The five exceptional simple Lie superalgebras of vector fields and their fourteen regradings
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- by Irina Shchepochkina
- Represent. Theory 3 (1999), 373-415
- DOI: https://doi.org/10.1090/S1088-4165-99-00012-6
- Published electronically: October 13, 1999
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Abstract:
The five simple exceptional complex Lie superalgebras of vector fields are described. One of them, $\mathfrak {fas}$, 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.References
- D. V. Alekseevskiĭ, D. A. Leĭtes, and I. M. Ščepočkina, Examples of simple infinite-dimensional Lie superalgebras of vector fields, C. R. Acad. Bulgare Sci. 33 (1980), no. 9, 1187–1190 (Russian). MR 620659
- J. N. Bernstein and D. A. Leĭtes, Invariant differential operators and irreducible representations of Lie superalgebras of vector fields, Selecta Math. Soviet. 1 (1981), no. 2, 143–160. Selected translations. MR 672426
- Shun-Jen Cheng and Victor G. Kac, A new $N=6$ superconformal algebra, Comm. Math. Phys. 186 (1997), no. 1, 219–231. MR 1462763, DOI 10.1007/BF02885679
- Grozman P., Leites D., Shchepochkina I., Lie superalgebras of string theories. hep-th 9702120
- Joaquim Gomis, Jordi París, and Stuart Samuel, Antibracket, antifields and gauge-theory quantization, Phys. Rep. 259 (1995), no. 1-2, 145. MR 1345328, DOI 10.1016/0370-1573(94)00112-G
- V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
- Yuri Yu. Kotchetkoff, Déformations des superalgèbres de Buttin et quantification, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 14, 643–645 (French, with English summary). MR 770453
- Kochetkov Yu. Deformations of Lie superalgebras. VINITI Depositions, Moscow (in Russian) 1985, 384–85
- Yuri Kochetkov and Dimitry Leites, Simple Lie algebras in characteristic $2$ recovered from superalgebras and on the notion of a simple finite group, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 59–67. MR 1175822, DOI 10.1090/conm/131.2/1175822
- D. A. Leĭtes, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk 35 (1980), no. 1(211), 3–57, 255 (Russian). MR 565567
- Leites D., New Lie superalgebras and mechanics. Soviet Math. Doklady, v. 18, N5, 1977, 1277–1280
- D. A. Leĭtes, Lie superalgebras, Current problems in mathematics, Vol. 25, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 3–49 (Russian). MR 770940, DOI 10.1007/BF02249121
- Leites D., Quantization. Supplement $3$. In: Berezin F., Shubin M. Schrödinger equation, Kluwer, Dordrecht, 1991, 483–522
- D. Leĭtes and V. Serganova, Metasymmetry and Volichenko algebras, Phys. Lett. B 252 (1990), no. 1, 91–96. MR 1084729, DOI 10.1016/0370-2693(90)91086-Q
- D. Leites and I. Shchepochkina, Quivers and Lie superalgebras, Czechoslovak J. Phys. 47 (1997), no. 12, 1221–1229. Quantum groups and integrable systems, II (Prague, 1997). MR 1608811, DOI 10.1023/A:1022873515587
- Leites D., Shchepochkina I., Deformations of simple vectorial Lie superalgebras (to appear)
- Leites D., Shchepochkina I., Automorphisms and real forms of simple vectorial Lie superalgebras (to appear)
- Leites D., Shchepochkina I., Classification of simple vectorial Lie superalgebras (to appear)
- Manin Yu. I., Gauge fields and complex geometry, 2nd ed, Springer, 1996
- I. M. Shchepochkina, Exceptional simple infinite-dimensional Lie superalgebras, C. R. Acad. Bulgare Sci. 36 (1983), no. 3, 313–314 (Russian). MR 709013
- 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)
- Shchepochkina I., Post G., Explicit bracket in an exceptional simple Lie superalgebra, Internat. Journal of Algebra and Computations (to appear); physics 9703022
- Shlomo Sternberg, Lectures on differential geometry, 2nd ed., Chelsea Publishing Co., New York, 1983. With an appendix by Sternberg and Victor W. Guillemin. MR 891190
- B. Ju. Veĭsfeĭler, Infinite dimensional filtered Lie algebras and their connection with graded Lie algebras, Funkcional. Anal. i Priložen. 2 (1968), no. 1, 94–95 (Russian). MR 0232811, DOI 10.1007/BF01075364
Bibliographic 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: mleites@matematik.su.se, lra@paramonova.mccme.ru
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: Represent. Theory 3 (1999), 373-415
- MSC (1991): Primary 17A70; Secondary 17B35
- DOI: https://doi.org/10.1090/S1088-4165-99-00012-6
- MathSciNet review: 1715110