On a new class of high-corank Kac-Moody algebras

Abstract

We present recursive constructions of several families of generalized Cartan matrices associated with Kac-Moody algebras, whose sizes and coranks grow exponentially. The constructions are encoded by connected multigraphs and by block-doubling operations on their associated symmetric generalized Cartan matrices. Equivalently, the corank problem is translated into a spectral graph-theoretic problem: the corank of $2\mathrm{Id}-\operatorname{Adj}(G)$ is the multiplicity of the adjacency eigenvalue $2$. We give two explicit recursive families, compute their spectra and coranks, and emphasize the difference between absolute exponential growth and relative asymptotic density. The resulting algebras are typically indefinite and singular of corank larger than one, and therefore contain several independent central directions and several isotropic radical directions in the root lattice. We also discuss alternative constructions and possible applications to the algebraic structures appearing in gravity, supergravity, string/M-theory and related generalized symmetry problems.