I.N. Zotov, V.M. Levchuk. The Mal’tsev correspondence and isomorphisms of niltriangular subrings of Chevalley algebras ... P. 135-145

Models of algebraic systems of a first-order language are called elementarily equivalent (we write $\equiv$) 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 $\equiv$ 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 $char  \, K\ne 2,3$. 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 $N\Phi(K)$.

Keywords: Chevalley algebra, niltriangular subalgebra, isomorphism, model-theoretic Mal'tsev correspondence

Received September 10, 2018

Revised November 20, 2018

Accepted November 26, 2018

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 16-01-00707).

Igor’ Nikolaevich Zotov, Siberian Federal University, Krasnoyarsk, 660041 Russia,
e-mail: zotovin@rambler.ru

Vladimir Mikhailovich Levchuk, Dr. Phys.-Math. Sci., Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: vlevchuk@sfu-kras.ru

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