The Mal'tsev correspondence and isomorphisms of niltriangular subrings of Chevalley algebras

Abstract

Models of algebraic systems of a first-order language are called elementarily equivalent (we write ) if every sentence that is true in one of the models is also true in the other model. The model-theoretic study of linear groups and rings initiated by A. I. Mal'tsev (1960, 1961) is closely related to isomorphism theory; as a rule, the relation of systems was transferred to fields (or rings encountered) of the coefficients. The Mal'tsev correspondence was analyzed for rings of niltriangular matrices and unitriangular groups (B. Rose, 1978; V. Weiler, 1980; K. Videla, 1988; O. V. Belegradek, 1999; V. M. Levchuk, E. V. Minakova, 2009). For unipotent subgroups of Chevalley groups over a field K, the correspondence was studied in 1990 by Videla for char K not equal 2, 3. Earlier the authors announced a weakening of the constraint on the field K in the Videla theorem. In the Chevalley algebra associated with a root system Phi and a ring K, the niltriangular subalgebra N Phi(K) is naturally distinguished. The main results of this paper establish the Mal'tsev correspondence (related with the description of isomorphisms) for the Lie rings N Phi(K) of classical types over arbitrary associative commutative rings with unity. A corollary is noted for (nonassociative) enveloping algebras to N Phi(K).