Siegel automorphic form corrections of some Lorentzian Kac--Moody Lie algebras

Abstract

We find automorphic form corrections which are generalized Lorentzian Kac--Moody superalgebras without odd real simple roots (see R. Borcherds \cite{Bo1} -- \cite{Bo7}, V. Kac \cite{Ka1} -- \cite{Ka3}, R. Moody \cite{Mo} and §~6 of this paper) for two elliptic Lorentzian Kac--Moody algebras of the rank $3$ with a lattice Weyl vector, and calculate multiplicities of their simple and arbitrary imaginary roots (see an appropriate general setting in \cite{N5}). These Kac--Moody algebras are defined by hyperbolic (i.e. with exactly one negative square) symmetrized generalized Cartan matrices $$G_1\ =\ \left(\matrix\hphantom{-} 2&-2&-2\\-2 &\hphantom{-}2&-2\\-2&-2& \hphantom{-} 2 \endmatrix\right)\hskip30pt \text{and} \hskip 30pt G_2\ =\ \left(\matrix\hphantom{-}4 & -4 & -12& -4 \\-4 &\hphantom{-}4 &-4&-12