extended Cartan matrix

Let A be the Cartan matrixMathworldPlanetmathPlanetmath of a complex, semi-simple, finite dimensional, Lie algebraMathworldPlanetmath 𝔤. Recall that A=(aij) where aij=αi,αj where the αi are simple roots for 𝔤 and the αj are simple coroots. The extended Cartan matrix denoted A^ is obtained from A by adding a zero-th row and column corresponding to adding a new simple root α0:=-θ where θ is the maximal (relative to {α1,,αn}) root for 𝔤. θ can be defined as a root of 𝔤 such that when written in terms of simple roots θ=imiαi the coefficient sum imi is maximal (i.e. it has maximal height). Such a root can be shown to be unique.

The matrix A^ is an example of a generalized Cartan matrix. The corresponding Kac-Moody Lie algerba is said to be of affine type.

For example if 𝔤=𝔰𝔩n then A^ is obtained from A by adding a zero-th row: (2,-1,0,,0,-1) and zero-th column (2,-1,0,,0,-1) simultaneously to the Cartan matrix for 𝔰𝔩n.


  • 1 Victor Kac, Infinite Dimensional Lie Algebras, Third edition. Cambridge University Press, Cambridge, 1990.
Title extended Cartan matrix
Canonical name ExtendedCartanMatrix
Date of creation 2013-03-22 15:30:14
Last modified on 2013-03-22 15:30:14
Owner benjaminfjones (879)
Last modified by benjaminfjones (879)
Numerical id 8
Author benjaminfjones (879)
Entry type Definition
Classification msc 17B67
Related topic GeneralizedCartanMatrix
Defines extended Cartan matrix