Kac-Moody algebra

Let A be an n×n generalized Cartan matrix. If n-r is the rank of A, then let 𝔥 be a n+r dimensional complex vector space. Choose n linearly independentMathworldPlanetmath elements α1,,αn𝔥* (called roots), and αˇ1,,αˇn𝔥 (called coroots) such that αi,αjˇ=aij, where , is the natural pairing of 𝔥* and 𝔥. This choice is unique up to automorphismsPlanetmathPlanetmathPlanetmath of 𝔥.

Then the Kac-Moody algebra associated to 𝔤(A) is the Lie algebraMathworldPlanetmath generated by elements X1,,Xn,Y1,,Yn and the elements of 𝔥, with the relations

[Xi,Yi] =αiˇ [Xi,Yj] =0
=αi(h)Xi [Yi,h] =-αi(h)Yi
[Xi,[Xi,,[Xi1-aij times,Xj]]] =0 [Yi,[Yi,,[Yi1-aij times,Yj]]] =0

for any h𝔥.

If the matrix A is positive-definite, we obtain a finite dimensional semi-simple Lie algebra, and A is the Cartan matrixMathworldPlanetmathPlanetmath associated to a Dynkin diagramMathworldPlanetmath. Otherwise, the algebraPlanetmathPlanetmath we obtain is infinite dimensional and has an r-dimensional center.

Title Kac-Moody algebra
Canonical name KacMoodyAlgebra
Date of creation 2013-03-22 13:31:52
Last modified on 2013-03-22 13:31:52
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 7
Author bwebste (988)
Entry type Definition
Classification msc 17B67