Prenilpotent Pairs in the E10 Root Lattice

Tits has defined Kac–Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumeration is impractical. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as (constant) · k as k →∞. Our purpose is to motivate alternate approaches to Tits’ groups, such as the one in [2]. Kac–Moody groups generalize reductive algebraic groups to include the infinite dimensional case. Various authors have defined them in many ways, the most comprehensive approach being due to Tits [15]. Given a generalized Cartan matrix A, he defined a functor G̃A assigning a group to each commutative ring R with unit. The main result of [15] is that any functor from commutative rings to groups, having some properties that are reasonable to expect of anything called a Kac–Moody group, must agree with G̃A over every field. (See [15, Theorems 1 and 1 ′].) (Actually Tits defined a group functor G̃D for a root datum D. For G̃A we use the root datum with generalized Cartan matrix A, which is “simply connected in the strong sense” [15, p. 551]. The difference between a root datum and its generalized Cartan matrix plays no role in this paper.) Tits defined G̃A(R) by a complicated implicitly described presentation. The key relations are his generalizations of the Chevalley relations. He begins with the free product ∗α(Uα), where α varies over all real roots and each Uα is isomorphic to the additive group of R. This step requires knowing all the real roots, which is nontrivial but reasonably accessible (lemma 2). He imposes relations of the form