extended Cartan matrix
Let be the Cartan matrix of a complex, semi-simple, finite dimensional, Lie algebra . Recall that where where the are simple roots for and the are simple coroots. The extended Cartan matrix denoted is obtained from by adding a zero-th row and column corresponding to adding a new simple root where is the maximal (relative to ) root for . can be defined as a root of such that when written in terms of simple roots the coefficient sum is maximal (i.e. it has maximal height). Such a root can be shown to be unique.
The matrix is an example of a generalized Cartan matrix. The corresponding Kac-Moody Lie algerba is said to be of affine type.
For example if then is obtained from by adding a zero-th row: and zero-th column simultaneously to the Cartan matrix for .
- 1 Victor Kac, Infinite Dimensional Lie Algebras, Third edition. Cambridge University Press, Cambridge, 1990.
|Title||extended Cartan matrix|
|Date of creation||2013-03-22 15:30:14|
|Last modified on||2013-03-22 15:30:14|
|Last modified by||benjaminfjones (879)|
|Defines||extended Cartan matrix|