A minicourse on n-algebras and its applications

This is the website for a minicourse on n-algebras taught by José A. de Azcárraga and José M. Figueroa-O'Farrill in October 2008 at the University of Valencia. The course consisted of four 2-hour lectures (with a 10m break), starting at 15:30 in the Seminar room of the Department of Theoretical Physics.

The recent proposal of Bagger–Lambert and Gustavsson for a field theory dual to a stack of coincident M2-branes has led to renewed interest in n-ary generalizations of Lie algebras. The present course is an introduction to this subject, highlighting recent results of relevance to Physics, particularly to the study of three-dimensional superconformal Chern–Simons theory.

Schedule of lectures

1. Monday, 20 October 2008
   José A. de Azcárraga
   n-ary generalizations of Lie algebras (I)
   From Lie algebras to n-algebras. Generalized Lie algebras and Filippov algebras. Examples. (Co)homology of generalized Lie algebras and Filippov algebras.
2. Tuesday, 21 October 2008
   José A. de Azcárraga
   n-ary generalizations of Lie algebras (II)
   (Co)homology of generalized Lie algebras and Filippov algebras. Nambu–Poisson algebras and the quantization problem of Nambu mechanics.
   José M. Figueroa-O'Farrill
   “3 is the new 4”
   Physical motivation for the study of metric 3-algebras. AdS/CFT and the Bagger–Lambert–Gustavsson theory. Superconformal Chern–Simons theories in three dimensions.
3. Wednesday, 22 October 2008
   José M. Figueroa-O'Farrill
   “3 is the new 2”
   Metric Lie algebras: examples and structure theorem. Metric Filippov 3-algebras: examples, structure theory, classification in low index.
4. Thursday, 23 October 2008
   José M. Figueroa-O'Farrill
   “2 strikes back”
   The Faulkner construction and the Lie algebraic origin of metric 3-algebras. Examples.