Simple Lie superalgebras and nonintegrable distributions in characteristic $p$

S. Bouarroudj   ; D. A. Leites

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 15-29

Résumé

Recently, Grozman and Leites returned to the original Cartan's description of Lie algebras to interpret the Melikyan algebras (for $p\le 5$) and several other little-known simple Lie algebras over algebraically closed fields for $p=3$ as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by $G(2)$, $O(7)$, $Sp(4)$, and $Sp(10)$. The description was performed in terms of Cartan–Tanaka–Shchepochkina prolongs using Shchepochkina's algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite dimensional Lie superalgebras over $\mathbb C$; for $p=3$, we obtain a series of new simple Lie superalgebras and an exceptional one.

Bibliographie

[1] D. Alekseevsky, D. Leites, I. Shchepochkina, “Examples of simple Lie superalgebras of vector fields”, C. R. Acad. Bulg. Sci., 34:9 (1980), 1187–1190

[2] M. Atiyah, E. Witten, “$M$-theory dynamics on a manifold of $G_2$ holonomy”, Adv. Theor. Math. Phys., 6:1 (2002), 1–106 | MR | Zbl

[3] A. Belov-Kanel, M. Kontsevich, “Automorphisms of the Weyl algebra”, Lett. Math. Phys., 74:2 (2005), 181–199 | DOI | MR | Zbl

[4] A. Belov-Kanel, L. Rowen, “Computational Aspects of Polynomial Identities”, Research Notes Math., 9 (2005), AK Peters, Ltd., Wellesley, MA | MR

[5] J. Bernstein, D. Leites, “Invariant differential operators and irreducible representations of Lie superalgebras of vector fields”, Selecta Math. Sov., 1:2 (1981), 143–160 | MR | Zbl

[6] É. Cartan, “Über die einfachen Transformationsgrouppen”, Leipziger Berichte, 1893, 395–420 ; ØE uvres complètes. Partie II: Algèbre, systèmes différentiels et problèmes d'équivalence, Second edition, Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1984 | Zbl

[7] A. Ciampella, L. A. Lomonaco, “The universal Steenrod algebra at odd primes”, Comm. Algebra, 32:7 (2004), 2589–2607 | DOI | MR | Zbl

[8] I. Cunha, A. Elduque, An extended Freudenthal magic square in characteristic $3$, arXiv: /math.RA/0605379 | MR

[9] W. Fulton, J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991 | MR | Zbl

[10] P. Grozman, SuperLie, http://www.equaonline.com/math/SuperLie

[11] P. Grozman, D. Leites, “Defining relations for classical Lie superalgebras with Cartan matrix”, Czech. J. Phys., 51:1 (2001), 1–22 ; arXiv: /hep-th/9702073 | DOI | MR

[12] P. Grozman, D. Leites, SuperLie and problems (to be) solved with it, Preprint MPIM-Bonn, 2003-39 | MR

[13] P. Grozman, D. Leites, “Structures of $G(2)$ type and nonintegrable distributions in characteristic $p$”, Lett. Math. Phys., 74:3 (2005), 229–262 ; arXiv: /math.RT/0509400 | DOI | MR | Zbl

[14] V. G. Kac, “Letter to the editors: “Classification of simple Lie superalgebras””, Funkts. Anal. Prilozhen, 9:3 (1975), 91–92 | MR | Zbl

[15] V. Kac, “Lie superagebras”, Adv. Math., 26 (1977), 8–96 | DOI | Zbl

[16] V. Kac, “Classification of supersymmetries”, Proceedings of the International Congress of Mathematicians, v. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 319–344 ; Shun-Jen Cheng, V. Kac, “Addendum: Generalized Spencer cohomology and filtered deformations of $\mathbb{Z}$-graded Lien superalgebras”, Adv. Theor. Math. Phys., 8:4 (2004), 697–709 ; N. Cantarini, S.-J. Cheng, V. Kac, “Errata to paper: Cheng and Kac, "Structure of some $\mathbb{Z}$-graded Lie superalgebras of vector fields"”, Transform. Groups, 9:4 (2004), 399–400 | MR | Zbl | MR | Zbl | DOI | MR

[17] A. I. Kostrikin, I. R. Shafarevich, “Graded Lie algebras of finite characteristic”, Izv. Akad. Nauk SSSR, Ser. Mat., 33 (1969), 251–322 | MR | Zbl

[18] A. Lebedev, Nondegenerate bilinear forms in characteristic 2, related contact forms, simple Lie algebras and simple superalgebras, arXiv: /math.AC/0601536

[19] D. Leites, “Indecomposable representations of Lie superalgebras”, Memorial Volume Dedicated to Misha Saveliev and Igor Luzenko (JINR, Dubna), eds. A. N. Sissakian et al., 2000, 126–131; arXiv: /math.RT/0202184

[20] D. Leites, I. Shchepochkina, Classification of the simple Lie superalgebras of vector fields, Preprint MPIM-Bonn, 2003-28

[21] V. Serganova, “Automorphisms of simple Lie superalgebras”, Izv. Akad. Nauk SSSR, Ser. Mat., 48:3 (1984), 585–598 | MR

[22] V. Serganova, “On generalizations of root systems”, Comm. Algebra, 24:13 (1996), 4281–4299 | DOI | MR | Zbl

[23] I. Shchepochkina, “How to realize Lie algebras by vector fields”, Theor. Math. Phys., 147:3 (2006), 821–838 ; arXiv: /math.RT/0509472 | DOI | MR | Zbl

[24] I. Shchepochkina, “Five exceptional simple Lie superalgebras of vector fields and their fourteen regradings”, Represent. Theory Electr. J. Amer. Math. Soc., 3 (1999), 373–415 ; arXiv: /hep-th/9702121 | DOI | MR | Zbl

[25] H. Strade, Simple Lie Algebras Over Fields of Positive Characteristic. I. Structure Theory, de Gruyter Expositions Math., 38, Walter de Gruyter Co, Berlin, 2004 | MR

[26] J. van de Leur, Contragredient Lie superalgebras of finite growth, Ph. D. Thesis, Utrecht, 1986; Comm. Algebra, 17 (1989), 1815–1841 | DOI | MR | Zbl

[27] R. N. W. Wood, “Problems in the Steenrod algebra”, Bull. London Math. Soc., 30:5 (1998), 449–517 | DOI | MR | Zbl

[28] K. Yamaguchi, “Differential systems associated with simple graded Lie algebras Progress in differential geometry”, Adv. Stud. Pure Math., 22, Math. Soc. Japan, Tokyo, 1993, 413–494 | MR | Zbl