Prenilpotent pairs in the E₁₀ root lattice

Abstract

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 E 10 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 7 as k → ∞. Our purpose is to motivate alternate approaches to Tits' groups, such as the one in [2].

Presentation of hyperbolic Kac–Moody groups over rings

“…So q = p − (α − 2α ′ ) p · (α − 2α ′ ) /6. Calculation reveals d(p, q) = cosh −1 4 3 · 3 + km − k 2 − m 2 4 − k 2 By applying ∂/∂m to the radicand and using m ≤ 0 and k ≥ 3, one checks that the right side is decreasing as a function of m. Therefore d(p, q) is at least what one would get by plugging in 0 for m: d(p, q) ≥ cosh −1 4 3 · 3 − k 2 4 − k 2 By differentiating one shows that the right side is decreasing in k. So d(p, q) is larger than the limit as k → ∞, which is cosh −1 4/3. This is a contradiction, proving the claim.…”

“…His definition, by generators and relations, is very complicated. Even enumerating his relations in non-affine examples is difficult and in some cases impracticable ( [10], [4]).…”

The Conway-Sloane calculus for 2-adic lattices

“…A similar common question is whether M occurs as a primitive sublattice of L. Under the same conditions as in the previous paragraph, this reduces to the problem of building a suitable candidate for the orthogonal complement of M ⊗ Z p in L ⊗ Z p , for each prime p. The case of odd p is no longer trivial, but still the p = 2 case usually dominates the analysis. See [2] for an extended calculation of this sort.…”

“…(3) their scales differ by a factor of 4 and they both have type I. Consider as well the fine symbol got by negating the signs of these terms, and also changing both from givers to receivers or vice-versa in case (2). Then the two fine symbols represent isometric lattices.…”

“…As an extended demonstration of sign walking, we determine the lattices M with the property that M ⊕ 2, 2 ∼ = L where L is from (6.1). Note that 2, 2 = 2 2 2 . Obviously we require M = 1…”