The structure ofE10at higherA9levels — a first algorithmic approach

The conjecture of a hidden $E_{10}$ symmetry of M-theory is supported by theclose connection between the dynamics of D=11 supergravity near a spacelikesingularity and a truncation of an one-dimensional $\sigma$-model with $E_{10}$symmetry where all representations beyond SL(10) level $\ell=3$ are omitted. Ifthis conjecture is right, higher-level representations should especiallycapture the dynamics of further M-theory degrees of freedom. Unfortunately, thelevel by level determination of $E_{10}$ commutators which is necessary toextend the model to higher levels is both an involved and toilsome task thatrequires computer aid. In this work, some of the relevant problems are exposedand algorithmic methods are developed which simplify key steps in thedetermination of explicit $E_{10}$ commutators at higher levels. As anapplication, we compute the commutator of the level-two six-form with itself.Comment: 26 pages, 1 figur