Algebra & Number Theory
- Algebra Number Theory
- Volume 10, Number 3 (2016), 533-556.
Presentation of affine Kac–Moody groups over rings
Tits has defined Steinberg groups and Kac–Moody groups for any root system and any commutative ring . We establish a Curtis–Tits-style presentation for the Steinberg group of any irreducible affine root system with rank , for any . Namely, is the direct limit of the Steinberg groups coming from the - and -node subdiagrams of the Dynkin diagram. In fact, we give a completely explicit presentation. Using this we show that is finitely presented if the rank is and is finitely generated as a ring, or if the rank is and is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac–Moody groups when is a Dedekind domain of arithmetic type.
Algebra Number Theory, Volume 10, Number 3 (2016), 533-556.
Received: 23 September 2014
Revised: 21 June 2015
Accepted: 15 October 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20G44: Kac-Moody groups
Secondary: 14L15: Group schemes 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05] 19C99: None of the above, but in this section
Allcock, Daniel. Presentation of affine Kac–Moody groups over rings. Algebra Number Theory 10 (2016), no. 3, 533--556. doi:10.2140/ant.2016.10.533. https://projecteuclid.org/euclid.ant/1510842495