Geodesic
Niltriangular subalgebras of the Chevalley algebras and their generalizations
Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 2, pp. 37-46 
V. M. Levchuk; A. V. Litavrin; N. D. Hodyunya; V. V. Tsigankov. Niltriangular subalgebras of the Chevalley algebras and their generalizations. Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 2, pp. 37-46. http://geodesic.mathdoc.fr/item/VMJ_2015_17_2_a5/
@article{VMJ_2015_17_2_a5,
     author = {V. M. Levchuk and A. V. Litavrin and N. D. Hodyunya and V. V. Tsigankov},
     title = {Niltriangular subalgebras of the {Chevalley} algebras and their generalizations},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {37--46},
     year = {2015},
     volume = {17},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2015_17_2_a5/}
}
TY  - JOUR
AU  - V. M. Levchuk
AU  - A. V. Litavrin
AU  - N. D. Hodyunya
AU  - V. V. Tsigankov
TI  - Niltriangular subalgebras of the Chevalley algebras and their generalizations
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2015
SP  - 37
EP  - 46
VL  - 17
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VMJ_2015_17_2_a5/
LA  - ru
ID  - VMJ_2015_17_2_a5
ER  - 
%0 Journal Article
%A V. M. Levchuk
%A A. V. Litavrin
%A N. D. Hodyunya
%A V. V. Tsigankov
%T Niltriangular subalgebras of the Chevalley algebras and their generalizations
%J Vladikavkazskij matematičeskij žurnal
%D 2015
%P 37-46
%V 17
%N 2
%U http://geodesic.mathdoc.fr/item/VMJ_2015_17_2_a5/
%G ru
%F VMJ_2015_17_2_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $UT(n,q)$.

[1] Carter R., Simple groups of Lie type, Wiley and Sons, N.Y., 1972, 364 pp. | MR | Zbl

[2] Levchuk V. M., “Svyazi unitreugolnoi gruppy s nekotorymi koltsami”, Algebra i logika, 1:5 (1976), 558–578

[3] Levchuk V. M., “Nekotorye lokalno nilpotentnye koltsa i ikh prisoedinennye gruppy”, Mat. zametki, 42:5 (1987), 631–641 | MR | Zbl

[4] Maltsev A. I., “Obobschenno nilpotentnye algebry i ikh prisoedinennye gruppy”, Mat. sb., 25(67):3 (1949), 347–366 | MR | Zbl

[5] Ado I. D., “O nilpotentnykh algebrakh i $p$-gruppakh”, Dokl. AN SSSR, 40:8 (1943), 339–342 | MR

[6] McLain D. H., “A characteristically-simple group”, Proc. Cambridge Phil. Soc., 50 (1954), 641–642 | DOI | MR | Zbl

[7] Merzlyakov Yu. I., “Ekvipodgruppy unitreugolnykh grupp: kriterii samonormalizuemosti”, Dokl. RAN, 339:6 (1994), 732–735 | Zbl

[8] Levchuk V. M., “Avtomorfizmy unipotentnykh podgrupp grupp Shevalle”, Algebra i logika, 29:3 (1990), 315–338 | MR | Zbl

[9] Kondratev A. S., “Podgruppy konechnykh grupp Shevalle”, Uspekhi mat. nauk, 41:1(247) (1986), 57–96 | MR | Zbl

[10] Videla C. R., “On the Mal'cev correspondence”, Proceed. AMS, 109:2 (1990), 493–502 | MR | Zbl

[11] Levchuk V. M., “Teoretiko-modelnye i strukturnye voprosy algebr i grupp Shevalle”, Gruppy i grafy, Itogi nauki. Yug Rossii. Ser. Mat. forum, 6, YuMI VNTs RAN i RSO-A, Vladikavkaz, 2012, 75–84

[12] Cao Y., Jiang D., Wang D., “Automorphisms of certain nilpotent algebras over commutative rings”, J. Algebra, 17:3 (2007), 527–555 | MR | Zbl

[13] Litavrin A. V., “Avtomorfizmy nilpotentnoi podalgebry $N\Phi(K)$ algebry Shevalle simplekticheskogo tipa”, Izv. IrkGU. Ser. mat-ka, 8:2 (2015), 43–58

[14] Levchuk V. M., “Chevalley groups and their unipotent subgroups”, Contemp. Math. (AMS), 131, Part 1 (1992), 227–242 | DOI | MR | Zbl

[15] Levchuk V. M., Suleimanova G. S., “Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups”, Proceed. Steklov Inst. Math., 267, Suppl. 1, Pleiades Publ., Ltd, 2009, 118–127 | DOI | MR

[16] Slowik R., “Bijective maps of infinite triangular and unitriangular matrices preserving commutators”, Linear and Multilinear Algebra, 61:8 (2013), 1028–1040 | DOI | MR | Zbl

[17] Egorychev G. P., Levchuk V. M., “Enumeration in the Chevalley algebras”, ACM SIGSAM Bulletin, 35:2 (2001), 439–452 | DOI

[18] Gorchakov Yu. M., Gruppy s konechnymi klassami sopryazhennykh elementov, Nauka, M., 1978, 119 pp. | MR

[19] Levchuk V. M., Martynova L. A., “Normalnoe stroenie unipotentnykh podgrupp grupp Shevalle i idealy assotsiirovannogo koltsa Li”, Konstruktsii v algebre i logike, TGU, Tver, 1990, 60–66 | MR

[20] Martynova L. A., Normalnoe stroenie i avtomorfizmy unipotentnykh podgrupp grupp lievykh tipov, Diss. $\dots$ k.f.-m.n., MGU, M., 1994

[21] Krivokolesko V. P., Levchuk V. M., “Perechislenie idealov isklyuchitelnykh nilpotentnykh matrichnykh algebr”, Trudy IMM UrO RAN, 21, no. 1, 2015, 166–171

[22] Egorychev G. P., Integral represantation and computation combinatorial sums, Transl. of Math. Monogr., 59, Americ. Math. Soc., 1984, 300 pp., 2-d ed. in 1989 | MR | MR

[23] Dubish R., Perlis S., “On total nilpotent algebras”, Amer. J. Math., 73 (1951), 439–452 | DOI | MR

[24] Levchuk V. M., “Podgruppy unitreugolnoi gruppy”, Izv. AN CCCR. Ser. mat., 38:6 (1974), 1202–1220 | MR | Zbl

[25] Egorychev G. P., Kuzucuoglu F., Levchuk V. M., “Enumeration of ideals of some nilpotent matrix rings”, J. Algebra and Its Applications, 12:1 (2013), 1250140, 11 pp. | DOI | MR | Zbl