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The five exceptional simple Lie
superalgebras of vector fields
and their fourteen regradings

Author: Irina Shchepochkina
Journal: Represent. Theory 3 (1999), 373-415
MSC (1991): Primary 17A70; Secondary 17B35
Published electronically: October 13, 1999
MathSciNet review: 1715110
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Abstract: The five simple exceptional complex Lie superalgebras of vector fields are described. One of them, ${\mathfrak{k}}{\mathfrak{as}}$, is new; the other four are explicitly described for the first time. All nonisomorphic maximal subalgebras of finite codimension of these Lie superalgebras, i.e., all other realizations of these Lie superalgebras as Lie superalgebras of vector fields, are also described; there are 14 of them altogether. All of the exceptional Lie superalgebras are obtained with the help of the Cartan prolongation or a generalized prolongation.

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  • [ALSh] Alekseevsky D., Leites D., Shchepochkina I., Examples of infinite-dimensional simple Lie superalgebras of vector fields. C.R. Acad. Bulg. Sci., v. 33, N 9, 1980, 1187-1190 (in Russian) MR 82k:17010
  • [BL] Bernstein J., Leites D., Invariant differential operators and irreducible representations of Lie superalgebras of vector fields. Serdika, v.7, 1981 320-334 (in Russian); Sel. Math. Sov., v. 1, N 2, 1981, 143-160 MR 84i:17015
  • [CK] Cheng, Shun-Jen; Kac, Victor G. A new $N=6$ superconformal algebra. Comm. Math. Phys. 186 (1997), no. 1, 219-231 MR 99f:17029
  • [GLS] Grozman P., Leites D., Shchepochkina I., Lie superalgebras of string theories. hep-th 9702120
  • [GPS] Gomis J., París J., Samuel S., Antibracket, antifields and gauge-theory quantization, Phys. Rep. 259 (1995), no. 1-2, 145 pp. MR 96e:81223
  • [K1] Kac V. G., Lie superalgebras. Adv. Math. v. 26, 1977, 8-96 MR 58:5803
  • [Ko1] Kotchetkoff Yu. Déformations de superalgèbres de Buttin et quantification. C.R. Acad. Sci. Paris, Sér. I, 299: 14 (1984), 643-645 MR 85k:17021
  • [Ko2] Kochetkov Yu. Deformations of Lie superalgebras. VINITI Depositions, Moscow (in Russian) 1985, 384-85
  • [KL] Kotchetkov Yu., Leites D., Simple Lie algebras in characteristic 2 recovered from superalgebras and on the notion of a simple finite group. In: Kegel O. (eds.) Proc. Internat. algebraic conference, Novosibirsk, August 1989, Contemporary Math. AMS, 1992, (Part 2), v. 131, 59-67 MR 93g:17035
  • [L] Leites D., Introduction to the theory of supermanifold. Uspekhi Mat. Nauk, v. 35, 1, 1980, 3-57 MR 81j:58003
  • [L1] Leites D., New Lie superalgebras and mechanics. Soviet Math. Doklady, v. 18, N5, 1977, 1277-1280
  • [L2] Leites D., Lie superalgebras. In: Current Problems of Mathematics. Recent developments, v. 25, VINITI, Moscow, 1984, 3-49 (English translation = JOSMAR, v. 30(6), 1985, 2481-2512) MR 86f:17019
  • [L3] Leites D., Quantization. Supplement $3$. In: Berezin F., Shubin M. Schrödinger equation, Kluwer, Dordrecht, 1991, 483-522
  • [LSe] Leites D., Serganova V., Metasymmetry and Volichenko algebras, Phys. Lett. B, 1990, v. 252(1), 91-96 MR 92b:17005
  • [LSh1] Leites D., Shchepochkina I., Quivers and Lie superalgebras, Czech. J. Phys. vol 47, n 12, 1997, 1221-1229 MR 99c:16012
  • [LSh2] Leites D., Shchepochkina I., Deformations of simple vectorial Lie superalgebras (to appear)
  • [LSh3] Leites D., Shchepochkina I., Automorphisms and real forms of simple vectorial Lie superalgebras (to appear)
  • [LSh4] Leites D., Shchepochkina I., Classification of simple vectorial Lie superalgebras (to appear)
  • [M] Manin Yu. I., Gauge fields and complex geometry, 2nd ed, Springer, 1996
  • [Sh1] Shchepochkina I., Exceptional simple infinite-dimensional Lie superalgebras. C. R. Bulg. Sci., 36, 3, 1983, 313-314 MR 85i:17024
  • [Sh2] Shchepochkina I., Maximal subalgebras of simple Lie superalgebras. In: Leites D. (ed.) Seminar on Supermanifolds vv.1-34, 1987-1990, v. 32/1988-15, Reports of Stockholm University, 1-43 (hep-th 9702120)
  • [ShP] Shchepochkina I., Post G., Explicit bracket in an exceptional simple Lie superalgebra, Internat. Journal of Algebra and Computations (to appear); physics 9703022
  • [St] Sternberg S., Lectures on differential geometry, Chelsea, 2nd edition, 1983 MR 88f:58001
  • [W] Weisfeiler B., Infinite dimensional filtered Lie algebras and their relation with the graded Lie algebras, Funkcional. Anal. i Prilozhen. 2, n.1, 1968, 94-95 (in Russian) MR 38:1134

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Additional Information

Irina Shchepochkina
Affiliation: On leave of absence from the Independent University of Moscow; Correspondence: c/o D. Leites, Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, S-106 91, Stockholm, Sweden
Email:, lra@paramonova,

Keywords: Lie superalgebra, Cartan prolongation, spinor representation
Published electronically: October 13, 1999
Additional Notes: I am thankful to D. Leites for raising the problem and help; to INTAS grant 96-0538 and NFR for financial support; University of Twente and Stockholm University for hospitality. Computer experiments by G. Post and P. Grozman encouraged me to carry on with unbearable calculations.
Article copyright: © Copyright 1999 American Mathematical Society

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