On Gradings Modulo $2$ of Simple Lie Algebras in Characteristic $2$

Andrey Krutov ; Alexei Lebedev

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

The ground field in the text is of characteristic $2$. The classification of modulo $2$ gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson–Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than $3$ indeterminates. Here we completely describe gradings modulo $2$ for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for ${\mathfrak{o}}_\Pi(8)$). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height $n>1$, there is an $(n-2)$-parametric family of modulo $2$ gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple $3|2$-dimensional Lie superalgebra (new result).

Détail

Keywords: modular vectorial Lie algebra; characteristic $2$; simple Lie algebra; simple Lie superalgebra.

Bibliographie

[1] Bahturin Y., Kochetov M., “Group gradings on restricted Cartan-type Lie algebras”, Pacific J. Math., 253 (2011), 289–319, arXiv: 1001.0191 | DOI

[2] Bouarroudj S., Grozman P., Lebedev A., Leites D., Derivations and central extensions of simple modular Lie algebras and superalgebras, arXiv: 1307.1858

[3] Bouarroudj S., Grozman P., Lebedev A., Leites D., Shchepochkina I., Simple vectorial Lie algebras in characteristic 2 and their superizations, arXiv: 1510.07255

[4] Bouarroudj S., Grozman P., Lebedev A., Leites D., Shchepochkina I., “New simple Lie algebras in characteristic 2”, Int. Math. Res. Not., 2016 (2016), 5695–5726, arXiv: 1307.1551 | DOI

[5] Bouarroudj S., Grozman P., Leites D., Deforms of symmetric simple modular Lie (super)algebras, arXiv: 0807.3054

[6] Bouarroudj S., Grozman P., Leites D., “Classification of finite dimensional modular Lie superalgebras with indecomposable Cartan matrix”, SIGMA, 5 (2009), 060, 63 pp., arXiv: 0710.5149 | DOI

[7] Bouarroudj S., Lebedev A., Leites D., Shchepochkina I., Classifications of simple Lie superalgebras in characteristic 2, arXiv: 1407.1695

[8] Bouarroudj S., Lebedev A., Leites D., Shchepochkina I., “Lie algebra deformations in characteristic 2”, Math. Res. Lett., 22 (2015), 353–402, arXiv: 1301.2781 | DOI

[9] Bouarroudj S., Lebedev A., Wagemann F., “Deformations of the Lie algebra ${\mathfrak o}(5)$ in characteristics 3 and 2”, Math. Notes, 89 (2011), 777–791, arXiv: 0909.3572 | DOI

[10] Chebochko N. G., Kuznetsov M. I., “Integrable cocycles and global deformations of Lie algebra of type $G_2$ in characteristic 2”, Comm. Algebra, 45 (2017), 2969–2977 | DOI

[11] Dolotkazin A. H., “Irreducible representations of a simple three-dimensional Lie algebra of characteristic $p=2$”, Math. Notes, 24 (1978), 588–590 | DOI

[12] Eick B., “Some new simple Lie algebras in characteristic 2”, J. Symbolic Comput., 45 (2010), 943–951 | DOI

[13] Elduque A., Kochetov M., Gradings on simple Lie algebras, Mathematical Surveys and Monographs, 189, Amer. Math. Soc., Providence, RI; Atlantic Association for Research in the Mathematical Sciences (AARMS), Halifax, NS, 2013 | DOI

[14] Grishkov A., Zusmanovich P., “Deformations of current Lie algebras. I. Small algebras in characteristic 2”, J. Algebra, 473 (2017), 513–544, arXiv: 1410.3645 | DOI

[15] Grozman P., SuperLie: a Mathematica package for calculations in Lie algebras and superalgebras, http://www.equaonline.com/math/SuperLie

[16] Grozman P., Leites D., “Structures of $G(2)$ type and nonintegrable distributions in characteristic $p$”, Lett. Math. Phys., 74 (2005), 229–262, arXiv: math.RT/0509400 | DOI

[17] Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press, Inc., New York–London, 1978

[18] Kochetov M., “Gradings on finite-dimensional simple Lie algebras”, Acta Appl. Math., 108 (2009), 101–127 | DOI

[19] Kochetov M., Parsons N., Sadov S., “Counting fine grading on matrix algebras and on classical simple Lie algebras”, Internat. J. Algebra Comput., 23 (2013), 1755–1781, arXiv: 1210.4589 | DOI

[20] Kostrikin A. I., “The beginnings of modular Lie algebra theory”, Group Theory, Algebra, and Number Theory (Saarbrücken, 1993), de Gruyter, Berlin, 1996, 13–52

[21] Kuznetsov M. I., “Maximal tori of a general Lie algebra of Cartan type”, Sb. Math., 188 (1997), 1317–1342 | DOI

[22] Lebedev A., “Analogs of the orthogonal, Hamiltonian, Poisson, and contact Lie superalgebras in characteristic 2”, J. Nonlinear Math. Phys., 17:1 (2010), 217–251 | DOI

[23] Permiakov D. S., “Derivations of classical Lie algebras over the field of characteristic 2”, Vestnik Lobachevsky State Univ. Nizhni Novgorod Ser. Math., 1 (2005), 123–134 http://www.vestnik.unn.ru/en/nomera?anum=1455

[24] Richardson Jr. R.W., “On the rigidity of semi-direct products of Lie algebras”, Pacific J. Math., 22 (1967), 339–344 | DOI

[25] Skryabin S., “Classification of Hamiltonian forms over divided power algebras”, Math. USSR-Sb., 69 (1991), 121–141 | DOI

[26] Skryabin S., “Toral rank one simple Lie algebras of low characteristics”, J. Algebra, 200 (1998), 650–700 | DOI

[27] Skryabin S., “On the automorphism group schemes of Lie algebras of Witt type”, Comm. Algebra, 29 (2001), 4047–4077 | DOI

[28] Strade H., Simple Lie algebras over fields of positive characteristic. I. Structure theory, De Gruyter Expositions in Mathematics, 38, Walter de Gruyter Co., Berlin, 2004 | DOI

[29] Tyurin S. A., “Classification of tori in the Zassenhaus algebra”, Russian Math. (Iz. VUZ), 42:2 (1998), 66–73