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


1.   Mal’tsev A.I. On a correspondence between rings and groups. Am. Math. Soc., Transl., Ser. II, 1965, vol. 45, pp. 221–231. doi: 10.1090/trans2/045/14

2.   Mal’tsev A.I. Elementary properties of linear groups. Nekotorye Problemy v Matematike i Mekhanike, Novosibirsk, Izd-vo AN SSSR, 1961, pp. 110–132 (in Russian).

3.   Chang C.C., Keisler H.J. Model theory. Studies in logic and the foundations of mathematics, vol. 73. Amsterdam; London: North-Holland Publ.; N Y: Elsevier, 1973, 550 p. ISBN(3rd ed.): 0-444-88054-2 . Translated to Russian under the title Teoriya modelei, Moscow: Mir Publ., 1977, 614 p.

4.   Hodges W. Model theory. Cambridge: Cambridge Univ. Press, 1993, 772 p. ISBN: 978-0-521-30442-9

5.   Remeslennikov V.N., Roman’kov V.A. Model-theoretic and algorithmic questions in group theory. J. Soviet Math., 1985, vol. 31, no. 3, pp. 2887–2939. doi: 10.1007/BF02106805

6.   Rose B.I. The $\chi_1$-categoricity of strictly upper triangular matrix rings over algebraically closed fields. J. Symbolic Logic, 1978, vol. 43, no. 2, pp. 250–259. doi: 10.2307/2272823

7.   Wheeler W.H. Model theory of strictly upper triangular matrix ring. J. Symbolic Logic, 1980, vol. 45, no. 3, pp. 455–463. doi: 10.2307/2273414

8.   Videla C.R. On the model theory of the ring NT(n,R). Pure Appl. Algebra, 1988, vol. 55, no. 3, pp. 289–302. doi: 10.1016/0022-4049(88)90120-X

9.   Belegradek O.V. Model theory of unitriangular groups. Amer. Math. Soc. Transl., 1999, vol. 195, no. 2, pp. 1–116.

10.   Levchuk V.M., Minakova E.V. Elementary equivalence and isomorphisms of locally nilpotent matrix groups and rings. Dokl. Math., 2009, vol. 79, no. 2, pp. 185–188. doi: 10.1134/S1064562409020100

11.   Videla C.R. On the Mal’cev correspondence. Proceed. AMS, 1990, vol. 109, no. 2, pp. 493–502. doi: 10.2307/2048013

12.   Bunin E.I., Mikhalev A.V., Pinus A.G. Elementarnaya i blizkaya k nei logicheskie ekvivalentnosti klassicheskikh i universal’nykh algebr (Elementary and close to it logical equivalences of classical and universal algebras). Moscow: MTsNMO Publ., 2015, 360 p. ISBN: 978-5-4439-2401-4 .

13.   Carter R.W. Simple groups of Lie type. New York: Wiley and Sons, 1972, 331 p. ISBN: 0471137359 .

14.   Levchuk V.M. Model-theoretic and structural problems of Chevalley groups and algebras. Mat. Forum. Vol. 6. Gruppy i Grafy, Vladikavkaz, SMI VSC RAS Publ., 2012, pp. 75–84 (in Russian).

15.   Kuzucuoglu F., Levchuk V.M. Isomorphisms of certain locally Nilpotent finitary groups and associated rings. Acta Applicandae Mathematicae, 2004, vol. 82, no. 2, pp. 169–181. doi: 10.1023/B:ACAP.0000027533.59937.14

16.   Levchuk V.M., Suleimanova G.S. Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type. J. Algebra, 2012, vol. 349, no. 1, pp. 98–116. doi: 10.1016/j.jalgebra.2011.10.025

17.   Levchuk V.M. Automorphisms of unipotent subgroups of Chevalley groups. Algebra Logic, 1990, vol. 29, no. 3, pp. 211–224. doi: 10.1007/BF01979936

18.   Levchuk V.M. Automorphisms of unipotent subgroups of Lie type groups of small ranks. Algebra Logic, 1990, vol. 29, no. 2, pp. 97–112. doi: 10.1007/BF02001355

19.   Levchuk V.M., Litavrin A.V. Hypercentral automorphisms of nil-triangular subalgebras in Chevalley algebras. Sib. Elektron. Mat. Izv., 2016, vol. 13, pp. 467–477 (in Russian). doi: 10.17377/semi.2016.13.040

20.   Gibbs J.A. Automorphisms of certain unipotent groups. J. Algebra, 1970, vol. 14, no. 2, pp. 203–208. doi: 10.1016/0021-8693(70)90123-7

21.   Cao Y., Jiang D., Wang D. Automorphisms of certain nilpotent Lie algebras over commutative rings. International Journal of Algebra and Computation. 2007, vol. 17, no. 3, pp. 527–555. doi: 10.1142/S021819670700372X

22.   Serre J.-P. Algebres de Lie semi-simple complexes. New York; Amsterdam: Benjamin, 1966, 130 p.

23.   Levchuk V.M. Connections between a unitriangular group and certain rings. Part 2. Groups of automorphisms. Siberian Mat. J., 1983, vol. 24, no. 4, pp. 543–557. doi: 10.1007/BF00969552 .

24.   Levchuk V.M., Minakova E.V. Automorphisms and model-theoretic problems for nilpotent matrix groups and rings. J. Math. Sci. (N.Y.), 2010, vol. 166, no. 5, pp. 675–681. doi: 10.1007/s10958-010-9883-3 .

25.   Levchuk V.M. Nil-triangular subalgebra of algebra: enveloping algebra, ideals and automorphisms. Dokl. Akad. nauk, 2018, vol. 478, no. 2, pp. 137–140 (in Russian). doi: 10.7868/S0869565218020032 .