PHYSICS 500-006

Advanced Topics in Applicable Mathematical Physics
Several Useful References

for Lie Algebras and Groups:
- Semi-Simple Lie Algebras and Their Representations, by Robert N. Cahn [Benjamin-Cummings, 1984]. Now out of print; however, it is available free, online,at this link. This starts the discussions very slowly and methodically, with good detail on examples; it is the text I used for a 1/2 semester course on group theory a few years ago.
- Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, by D.H. Sattinger and O.L. Weaver, [Springer-Verlag, 1993].
- Representation Theory, by William Fulton and Joe Harris [Springer, 1991]
  A standard and very thorough reference for this subject, which straddles well the line between understanding and rigor.
- Symmetries, Lie Algebras and Representations, A graduate course for physicists, by Jurgen Fuchs and Christoph Schweigert [Cambridge Univ. Press, 1997]. This book is very detailed, and a good source for nomenclature in forms as usually understood by physicists. It discusses finite-dimensional algebras and Kac-Moody ones as well.
- Lectures on Lie Groups and Lie Algebras, by Roger Carter, Graeme Segal, and Ian MacDonald [Cambridge Univ. Press, 1995; London Math. Soc. Student Texts, #32].
  This is three separate sets of lectures given by the 3 people listed as authors, and are really very pleasant summaries. In order by author above, the first is on Lie algebras, including so-called Lie algebras of finite type, the second is on Lie groups with a concentration on understanding of the group structure (rather than the general classification facts), and the third is on linear algebraic groups, a much broader set, and requires some knowledge of algebraic geometry.
- Infinite-Dimensional Lie Algebras, by Minoru Wakimoto [Amer. Math. Soc., Translations, Vol. 195, 2001]
- The ones below are more mathematical in their presentation.
- Introduction to Lie Algebras and Representation Theory, by James E. Humphreys, [Springer-Verlag, 1987].
- Infinite dimensional Lie algebras, by V.G. Kac, [Cambridge Univ. Press, 1990 for 3rd Edn.]
- Simple Irreducible Graded Lie Algebras of Finite Growth, by V.G. Kac, Math. USSR--Izvestija, 2, 1271-1311 (1968).
  This is Kac's original paper on this subject, which includes not only the now-called Kac-Moody algebras, but also general contravariant ones, the notorious algebra K₂, and the 4 countable sequences of different sorts of transformations of n-dimensional spaces.
- Lie Groups, Lie Algebras, and Their Representations, by V.S. Varadarajan [Springer-Verlag, 1984].
- Vertex Operator Algebras and the Monster, by I. Frenkel, J. Lepowsky & A. Meurman, [Academic Press, 1988]
  The first serious "textbook" on vertex operator algebras; really quite good, if dense. Good explanations of formal delta functions, and twisted algebras.
  Also introduction to relations to the monster, although that somewhat old now.
for Finite Groups and Lattices
- The Theory of Finite Groups, by W. Lederman [Oliver and Boyd, Edinburgh, 1961]
- Sphere Packings, Lattices, and Groups, by J.H. Conway and N.J.A. Sloane [Springer-Verlag, 1998 (3rd Edn.)]
  This is a very complicated and sophisticated book, basically on sphere packings, but with many applications to other things and drawing in many things from other subjects.
- moonshine
for Differential Geometry
- Tensor Analysis on Manifolds, by R.L. Bishop and S.I. Goldberg [Macmillan, 1968] A standard, older text with many useful results.
- Applied differential geometry, by William L. Burke [Cambridge Univ. Press, 1985]
- Applicable Differential Geometry, by M. Crampin and F.A.E. Pirani [Cambridge Univ. Press, 1987]
- Gravitation, Gauge Theories and Differential Geometry, by T. Eguchi, Peter B. Gilkey, and A.J. Hanson, Physics Reports, 66, 213-393 (1980).
- Geometrical methods of mathematical physics, by Bernard Schutz [Cambridge Univ. Press, 1980 (reprinted 1993)]
- Elementary Topics in Differential Geometry, by J.A. Thorpe [Springer-Verlag, 1979]
- Foundations of Differential Geometry, by S. Kobayashi and K. Nomizu, Vols. 1 and 2 [Interscience Publishers (John Wiley), 1963]
  A very rigorous and careful mathematical treatment of the subjects.
- Exterior Differential Systems and Equivalence Problems, by Kichoon Yang [Kluwer Acad., 1992]
  The equivalence problem is an important, but very subtle, one. This is a simpler introduction to it than many.
- Foundations of Differentiable Manifolds and Lie Groups, by Frank W. Warner [Springer-Verlag, 1983]
  This a very thorough and authoritative book; also has good material on jet bundles, which most of the others, above, do not.
- The Geometry of Jet Bundles, by D.J. Saunders, [Cambridge Univ. Press, 1989] A good source for an in detail study of (different sorts of) jet bundles.
for algebraic function theory
- Abelian Functions, by H.F. Baker [Cambridge Univ. Press, 1897, reprinted 1995]
- Introduction to the Classical Theory of Abelian Functions, by A.I. Markushevich [Amer. Math. Soc., Rhode Island, 1992]
  A very nicely-presented introduction to this theory, in several complex variables, over (compact) Riemann surfaces of arbitrary genus. Utilizes theta functions in very nice ways, but does not resolve the standard Jacobi inversion problem.
- Abstract Algebra and Solution by Radicals, by John E. and Margaret W. Maxfield [Dover Pubs., 1971]
  Begins in a very, very elementary way. Nonetheless it proceeds, via theta functions, for example, to explain how to determine the roots of polynomials of arbitrary order.
- Complex Variables: Introduction and Applications, by Mark J. Ablowitz and A.S. Fokas [Cambridge Univ. Press, 1997]
  This is a very useful book, with very much more than its brief title might suggest, although in some cases it is more useful for the guide it gives and the references than the actual details. Among other unexpected things it has discussions of Riemann-Hilbert problems, Schwartz-Christoffel transformations, and the so-called dbar problem.

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finley@tagore.phys.unm.edu		Last updated/modified: 6 September, 2005

PHYSICS 500-006

Advanced Topics in Applicable Mathematical Physics Several Useful References

Advanced Topics in Applicable Mathematical Physics
Several Useful References