Научная электронная библиотека ULRICH'S Periodicals Directory

Сибирский государственный университет
науки и технологий имени академика М.Ф. Решетнева

Сибирский аэрокосмический журнал
ISSN 2712-8970 (Print) ISSN 2782-5760 (on-line)

Сведения об образовательной организации

UDK 512.554
AUTOMORPHISMS OF NIL-TRIANGULAR SUBRINGS IN CHEVALLEY ALGEBRA OF ORTHOGONAL TYPE
V. M. Levchuk, A. V. Litavrin
Siberian Federal University; 79, Svobodny Av., Krasnoyarsk, 660041, Russian Federation
Any Chevalley algebra over an associative commutative ring K with the identity is characterized by Chevalley base that correspondents to each indecomposable root system Ф. All elements er (r  Ф+ ) of Chevalley base give a base of subalgebra NФ(K) which is said to be nil-triangular. Automorphisms of algebras NФ(K) were described by Y. Cao, D. Jiang and D. Wang (J. Algebra, 2007) at K = 2K for Lie type Bn, Cn or F4 and under similar restrictions for other types. Their description uses only non-standard Gibbs’s automorphisms; in our terminology it is a hypercentral automorphisms of height 2 or 3 (for type Cn). Our main purpose is to describe the automorphism group A of the Lie ring NФ(K). The algebra NФ(K) of Lie type An–1 can be represented as Lie algebra which associated to the algebra NT(n, K) of all nil-triangular n n matrices over K. The automorphism group of the ring NT(n, K) and of its associated Lie ring (i. e., A for the type An) described earlier V. M. Levchuk (1983). A. V. Litavrin has described the automorphism group A for Lie type Cn recently. In the present paper we find non-standard automorphisms of the algebra NФ(K) for orthogonal types, when the condition K = 2K isn’t satisfied. It seems that if annihilator of element 2 in K is non-zero, then the largest height of hypercentral automorphisms grows together with the Lie rank. Also, we find automorphisms of the algebra NФ(K) of type Dn which are non-standard module second member of lower central series and generate the subgroup of A that isomorphic to certain subgroup S in SL(2, K); in particularly, S = SL(2, K) at 2K = 0. The standard automorphisms together with constructed non-standard automorphisms generate every automorphisms of the algebra NФ(K). For all classical types of rank > 4 our results show that the automorphism group A is the product of subgroups of the central and induced ring automorphisms and the automorphism group of the algebra NФ(K). We use developed earlier methods, in particularly, a special representation of the algebras NФ(K) of classical types. The results can be used in development of cryptographic methods.
Keywords: Chevalley algebra; nil-triangular subalgebra; automorphism of Lie ring; height of hypercentral automorphism.
References

1. Carter R. Simple groups of Lie type. New York: Wiley and Sons. 1972, 346 p.

2. Stein M. R. Generators, relations and coverings of Chevalley groups over commutative rings. Amer. J. Math. 1971, Vol. 93, No. 4, P. 965–1004.

3. Hurley J. F. Ideals in Chevalley algebras. Trans. Amer. Math. Soc. 1969, Vol. 137, No. 3, P. 245–258.

4. Levchuk V. M. [Communication unitriangular group with some rings. Part 2. Groups of automorphisms]. Sibirskiy matematicheskiy zhurnal. 1983, Vol. 24, No. 4, P. 543–557 (In Russ.).

5. Cao Y., Jiang D., Wang D. Automorphisms of certain nilpotent algebras over commutative rings. J. Algebra. 2007, Vol. 17, No. 3, P. 527–555.

6. Gibbs J. Automorphisms of certain unipotent groups. J. Algebra. 1970, Vol. 14, No. 2, P. 203–228.

7. Kondrat’ev A. S. [Subgroups of finite Chevalley groups]. Uspekhi matematicheskikh nauk. 1986, Vol. 41, No. 1, P. 57–96 (In Russ.).

8. Levchuk V. M. [Automorphisms of unipotent subgroups of Chevalley groups]. Algebra i logika. 1990, Vol. 29, No. 3, P. 316–338 (In Russ.).

9. Levchuk V. M. Chevalley groups and their unipotent subgroups. Contemp. Math., AMS. 1992, Vol. 131, Part 1, P. 227–242.

10. Levchuk V. M. [Model-theoretic and structural problems of algebra and Chevalley groups]. Matematicheskiy forum, gruppy i grafy.-Vladikavkaz: YuMI VNTs RAN i RSO-A. 2011, Vol. 6, P. 71–80 (In Russ.).

11. Mal’tsev A. I. [A correspondence between the rings and groups]. Matematicheskiy sbornik.1960, Vol. 50, P. 257–266 (In Russ.).

12. Litavrin A. V. [Automorphisms of the nilpotent subalgebra NΦ(K) Chevalley algebra of symplectic type]. Izvestiya IrkGU, ser. matem. 2015, Vol. 13, No. 3, P. 41–55 (In Russ.).

13. Levchuk V. M., Litavrin A. V., Khodyunya N. D., Tsygankov V. V. [Niltriangular subalgebras of the Chevalley algebras and their generalizations]. Vladikavkazskiy matematicheskiy zhurnal. 2015, Vol. 17, No. 2, P. 37–46 (In Russ.).

14. Seligman G. B. On automorphisms of Lie algebras of classical type. III. Trans. Amer. Math. Soc. 1960, Vol. 97, P. 286–316.

15. Steinberg R. Lections on Chevalley groups. Yale University, 1967, 151 p.

16. Bourbaki N. Gruppy i algebry Li [Groups and Lie algebra]. Moscow, Mir Publ., 1972, 334 p.


Levchuk Vladimir Mikhaylovich – Dr. Sc., professor, Siberian Federal University. Е-mail: vlevchuk@sfu-kras.ru.

Litavrin Andrey Viktorovich – postgraduate student, Siberian Federal University. Е-mail: anm11@rambler.ru.