Generalized Kac-Moody algebras (also known as Borcherds Kac-Moody algebras) are infinite-dimensional Lie algebras that are natural generalizations of semi-simple Lie algebras and affine Kac-Moody algebras. Here is the Wikipedia entry for Generalized Kac-Moody Algebras.

**As and when time permits, I will try to provide as many examples as I can write about. Meanwhile, you can refer to the Wikipedia entry that is given above.**

# Examples

It appears that a large class of BKM algebras appear from string theory. In fact, the first example (**the fake monster Lie algebra**) was constructed by Borcherds making use of chiral vertex operators in bosonic string theory as we will describe below. Another example (**the fake monster Lie superalgebra**) due to Scheithauer is a natural generalization of Borcherds construction with the bosonic string being replaced by the superstring.

## The fake Monster algebra

Consider the bosonic string compactifed to two-dimensions on a 24-dimensional torus. The momenta and winding take values in an even self-dual lattice, $\Gamma_{24,24}$, of signature $(24,24)$. The precise lattice depends on the choice of torus. Choose the torus to be $\mathbb{R}^{24}/\Gamma$ where $\Lambda$ is the Leech lattice. One then has $\Gamma_{24,24}=\Lambda \oplus (-\Lambda)$. Let the momenta in the two-dimensional spacetime take values in the unique even two-dimensional unimodular Lorentzian lattice $II_{1,1}$. Then all the momenta associated with a chiral half (say, left-movers) take values in the lattice

(1)with vertex operators given by

(2)where $\mathbf{X}\in \mathbb{R}^{25,1}$.

## The fake Monster superalgebra

The fake monster superalgebra is constructed by compactifying the superstring to two dimensions on a special eight-dimensional torus.

## Lorentzian Kac-Moody algebras

### The rank three Lie superalgebras of Gritsenko and Nikulin

**More to come**