Presentation of hyperbolic Kac–Moody groups over rings

Abstract

Abstract. Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis-Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac-Moody group when the ground ring is a Dedekind domain of arithmetic type. These finite-presentation results need slightly stronger hypotheses when the rank is smallest possible, namely 4. The presentations simplify considerably when the ground ring is Z, a case of special interest because of the conjectured role of the Kac-Moody group E 10 (Z) in superstring theory.

Presentation of affine Kac–Moody groups over rings

“…So theorem 1.1 extends the isomorphism PSt A ∼ = St A from the spherical case to the affine case, except for the two affine diagrams of rank 2. See [3] for a further extension, to the simply-laced hyperbolic case.…”

Untitled

“…By considering the list of affine Dynkin diagrams, one sees that these cases imply case (ii) except in rank 3 when R has a forbidden ‫ކ‬ 2 or ‫ކ‬ 3 quotient. Proving (ii) requires removing this restriction on R, for which we refer to [Allcock 2016]. An early version of the present paper was used in [Allcock and Carbone 2016] to establish Theorem 1.1 for certain hyperbolic Dynkin diagrams.…”

“…Proving (ii) requires removing this restriction on R, for which we refer to [Allcock 2016]. An early version of the present paper was used in [Allcock and Carbone 2016] to establish Theorem 1.1 for certain hyperbolic Dynkin diagrams. Those diagrams are now covered by case (iv).…”

Prenilpotent pairs in the E10 root lattice

“…The first is due to Abramenko and Mühlherr [1][5] [13], and applies to Kac-Moody groups associated to 2-spherical Dynkin diagrams, over fields, with some exceptions over F 2 and F 3 . The second approach is due to the author [2] [3]; see also [4]. It works over general rings, but requires some conditions on the diagram.…”

“…So one may discard almost all of Tits' Chevalleystyle relations, without changing the resulting group. In the author's approach, one even obtains an explicit presentation (often finite) given in terms of the Dynkin diagram, for exampleG E 10 (R) andG E 10 (Z) in theorem 1 and corollary 2 of [4]. Now we begin the E 10 -specific material.…”