extended Cartan matrix
Let $A$ be the Cartan matrix^{} of a complex, semi-simple, finite dimensional, Lie algebra^{} $\U0001d524$. Recall that $A=({a}_{ij})$ where ${a}_{ij}=\u27e8{\alpha}_{i},{\alpha}_{j}^{\vee}\u27e9$ where the ${\alpha}_{i}$ are simple roots for $\U0001d524$ and the ${\alpha}_{j}^{\vee}$ are simple coroots. The extended Cartan matrix denoted $\widehat{A}$ is obtained from $A$ by adding a zero-th row and column corresponding to adding a new simple root ${\alpha}_{0}:=-\theta $ where $\theta $ is the maximal (relative to $\{{\alpha}_{1},\mathrm{\dots},{\alpha}_{n}\}$) root for $\U0001d524$. $\theta $ can be defined as a root of $\U0001d524$ such that when written in terms of simple roots $\theta ={\sum}_{i}{m}_{i}{\alpha}_{i}$ the coefficient sum ${\sum}_{i}{m}_{i}$ is maximal (i.e. it has maximal height). Such a root can be shown to be unique.
The matrix $\widehat{A}$ is an example of a generalized Cartan matrix. The corresponding Kac-Moody Lie algerba is said to be of affine type.
For example if $\U0001d524=\U0001d530{\U0001d529}_{n}\u2102$ then $\widehat{A}$ is obtained from $A$ by adding a zero-th row: $(2,-1,0,\mathrm{\dots},0,-1)$ and zero-th column $(2,-1,0,\mathrm{\dots},0,-1)$ simultaneously to the Cartan matrix for $\U0001d530{\U0001d529}_{n}\u2102$.
References
- 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 |