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 Mat. Zametki, 2011, Volume 89, Issue 6, Pages 808–824 (Mi mz8779)

Deformations of the Lie Algebra $\mathfrak{o}(5)$ in Characteristics $3$ and $2$

S. Bouarroudja, A. V. Lebedevb, F. Vagemannc

a United Arab Emirates University
b N. I. Lobachevski State University of Nizhni Novgorod
c Université de Nantes, France

Abstract: All finite-dimensional simple modular Lie algebras with Cartan matrix fail to have deformations, even infinitesimal ones, if the characteristic $p$ of the ground field is equal to $0$ or exceeds $3$. If $p=3$, then the orthogonal Lie algebra $\mathfrak o(5)$ is one of two simple modular Lie algebras with Cartan matrix that do have deformations (the Brown algebras $\mathfrak{br}(2;\alpha)$ appear in this family of deformations of the $10$-dimensional Lie algebras, and therefore are not listed separately); moreover, the $29$-dimensional Brown algebra $\mathfrak{br}(3)$ is the only other simple Lie algebra which has a Cartan matrix and admits a deformation. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of an algebraic group $O(5)$ of automorphisms of the Lie algebra $\mathfrak o(5)$ on the space $H^2(\mathfrak o(5);\mathfrak o(5))$ of infinitesimal deformations and presented representatives of the isomorphism classes. We give here an explicit description of the global deformations of the Lie algebra $\mathfrak o(5)$ and describe the deformations of a simple analog of this orthogonal algebra in characteristic $2$. In characteristic $3$, we have found the representatives of the isomorphism classes of the deformed algebras that linearly depend on the parameter.

Keywords: finite-dimensional simple modular Lie algebra, Brown algebra, infinitesimal deformation, global deformation, Cartan matrix, Jacobi identity, Massey bracket, Maurer–Cartan equation, Chevalley basis

DOI: https://doi.org/10.4213/mzm8779

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English version:
Mathematical Notes, 2011, 89:6, 777–791

Bibliographic databases:

UDC: 512.544.3
Revised: 26.05.2010

Citation: S. Bouarroudj, A. V. Lebedev, F. Vagemann, “Deformations of the Lie Algebra $\mathfrak{o}(5)$ in Characteristics $3$ and $2$”, Mat. Zametki, 89:6 (2011), 808–824; Math. Notes, 89:6 (2011), 777–791

Citation in format AMSBIB
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\paper Deformations of the Lie Algebra $\mathfrak{o}(5)$ in Characteristics~$3$ and~$2$
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\vol 89
\issue 6
\pages 808--824
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\jour Math. Notes
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\pages 777--791
\crossref{https://doi.org/10.1134/S0001434611050191}
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• http://mi.mathnet.ru/eng/mz8779
• https://doi.org/10.4213/mzm8779
• http://mi.mathnet.ru/eng/mz/v89/i6/p808

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This publication is cited in the following articles:
1. Bouarroudj S., Lebedev A., Leites D., Shchepochkina I., “Lie Algebra Deformations in Characteristic 2”, Math. Res. Lett., 22:2 (2015), 353–402
2. Bouarroudj S., Grozman P., Lebedev A., Leites D., Shchepochkina I., “New Simple Lie Algebras in Characteristic 2”, Int. Math. Res. Notices, 2016, no. 18, 5695–5726
3. S. Bouarroudj, A. O. Krutov, A. V. Lebedev, D. A. Leites, I. M. Shchepochkina, “Restricted Lie (Super)Algebras in Characteristic 3”, Funct. Anal. Appl., 52:1 (2018), 49–52
4. Bouarroudj S., Krutov A., Leites D., Shchepochkina I., “Non-Degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras”, Algebr. Represent. Theory, 21:5 (2018), 897–941
5. Andrey Krutov, Alexei Lebedev, “On Gradings Modulo $2$ of Simple Lie Algebras in Characteristic $2$”, SIGMA, 14 (2018), 130, 27 pp.
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