that are relevant for the theory of matrix representations of groups. Some of the topics included are: changes of basis, similarity transformations, and matrix diagonalization. [PDF | Postscript].
7. Further details on the group algebra and its idempotents and the reduction of the regular representation can be found in Appendix III of Wu-Ki Tung, Group Theory in Physics, which is cited as one of the recommended textbooks in Section I of this website. For your convenience, this appendix is provided here: [PDF]
8. These notes provide a detailed treatment of the properties of the most general three-dimensional proper and improper rotation. The general form for the corresponding 3x3 orthogonal matrix is derived and is used to provide a simple method for determining the axis and angle of rotation and the equation for the reflection plane, if present. In an appendix, the Euler angles are introduced and the Euler angle representation of a three dimensional rotation is explicitly given in terms of the corresponding angle-axis representation. [PDF | Postscript].
9. A collection of important results involving the matrix exponential that are especially useful in the theory of Lie algebras (together with a proof or derivation of each result) are provided in this class handout. Included is a statement of the Baker-Campbell-Hausdorff (BCH) formula and its proof. For completeness, I also include a collection of corresponding results involving the matrix logarithm. [PDF | Postscript].
10. I am providing a table of the real Lie algebras corresponding to the classical matrix Lie groups, taken from Group Theory in Physics: An Introduction, by J.F. Cornwell (Academic Press Inc., San Diego, CA, 1997). The table provides the definition of each matrix Lie group and the corresponding Lie algebra, along with its dimension. [PDF]
11. In this class handout, I describe details of the local properties of a Lie group using the techniques first introduced by Sophus Lie. For example, I show that the existence and the properties of the structure constants can be derived by studying the local properties of the group multiplication law. I also discuss the concept of generators of an infinitesimal Lie group transformation, which can be applied to an action of a Lie group on a manifold or to an action of a Lie group on the group manifold itself. These concepts are then illustrated in the case of SO(3). [PDF | Postscript].
12. In this class handout, I discuss the properties of the Cartan-Killing form on a real Lie algebra and related theorems. Some applications are presented: the construction of real forms of a complex Lie algebra and the construction of a completely antisymmetric third rank tensor related to the structure constants of the Lie algebra. [PDF | Postscript].
13. The Gell-Mann matrices are a set of traceless hermitian matrices that generate the Lie algebra of SU(3). The properties of the Gell-Mann matrices, along with an explicit list of the structure constants f_{abc} and the totally symmetric tensor d_{abc} are provided in this class handout. The d_{abc} can be used to construct a cubic Casimir operator in the SU(3) Lie algebra. [PDF | Postscript].
14. There are many useful relations that can be derived involving the generators of SU(n), the structure constants f_{abc} and the symmetric tensor d_{abc} defined in the previous handout (and generalized to arbitrary n). In 2021, I published a review in SciPost Phys. Lect. Notes 21 (2021) that provided many useful relations among the generators in the defining and adjoint representations of SU(n). A link to this paper is provided here: [PDF].
15. On problem 5 of Problem Set 3, the pfaffian was introduced. I have expanded the treatment of the pfaffian to a set of notes, which are provided as a class handout. In these notes, I first show that any even-dimensional complex invertible antisymmetric matrix is congruent to a block diagonal matrix consisting of multiple copies of iσ_{2} (where σ_{2} is one of the Pauli matrices). Using this result, I prove that the square of the pfaffian is equal to the determinant. [PDF | Postscript].
16. The root diagrams of the rank-two semisimple Lie algebras are nicely presented in Brian G. Wybourne, Classical Groups for Physicists (John Wiley & Sons, New York, 1974). I have scanned in two pages from this book which exhibit the root diagrams of the rank-two semisimple Lie algebras. [PDF]
17. In Brian G. Wybourne, Classical Groups for Physicists (John Wiley & Sons, New York, 1974), you will also find a very nice treatment of the Dynkin techniques for analyzing semisimple Lie algebras, albeit with a number of typographical errors. In this handout, I have scanned in three tables from Chapter 7 of Wybourne's book. The first table provides the Dynkin diagrams and root structure of the semisimple Lie algebras (with the typographical errors corrected). The second table lists the scalar product of the roots, and the third table provides the Cartan matrices of the semisimple Lie algebras. [PDF]
18. I have written up a set of notes on the quadratic Casimir operator and second-order index of a simple Lie algebra. The relation of these quantities to the dual Coxeter number is clarified. [PDF | Postscript].
1. Lie Algebras in Particle Physics, Second edition, by Howard Georgi.
2. Quantum Theory, Groups and Representations: An Introduction, by Peter Woit (final draft version) [PDF]
3. Semi-Simple Lie Algebras and Their Representations, by Robert N. Cahn.
4. Group Theory: Birdtracks, Lie's, and Exceptional Groups, by Predrag Cvitanović. [PDF]
5. Lie Groups, Lie Algebras, and Representations, Second edition, by Brian C. Hall. A preliminary version of this book, which was subsequently published by Springer, can be found here.
