Kac-Moody algebra

Let $A$ be an $n\times n$ generalized Cartan matrix. If $n-r$ is the rank of $A$ , then let $\mathfrak{h}$ be a $n+r$ dimensional complex vector space. Choose $n$ linearly independent elements $\alpha_{1},\ldots,\alpha_{n}\in\mathfrak{h}^{*}$ (called roots), and $\check{\alpha}_{1},\ldots,\check{\alpha}_{n}\in\mathfrak{h}$ (called coroots) such that $\langle\alpha_{i},\check{\alpha_{j}}\rangle=a_{ij}$ , where $\langle\cdot,\cdot\rangle$ is the natural pairing of $\mathfrak{h}^{*}$ and $\mathfrak{h}$ . This choice is unique up to automorphisms of $\mathfrak{h}$ .

Then the Kac-Moody algebra associated to $\mathfrak{g}(A)$ is the Lie algebra generated by elements $X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}$ and the elements of $\mathfrak{h}$ , with the relations

$\displaystyle[X_{i},Y_{i}]$	$\displaystyle=\check{\alpha_{i}}$	$\displaystyle[X_{i},Y_{j}]$	$\displaystyle=0$
	$\displaystyle=\alpha_{i}(h)X_{i}$	$\displaystyle[Y_{i},h]$	$\displaystyle=-\alpha_{i}(h)Y_{i}$
$\displaystyle\underbrace{[X_{i},[X_{i},\cdots,[X_{i}}_{1-a_{ij}\text{ times}},% X_{j}]\cdots]]$	$\displaystyle=0$	$\displaystyle\underbrace{[Y_{i},[Y_{i},\cdots,[Y_{i}}_{1-a_{ij}\text{ times}},% Y_{j}]\cdots]]$	$\displaystyle=0$

for any $h\in\mathfrak{h}$ .

If the matrix $A$ is positive-definite, we obtain a finite dimensional semi-simple Lie algebra, and $A$ is the Cartan matrix associated to a Dynkin diagram. Otherwise, the algebra 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