E11 as E10 representation at low levels

Abstract

The Lorentzian Kac-Moody algebra E 11 , obtained by doubly overextending the compact E 8 , is decomposed into representations of its canonical hyperbolic E 10 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 E 11 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 E 10 occuring up to height 340 and for E 11 up to height 240.

Symmetries of M-theory and free Lie superalgebras

“…This can be achieved in an E 11 -covariant fashion by allowing non-trivial commutation relations for the elements of the ℓ 1 representation. One possibility would be an embedding in E 12 along the lines of [58], see also [64]. This embedding embeds the semi-direct sum E 11 ⊕ ℓ 1 as levels 0 and 1 of a graded decomposition of E 12 under its obvious E 11 subalgebra.…”

Very extended Kac–Moody algebras and their interpretation at low levels

“…There is a nice construction due to Feingold and Frenkel [21,22] (see also [23,20]) which allows one to abstractly characterise the decomposition of g into representations of a regular co-rank 1 subalgebra at all levels. The coefficient of the node which has been removed will be referred to as l (without index since j can only take one value in this case).…”

MAXIMAL SUPERGRAVITIES AND THE E10 COSET MODEL

“…By performing such calculations on a computer one can obtain a good picture of the root structure of a Kac-Moody algebra. 32,33 Due to the Lorentzian structure of the Cartan matrix of e 10 and the condition (11) we end up with the following picture: all lattice points α inside the solid hyperboloid {α 2 ≤ 2} ⊂ Q in the Lorentzian space h * are roots of e 10 , and each such point represents the vector space g α associated with the corresponding root α. We imagine the lattice points as being labelled in addition by the dimension (= multiplicity) of the root space.…”