## Algebra & Number Theory

### On abstract representations of the groups of rational points of algebraic groups and their deformations

Igor Rapinchuk

#### Abstract

In this paper, we continue our study, begun in an earlier paper, of abstract representations of elementary subgroups of Chevalley groups of rank $≥2$. First, we extend the methods to analyze representations of elementary groups over arbitrary associative rings and, as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form $SLn,D$, where $D$ is a finite-dimensional central division algebra over a field of characteristic $0$. Second, we apply the previous results to study deformations of representations of elementary subgroups of universal Chevalley groups of rank $≥2$ over finitely generated commutative rings.

#### Article information

Source
Algebra Number Theory, Volume 7, Number 7 (2013), 1685-1723.

Dates
Received: 9 June 2012
Revised: 15 June 2012
Accepted: 7 September 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730054

Digital Object Identifier
doi:10.2140/ant.2013.7.1685

Mathematical Reviews number (MathSciNet)
MR3117504

Zentralblatt MATH identifier
1285.20046

Subjects
Primary: 20G15: Linear algebraic groups over arbitrary fields
Secondary: 14L15: Group schemes

#### Citation

Rapinchuk, Igor. On abstract representations of the groups of rational points of algebraic groups and their deformations. Algebra Number Theory 7 (2013), no. 7, 1685--1723. doi:10.2140/ant.2013.7.1685. https://projecteuclid.org/euclid.ant/1513730054

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