E11 as E10 representation at low levels

TL;DR

This paper decomposes the Lorentzian Kac-Moody algebra E11 into E10 representations, providing recursive calculations and multiplicities up to significant heights, advancing understanding of its structure.

Contribution

It introduces algorithms combining Weyl orbit and classical methods to compute high-level representations of E11 and E10, extending known results to higher heights.

Findings

Decomposition of E11 into E10 representations up to height 120.

All multiplicities of E10 up to height 340 and E11 up to height 240.

Recursive methods enable computation of higher-level representations.

Abstract

The Lorentzian Kac-Moody algebra E11, obtained by doubly overextending the compact E8, is decomposed into representations of its canonical hyperbolic E10 subalgebra. Whereas the appearing representations at levels 0 and 1 are known on general grounds, higher level representations can currently only be obtained by recursive methods. We present the results of such an analysis up to height 120 in E11 which comprises representations on the first five levels. The algorithms used are a combination of Weyl orbit methods and standard methods based on the Peterson and Freudenthal formulae. In the appendices we give all multiplicities of E10 occuring up to height 340 and for E11 up to height 240.