Aspuru-Guzik research group - Algebras and Groups

Lie Algebras

Lie algebras are extremely important in the study of quantum mechanics. I am currently concerned with finite dimensional matrix Lie algebras and Lie groups.

Lie algebra g (Representation of a Lie algebra)
|
exp(g) (see also ln(G))
|
V
Lie group G (Representation of a Lie group)

John Baez on the ubiquity of Dynkin diagrams in mathematics

Web tool for making publication-quality Dynkin diagrams with labels (written in JavaScript)

M. Hazewinkel, W. Hesselink, D. Siermsa, and F. D. Veldkamp, The ubiquity of Coxeter-Dynkin diagrams (an introduction to the ADE problem), Niew. Arch. Wisk., 25 (1977), 257-307. (available here and here)

O'Connor, J. J. & Robertson, E.F. Biography of Sophus Lie, MacTutor History of Mathematics Archive.
O'Connor, J. J. & Robertson, E.F. Biography of Wilhelm Killing, MacTutor History of Mathematics Archive.

Online books/papers:

Introduction to matrix groups
Kac, Victor G. et al. Course notes for MIT 18.745: Introduction to Lie Algebras
Good (not free) book: http://www.maa.org/reviews/matrixgroups.html
Elementary introduction to Lie groups, Lie algebras, and their representations

Geometric (aka Clifford) Algebras

Online introductions

Primer for Geometric Algebra (David Hestenes, July 2005).
John Baez entry on Clifford Algebra
Wikipedia entry on Geometric algebra
Imaginary Numbers are not Real - the Geometric Algebra of Spacetime. Introduction (Cambridge GA group).
Physical Applications of Geometric Algebra. Final-year undergraduate course (Cambridge GA group; see also).
Maths for (Games) Programmers: 5 - Multivector methods. Comprehensive introduction and reference for programmers, from Ian Bell (programmer).
A Geometric Algebra Primer, especially for computer scientists.

Groups working on Geometric Algebra

Cambridge University Geometric Algebra Research Group home page

Boolean algebras!

Boolean algebra applications

I. Wegener, The Complexity of Boolean Functions, Wiley-Teubner, 470 pages, (1987).
Scribe notes of the course , Boolean Circuit Complexity, Lecturer: Uri Zwick.
E. Boros & P.L. Hammer, Pseudo-Boolean Optimization, Discrete Applied Mathematics, 123(1-3) (2002) pp. 155-225.

Boolean basics

List of Boolean algebra topics
Stone's representation theorem for Boolean algebras
The Theory of Representations of Boolean Algebras, ''Transactions of the American Mathematical Society 40'': 37-111.

Galois Fields

Knot Theory

More online books!

Elements of Abstract and Linear Algebra, by Edwin H. Connell.
(great book!) Linear mathematics in infinite dimensions, by U. H. Gerlach.
Linear Algebra Done Right, by Sheldon Axler. (2 good chapters free online, my interest in this book stems from the following, award winning, paper)
Down with Determinants!, by Sheldon Axler. (see also this blog entry and this one)

Functionals

Wikipedia
Functional analysis
Functional Analysis, by Gerald Teschl, University of Vienna.
Lecture Notes on Functional Analysis, (in one nice PDF) by Yevgeny Vilensky, New York University.
see also: James Binney's Lecture Course on Mechanics.

Getting started in Algebra

List of AMS free online books

Fredrick M. Goodman: Algebra: Abstract and Concrete.

John Beachy: Abstract Algebra On Line, Comprehensive list of definitions and theorems.

Universal Algebras

Hobby, David, and Ralph McKenzie, 1988. The Structure of Finite Algebras, American Mathematical Society. ISBN 0-8218-3400-2. (Available free online as of May 31, 2006).

Burris, Stanley N., and H.P. Sankappanavar: A Course in Universal Algebra, Springer-Verlag, 1981 ISBN 3-540-90578-2. Free online edition.

Bergman, George M.: An Invitation to General Algebra and Universal Constructions, (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 1998, 398 pp. ISBN 0-9655211-4-1. Free online edition.

Freese, Ralph, and Ralph McKenzie, 1987. Commutator Theory for Congruence Modular Varieties, 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. ISBN 0-521-34832-3. Free online second edition.