6. A detailed elementary treatment of various topics in abstract algebra, including the theory of groups, rings, vector spaces and fields, can be found in A Course on Algebra, by Ahmet Feyzioglu. [Volume 1 | Volume 2]
7. Classical and quantum mechanics via Lie algebras (draft version), by Arnold Neumaier and Dennis Westra provides numerous physics applications of the theory of Lie algebras. [PDF]
8. Geometric Mechanics, Part I and Part II, by Darryl D. Holm. These two books provide numerous applications of the theory of Lie groups and Lie algebras in classical mechanics. [PDF-I | [PDF-II]
1. One very nice treatment of finite group theory from a physicist's point of view can be found in Chapter 10 of Frederick W. Byron and Robert C. Fuller, Mathematics of Classical and Quantum Physics (Dover Publications, Inc., New York, 1992), originally published by the Addison-Wesley Publishing Company in 1970, but has now been reprinted in an inexpensive paperback edition by Dover Publications. For your convenience, I am providing a link to Chapter 10 here. [PDF]
2. Why is the cross product only defined in Euclidan spaces of 3 and 7 dimensions? Check out the following papers for some insight:
3. Are you interested in learning more about quaternions? Check out Visualizing Quaternions, by Andrew J. Hanson (Elsevier, Inc., Amsterdam, 2006), which provides a very readable account of their origin, mathematical properties and applications in visual representations.
4. The only real division algebras are the real numbers, the complex numbers, the quaternions and the octonions (the latter is non-associative). For more information on these issues, have a look at a fantastic book entitled Numbers, by H.-D. Ebbinghaus et al. The connection to the possible vector products in Eucliean spaces is mentioned on pp. 278--279.
5. The geometry of quaternions and octonions is discussed in a very readable book entitled The Geometry of Octonians, by Tevian Dray and Corinne A Manogue. This book also includes very nice material on related group theory topics.
6. Since quaternions are non-commuting, it is not clear how to define the determinant of a quaternionic matrix. Over the years, many people have given differen definitions. For a very clear introduction to this subject, see Quaternionic determinants, by Helmer Aslaksen in The Mathematical Intelligencer, 18 (1996) pp. 57--65. [PDF]
7. An elementary introduction to equivalence relations can be found on pp. 7--14 of A Course on Algebra, by Ahmet Feyzioglu. [PDF]
8. If the largest finite simple sporatic group, a.k.a. The Monster, intrigues you, then check out Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, by Mark Ronan (Oxford University Press, Oxford, UK, 2006) for an exciting, fast-paced historical narrative that describes the quest for the classification of the finite simple groups.
9. Related to the previous item, and introduction to Monstrous Moonshine can be found at arXiv:1902.03118 [PDF]. Although this subject seems highly mathematical with no connection to physics, the mathematics of the Monstrous Moonshine has recently found applications in theoretical physics as the following INSPIRE search reveals.
10. Conjugation and conjugacy classes in finite group theory is treated in a very nice set of notes by Keith Conrad, entitled Conjugaation in a Group. In these notes, you will find a proof of the statement that elements of S_{n} with the same cycle structure belong to the same conjugacy class. [PDF]
11. Given a finite group G, how many proper subgroups does it possess? How
many of these are normal subgroups? For an arbitrary group G, a
general formula for these numbers
is not generally known. Remarkably,a general formula is known for
the dihedral groups D_{n}. In contrast, no general formula is
known for the permutation groups S_{n}. Moreover,
explicit calculations of the number of subgroups of
S_{n} has only been achieved for n = 1,2,...,18.
Two relevant papers can be found below.
12. Schur's first lemma applies to the case where the representation space is taken over either the real or complex numbers. But, Schur's second lemma as stated in class applies in general in the case where the representation space is taken over the complex numbers. It can fail if the representation space is taken over the real numbers, as in the case of the representation of SO(2) by 2x2 real orthogonal matrices. A paper entitled A Basic Note on Group Representations and Schur's Lemma by Alen Alexanderian discusses the modification to Schur's second lemma in the case of a representation space over the real numbers. This paper is written in a scholarly mathematical style, which may appear too abstract to physics students at first glance. However, the basic results are quite easy to extract, and I encourage you to do so. [PDF]
13. The hook length formula provides a very simple way to compute the dimension of the irreducible representations of the permutation group. An unusual but simple proof of this formula is given by Kenneth Glass and Chi-Keung Ng, "A Simple Proof of the Hook Length Formula," American Mathematical Monthly 111, 700 (2004) [PDF]
14. The group SU(2) is isomorphic to the group of quaternions of unit length. This isomorphism provides a simple framework for understanding the isomorphism SO(3) ≅ SU(2) / Z_{2}. It also yields a simple proof that the automorphism group of SU(2) is isomorphic to SO(3). For further details, see Molecular Symmetry with Quaternions by Harald P. Fritzer, Spectrochimica Acta Part A 57 (2001) 1919–1930. [PDF]
15. Aaron Wangberg and Tevian Dray have provided a pedagogical introduction to root and weight diagrams of Lie algebras. In particular, they show how to construct root and weight diagrams from Dynkin diagrams, and how the root and weight diagrams can be used to identify subalgebras. Have a look at Aaron Wangberg and Tevian Dray, Visualizing Lie Subalgebras using Root and Weight Diagrams. [PDF | HTML].
