Generalized Kac-Moody Algebras

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)
\begin{align} \mathbf{k}=(\mathbf{p},\mathbf{k}_L)\in II_{1,1}\oplus \Lambda \ , \end{align}

with vertex operators given by

(2)
\begin{align} V_{\mathbf{k}}(\mathbf{X})=(\textrm{ oscillator terms })\ e^{i\mathbf{k}\cdot \mathbf{X}}\ , \end{align}

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

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