V.M. Levchuk, G.S. Suleimanova, N.D. Khodyunya. Nonassociative enveloping algebras of Chevalley algebras ... P. 91-100

An algebra $R$ is said to be an  exact enveloping algebra for a Lie algebra $L$ if $L$ is isomorphic to the algebra $R^{(-)}$ obtained by replacing the multiplication in $R$ by the commutation: $a*b:= ab- ba$. We study exact enveloping algebras of certain subalgebras of a Chevalley algebra over a field $K$ associated with an indecomposable root system $\Phi$. The structure constants of the Chevalley basis of this algebra are chosen with a certain arbitrariness for the niltriangular subalgebra $N\Phi(K)$ with the basis $\{e_r\ |\ r\in\Phi^+\}$. The exact enveloping algebras $R$ for $N\Phi(K)$, which were found in 2018, depend on this choice. The notion of standard enveloping algebra is introduced. For the type $A_{n-1}$, one of the exact enveloping algebras $R$ is the algebra $NT(n,K)$ of all niltriangular $n\times n$ matrices over $K$. The theorem of R. Dubish and S. Perlis on the ideals of $NT(n,K)$ states that $R$ is standard in this case. We prove that an associative exact enveloping algebra $R$ of a Lie algebra $NT(n,K)$ of type $A_{n-1}$ $(n>3)$ is unique and isomorphic to $NT(n,K)$ up to passing to the opposite algebra $R^{({\rm op})}$. Standard enveloping algebras $R$ are described. The existence of a standard enveloping algebra is proved for the Lie algebras $N\Phi(K)$ of all types excepting $D_{n}$ $(n\geq 4)$ and $E_{n}$ $(n=6,7,8)$.

Keywords: Lie algebra, exact enveloping algebra, Chevalley algebra, niltriangular subalgebra, standard ideal

Received December 11, 2019

Revised May 12, 2020

Accepted August 3, 2020

Funding Agency: This work was supported by the Krasnoyarsk Mathematical Center, which is financed by the Ministry of Science and Higher Education of the Russian Federation within the project for the establishment and development of regional centers for mathematical research and education (agreement no. 075-02-2020-1534/1).

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

Galina Safiullanovna Suleimanova, Dr. Phys.-Math. Sci., Khakass Technical Institute — Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: suleymanova@list.ru

Nikolay Dmitrievich Khodyunya, doctoral student, Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: nkhodyunya@gmail.com

REFERENCES

1.   Kaplansky I. Lie algebras and locally compact groups, Chicago and London, The University of Chicago Press, 1971, 152  p. Translated to Russian under the title Algebry Li i lokal’no kompaktnye gruppy, Moscow: Mir Publ., 152 p.

2.   Levchuk V.M. Niltriangular subalgebra of Chevalley algebra: the enveloping algebra, ideals and automorphisms. Dokl. Math., 2018, vol. 97, no. 1, pp. 23–27. doi: 10.1134/S1064562418010088 

3.   Chevalley C. Sur certain groups simples. Tohoku Math. J., 1955, vol. 7, no. 1-2, pp. 14–66. doi: 10.2748/tmj/1178245104 

4.   Carter R. Simple groups of Lie type. N Y: Wiley and Sons, 1972, 331 p. ISBN: 0471137359 .

5.   Dubish R., Perlis S. On total nilpotent algebras. Amer. J. Math., 1951, vol. 73, no. 2, pp. 439–452. doi: 10.2307/2372186 

6.   Egorychev G.P., Levchuk V.M. Enumeration in the Chevalley algebras. ACM SIGSAM Bulletin, 2001, vol. 35, no. 2, pp. 20–34.

7.   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.algebra.2011.10.025 

8.   Serre J-P. Lie algebras and Lie groups. N Y; Amsterdam: Benjamin, 1965, 376 p. Translated to Russian under the title Algebry Li i gruppy Li. Moscow: Mir Publ., 1969, 376 p.

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

10.   Hodyunya N.D. Enumerations of ideals in niltriangular subalgebra of Chevalley algebras. J. SFU Math. and Phys, 2018, vol. 11, no. 3, pp. 271–277. doi: 10.17516/1997-1397-2018-11-3-271-277 

Cite this article as: V.M. Levchuk, G.S. Suleimanova, N.D. Khodyunya. Nonassociative enveloping algebras of Chevalley algebras, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 91–100.