The local structure of groups of triangular automorphisms of relatively free algebras

Abstract

Listen

Translate article

Let K be an arbitrary field and C n a relatively free algebra of rank n. In particular, as C n we may treat a polynomial algebra P n , a free associative algebra A n , or an absolutely free algebra F n . For the algebras C n = P n , A n , F n , it is proved that every finitely generated subgroup G of a group TC n of triangular automorphisms admits a faithful matrix representation over a field K; hence it is residually finite by Mal’tsev’s theorem. For any algebra C n , the triangular automorphism group TC n is locally soluble, while the unitriangular automorphism group UC n is locally nilpotent. Consequently, UC n is local (linear and residually finite). Also it is stated that the width of the commutator subgroup of a finitely generated subgroup G of UC n can be arbitrarily large with increasing n or transcendence degree of a field K over the prime subfield.