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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 17: Nonassociative rings and algebras

Introduction

Here are a few notes on nonassociative rings; associative rings are treated in a separate section. There is a long FAQ on sets with products (rings), a particular emphasis of which is the study of division rings over the reals, including the nonassociative ones. For detailed expository information you are welcome to to peruse an on-line textbook made freely available by a colleague.

In nonassociative ring theory we widen the scope of rings to be studied. Here the general theory is much weaker, but special cases of such rings are of key importance: Lie algebras in particular, as well as Jordan algebras and other types.

History

Applications and related fields

Subfields

• 17A: General nonassociative rings
• 17B: Lie algebras, For Lie groups, see 22EXX
• 17C: Jordan algebras (algebras, triples and pairs)
• 17D: Other nonassociative rings and algebras

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

"Reviews in Ring Theory, 1980-1984", American Mathematical Society, 1986 ISBN 0-8218-0097-3. Also available: reviews 1940-1979.

Bourbaki, Nicolas, "Lie groups and Lie algebras. Chapters 1--3" Springer-Verlag, Berlin-New York, 1989. 450 pp. ISBN3-540-50218-1 MR89k:17001

Chari, Vyjayanthi; Pressley, Andrew: "A guide to quantum groups", Corrected reprint of the 1994 original. Cambridge University Press, Cambridge, 1995. 651 pp. ISBN 0-521-55884-0 MR96h:17014

A survey: Rittenberg, V.: "A guide to Lie superalgebras", Group theoretical methods in physics (Sixth Internat. Colloq., Tübingen, 1977), pp. 3--21; Lecture Notes in Phys., 79, Springer, Berlin-New York, 1978. MR80d:17012

Software and tables

McKay, W. G.; Patera, J.; Rand, D. W., "Tables of representations of simple Lie algebras", Université de Montréal, Centre de Recherches Mathématiques, Montréal, 1990, ISBN 2-921120-06-2 [Only Vol. I, on the exceptional algebras, was published.]

http://www.riaca.win.tue.nl/CAN/SystemsOverview/Special/Algebra/FELIX/

Other web sites with this focus

• Preprint server (includes Quantum Algebra)
• Here are the AMS and Goettingen resource pages for area 17.
• Algebras and other algebraic information for physicists

Selected topics at this site

• How do Lie groups and algebras fit in among the branches of mathematics?
• Comparison of semisimple, reductive, etc. for Lie algebras and groups
• Construction of octonions and related real algebras