## Algebra & Number Theory

### Presentation of affine Kac–Moody groups over rings

Daniel Allcock

#### Abstract

Tits has defined Steinberg groups and Kac–Moody groups for any root system and any commutative ring $R$. We establish a Curtis–Tits-style presentation for the Steinberg group $St$ of any irreducible affine root system with rank $≥ 3$, for any $R$. Namely, $St$ is the direct limit of the Steinberg groups coming from the $1$- and $2$-node subdiagrams of the Dynkin diagram. In fact, we give a completely explicit presentation. Using this we show that $St$ is finitely presented if the rank is $≥ 4$ and $R$ is finitely generated as a ring, or if the rank is $3$ and $R$ is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac–Moody groups when $R$ is a Dedekind domain of arithmetic type.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 3 (2016), 533-556.

Dates
Revised: 21 June 2015
Accepted: 15 October 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842495

Digital Object Identifier
doi:10.2140/ant.2016.10.533

Mathematical Reviews number (MathSciNet)
MR3513130

Zentralblatt MATH identifier
1348.20058

#### Citation

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

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