1. Gregory Moore is teaching a course at Princeton University entitled Applied Group Theory. For this class, he has written an extensive set of lecture notes. Links to these notes (in PDF format) are provided below. These notes are still under construction and will be continually updated over time.
1. Dongyi Liu recommends a course on visual group theory that appears
on YouTube, which can be accessed here:
[HTML]
This course has been inspired by a book by Nathan Carter entitled
Visual Group Theory.
2. Confused on how to multiply two permutations together? Check out the following step-by-step instructions.
3. An elementary introduction to the group of units in the integers
mod n (denoted by U_{n}) can be found
here [HTML]
A table that lists the groups U_{n} for all
n ≤ 30 can be found here: [HTML]
4. A list of all groups of sixteen or fewer elements along with some their properties and additional links can be found at Wikipedia.
5. GroupNames.org is a database, under construction, of names, extensions, properties and character tables of finite groups of order 500 or less.
6. Groupprops, the group properties wiki, provides links to over 7000 articles, including most basic group theory material. It is managed by Vipul Naik, a Ph.D. in Mathematics at the University of Chicago.
7. Space groups in two dimensions are called wallpaper groups. You can find very nice computer images of these groups here. I can also recommend a marvelous book by William Barker and Roger Howe, Continuous Symmetry: From Euclid to Klein (American Mathematical Society, Providence, RI, 2007). In particular, Chapter VIII of this book provides a superb introduction to the wallpaper groups. For the three-dimensional case, one can find a very useful summary table in the Wikipedia article on space groups.
8. If the origin is fixed, then the space groups reduce to point groups. For further details, see the Wikipedia article Point groups in three dimensions. Point groups in other dimensions, along with many useful tables, are discussed in the Wikipedia article on the concept of a Point group.
9. A very nice set of notes on the symmetric group (also called the permutation group) can be found in an article on Wikipedia. Note that this article also provides references that treat symmetric groups on infinite sets, although the main focus of this article is on the group S_{n} where n is a positive (finite) integer. [HTML]
10. For further details on the Monster group, check out the Wikipedia article, which also provides links to the classification of the finite simple groups.
11. A musical piece entitled Finite simple group (of order two).
12. nLab is a wiki-lab for collaborative work on Mathematics, Physics
and Philosophy. Here is a link to the nLab group theory webpage,
which contains a lot of useful information and interesting links.
[HTML]
From the nLab grop theory webpage, you can find a direct link to the
the nLab Lie group webpage, with many useful links to related topics.
[HTML]
13. On my Quora feed today, the following question appeared: why do the sporadic simple groups exist? Alon Amit provides the following answer here: [HTML]
14. The connection between symplectic matrices and unitary quaternionic matrices is very nicely explained in some notes by John Baez that can be found here.
15. A Table of Lie groups and their associated Lie algebras can be found in Wikipedia. [HTML]
16. In part (h) of problem 4 on Problem Set 3, you were asked to establish the topological equivalence of the real two dimensional sphere S^{2} and the complex manifold CP^{1}. This is most easily understood via the representation of the extended complex plane (which includes the point of infinity) by the stereographic projection of the Riemann sphere. The two Wikipedia links just given will provide you with a good starting point for further study. In particular, the Wikipedia article on the Riemann sphere discusses the relation between the Riemann sphere and the complex projective line (i.e. CP^{1}).
17. Tevian Dray at Oregon State University has created an interesting website that provides a nice introduction to root and weight diagrams of Lie algebras and Dynkin diagrams. [HTML]
This website is part of a larger effort associated with a paper by Aaron Wangberg and Tevian Dray entitled Visualizing Lie Subalgebras using Root and Weight Diagrams. [PDF | HTML].
Access to the complete website can be found here. [HTML]
18. Notes on the classification of complex Lie algebras have been provided by Terence Tao at this website. [HTML]
Additional notes by Terence Tao on other topics in the theory of Lie groups and algebras can be accessed here. [HTML]