Standard and Non-standard Extensions of Lie algebras

Abstract

We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root $\alpha_{+4}$, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no loops between the dots, while it is possible if $\alpha_{+4}$ is a Borcherds imaginary simple root. We also comment on the root lattices of these new algebras. The folding procedure is applied to the simply-laced triple extended Lie algebras, obtaining all the non-simply laced ones. Non- standard extension procedures for a class of Lie algebras are proposed. It is shown that the 2-extensions of $E_{8}$, with a dot simply linked to the Dynkin-Kac diagram of $E_{9}$, are rank 10 subalgebras of $E_{10}$. Finally the simple root systems of a set of rank 11 subalgebras of $E_{11}$, containing as sub-algebra $E_{10}$, are explicitly written.