PlanetMath: Kac-Moody algebra

(more info)

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Kac-Moody algebra

(Definition)

Let be an $n\times n$ generalized Cartan matrix. If is the rank of , then let $\mathfrak{h}$ be a dimensional complex vector space. Choose 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

$\displaystyle [X_i,Y_i]$	$\displaystyle =\check{\alpha_i}$	$\displaystyle [X_i,Y_j]$	$\displaystyle =0$
$\displaystyle [X_i,h]$	$\displaystyle =\alpha_i(h)X_i$	$\displaystyle [Y_i,h]$	$\displaystyle =-\alpha_i(h)Y_i$
$\displaystyle \underbrace{[X_i,[X_i,\cdots,[X_i}_{1-a_{ij} \text{ times}},X_j]\cdots]]$	$\displaystyle =0$	$\displaystyle \underbrace{[Y_i,[Y_i,\cdots,[Y_i}_{1-a_{ij} \text{ times}},Y_j]\cdots]]$	$\displaystyle =0$