Algebra & Number Theory

Presentation of affine Kac–Moody groups over rings

Daniel Allcock

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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

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

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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

affine Kac–Moody group Steinberg group Curtis–Tits presentation


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.

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