Abstract: We study some problems concerned with ideals and automorphisms of niltriangular subalgebras of classical Lie type Chevalley algebras over a field \(K\) and of their non-finitary generalizations and also automorphisms of adjoint group. We characterize (for Lie type \(A_ {n-1})\) every Lie ideal of algebra \(NT(n, K)\) of all niltriangular \(n \times n\) matrices by a selection of constants from \(K\). When \(K = GF(q)\), this gives a combinatorial expression of number of Lie ideals and, for a simple \(q\), also the number of normal subgroups in unitriangular group \(UT(n,q)\).
For citation: Levchuk V. M., Litavrin A. V., Hodyunya N. D., Tsigankov V. V. Niltriangular subalgebras of the Chevalley algebras and their generalizations. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol.
17, no. 2, pp.37-46.
DOI 10.23671/VNC.2015.2.7278
1. Carter R. Simple Groups of Lie Type, N.Y., Wiley and Sons, 1972,
364 p.
2. Levchuk V. M. Svjazi unitreugol'noj gruppy s nekotorymi
kol'cami. Algebra i Logika [Algebra and Logic], 1976, vol. 1, no. 5,
pp. 558-578 (Russian).
3. Levchuk V. M. Some locally nilpotent rings and their adjoined
groups. Mathematical Notes [Mat. Zametki], 1987,
vol. 42, no. 5, pp. 848-853.
4. Mal'cev A. I. Generalized nilpotent algebras and their
associated groups. Mat. Sb. (N.S.) [Sbornik: Mathematics], 1949,
vol. 25(67), no. 3, pp. 347-366 (Russian).
5. Ado I. D. O nil'potentnyh algebrah i p-gruppah. Dokl. AN SSSR
[Soviet Math. Dokl.], 1943, vol. 40, no. 8, pp. 339-342 (Russian).
6. McLain D. H. A characteristically-simple group. Proc. Cambridge
Phil. Soc., 1954, vol. 50, pp. 641-642.
7. Merzljakov Ju. I. Jekvipodgruppy unitreugol'nyh grupp: kriterij
samonormalizuemosti. Dokl. AN [Doklady Mathematics], 1994, vol. 339,
no. 6, pp. 732-735 (Russian).
8. Levchuk V. M. Avtomorfizmy unipotentnyh podgrupp grupp Shevalle.
Algebra i Logika [Algebra and Logic], 1990, vol. 29, no. 3, pp.
315-338 (Russian).
9. Kondrat'ev A. S. Subgroups of finite Chevalley groups. Russian
Mathematical Surveys [Uspekhi Mat. Nauk], 1986,
vol. 41, no. 1, pp. 65-118.
10. Videla C. R. On the Mal'cev correspondence. Proc. Amer. Math.
Soc., 1990, vol. 109, no. 2, pp. 493-502.
11. Levchuk V. M. Model-theoretic and structural problems of
Chevalley groups and algebras. Mat. Forum. Vol. 6. Gruppy i Grafy
[Math. Forum. Vol. 6. Groups and Grafs], Vladikavkaz, SMI VSC RAS,
2012, pp. 75-84 (Russian).
12. Cao Y., Jiang D., Wang D. Automorphisms of certain nilpotent
algebras over commutative rings. J. Algebra, 2007, vol. 17, no. 3,
pp. 527-555.
13. Litavrin A. V. Automorphisms of the nilpotent subalgebra N?(K)
Chevalley algebra of symplectic type. Izv. IrkGU. Ser. Matematika,
2015, vol. 13, pp. 41-55 (Russian).
14. Levchuk V. M. Chevalley groups and their unipotent subgroups.
Contemp. Math., AMS, 1992, vol. 131, part 1, pp. 227-242.
15. Levchuk V. M., Suleimanova G. S. Automorphisms and normal
structure of unipotent subgroups of finitary Chevalley groups.
Proceed. Steklov Inst. Math, Pleiades Publ., Ltd, 2009, vol. 3, pp.
118-127.
16. Slowik R. Bijective maps of infinite triangular and
unitriangular matrices preserving commutators. Linear and
Multilinear Algebra, 2013, vol. 61, no. 8, pp. 1028-1040.
17. Egorychev G. P., Levchuk V. M. Enumeration in the Chevalley
algebras. ACM SIGSAM Bulletin, 2001, vol. 35, no. 2, pp. 439-452.
18. Gorchakov Ju. M. Gruppy s Konechnymi Klassami Soprjazhennyh
Jelementov, M., Nauka, 1978, 119 p (Russian).
19. Levchuk V. M., Martynova L. A. Normal'noe stroenie unipotentnyh
podgrupp grupp Shevalle i idealy associirovannogo kol'ca Li,
Konstrukcii v Algebre i Logike, Tver', TGU, 1990, pp. 60-66
(Russian).
20. Martynova L. A. Normal'noe Stroenie i Avtomorfizmy Unipotentnyh
Podgrupp Grupp Lievyh Tipov: Diss. ... k.f.-m.n., M., MGU, 1994
(Russian).
21. Krivokolesko V. P., Levchuk V. M. Enumeration of ideals of
exceptional nilpotent matrix algebras. Trudy Inst. Mat. i Mekh. UrO
RAN [Proceedings of the Steklov Institute of Mathematics], 2015,
vol. 21, no. 1, pp. 166-171 (Russian).
22. Egorychev G. P. Integral Represantation and Computation
Combinatorial Sums [Integral'noe Predstavlenie i Vychislenie
Kombinatornyh Summ], Americ. Math.
Soc., 1984, 300 p. (Transl. of Math. Monogr., vol. 59); 2-d ed. in
1989.
23. Dubish R., Perlis S. On total nilpotent algebras. Amer. J.
Math., 1951, vol. 73, pp. 439-452.
24. Levchuk V. M. Subgroups of the unitriangular group. Mathematics
of the USSR-Izvestiya [Izv. AN SSSR. Ser. Mat., 1974, vol. 38, no.
6, pp. 1202-1220], 1974, vol. 8, no. 6, pp. 1191-1208.
25. Egorychev G. P., Kuzucuoglu F., Levchuk V. M. Enumeration of
ideals of some nilpotent matrix rings. J. Algebra and its
Applications, 2013, vol. 12, no. 1, 1250140 [11 pages].
