Generalized Kac–Moody algebra

Definition

Let $I$ be a countable index set and let $A = (a_{ij})_{i, j \in I}$ be a real symmetric matrix such that:

$a_{ij} \le 0$ for all $i \neq j$ ;
if $a_{ii} > 0$ , then $2 a_{ij} / a_{ii} \in \Z$ for all $j \in I$ .

The generalized Kac–Moody algebra of $A$ is the real Lie algebra generated by elements $e_i, f_i, h_i$ for $i \in I$ , subject to the relations:

$[h_i, h_j] = 0$ for all $i, j$ ;
$[e_i, f_j] = \delta_{ij}\, h_i$ for all $i, j$ , where $\delta_{ij}$ is the Kronecker delta;
$[h_i, e_j] = a_{ij}\, e_j$ and $[h_i, f_j] = -a_{ij}\, f_j$ for all $i, j$ ;
$(\operatorname{ad} e_i)^{1 - 2a_{ij}/a_{ii}}(e_j) = 0$ and $(\operatorname{ad} f_i)^{1 - 2a_{ij}/a_{ii}}(f_j) = 0$ whenever $a_{ii} > 0$ and $i \neq j$ ;
$[e_i, e_j] = 0$ and $[f_i, f_j] = 0$ whenever $a_{ij} = 0$ .

Here $\operatorname{ad} x$ is the map $y \mapsto [x, y]$ .

Remarks

Example: let $A$ be a symmetric generalized Cartan matrix, so $a_{ii} = 2$ and $a_{ij} \in \Z_{\le 0}$ . Both conditions on $A$ hold, the exponent in relation 4 is $1 - a_{ij}$ , and relation 5 follows from relation 4 because $a_{ij} = 0$ gives $(\operatorname{ad} e_i)(e_j) = 0$ . So the relations are the relations of the Kac–Moody algebra among $e_i$ , $f_i$ and $h_i = \alpha_i^\vee$ . When $A$ is singular, the Kac–Moody algebra also contains extra elements of its larger Cartan subalgebra, which are not among the generators here.

Non-example: the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ is symmetric, but its off-diagonal entry $1$ is positive, so it fails condition 1 on $A$ .

The index $i$ is called real if $a_{ii} > 0$ and imaginary if $a_{ii} \le 0$ . Many authors, following Borcherds, use the name more widely. They also allow Lie algebras obtained from this one by adding commuting outer derivations to the Cartan part or by dividing by a central subalgebra.

Tag	`KB2875ZC7P`
URI	`m2://KB2875ZC7P`
Kind	definition
Writing	W0 stub
MSC	17B67, 17B65, 11F22
Defines	Generalized Kac–Moody algebra
Uses	Lie algebra, Symmetric matrix, Kac–Moody algebra, Generalized Cartan matrix, Adjoint representation of a Lie algebra
Changed	3 November 2026

Tag

KB2875ZC7P

URI

m2://KB2875ZC7P

Kind

definition

Writing

W0 stub

MSC

17B67, 17B65, 11F22

Defines

Generalized Kac–Moody algebra

Uses

Lie algebra, Symmetric matrix, Kac–Moody algebra, Generalized Cartan matrix, Adjoint representation of a Lie algebra

Changed

3 November 2026

Generalized Kac–Moody algebra

Definition

Remarks

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