Сибирский федеральный университет

Тип публикации: статья из журнала

Год издания: 2020

Идентификатор DOI: 10.21538/0134-4889-2020-26-3-91-100

Ключевые слова: chevalley algebra, exact enveloping algebra, lie algebra, niltriangular subalgebra, standard ideal

Аннотация: 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 Φ. The structure constants of the Chevalley basis of this algebra are chosen with a certain arbitrariness for the niltriangular subalgebra NΦ(K) with the basis {er | r ∈ Φ+}. The exact enveloping algebras R for NΦ(K), which were found in 2018, depend on this choice. The notion of standard enveloping algebra is introduced. For the type An−1, one of the exact enveloping algebras R is the algebra NT (n, K) of all niltriangular n × 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 An−1 (n > 3) is unique and isomorphic to NT (n, K) up to passing to the opposite algebra R(op). Standard enveloping algebras R are described. The existence of a standard enveloping algebra is proved for the Lie algebras NΦ(K) of all types excepting Dn (n ≥ 4) and En (n = 6, 7, 8). © Krasovskii Institute of Mathematics and Mechanics.

Журнал: Trudy Instituta Matematiki i Mekhaniki UrO RAN

Выпуск журнала: Vol. 26, Is. 3

Номера страниц: 91-100

ISSN журнала: 01344889

Издатель: Krasovskii Institute of Mathematics and Mechanics

• Levchuk V.M. (Siberian Federal University, Krasnoyarsk, 660041, Russian Federation)
• Suleimanova G.S. (Khakass Technical Institute, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation)
• Khodyunya N.D. (Siberian Federal University, Krasnoyarsk, 660041, Russian Federation)

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