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Finite-dimensional odd Hamiltonian superalgebras over a field of prime characteristic

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Wende Liu  and
Yongzheng Zhang
Wende Liu
Affiliation:
Department of MathematicsHarbin Normal UniversityHarbin 150080ChinaandDepartment of MathematicsNortheast Normal UniversityChangchun 130024China e-mail: [email protected]
Yongzheng Zhang
Affiliation:
Department of MathematicsHarbin Normal UniversityHarbin 150080China e-mail: [email protected]
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Abstract

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Let ℋ(m;t) be the finite-dimensional odd Hamiltonian superalgebra over a field of prime characteristic. By determining ad-nilpotent elements in the even part, the natural filtration of ℋ (m;t) is proved to be invariant in the following sense: If ϕ: ℋ (m;t) → ℋ (m′t′) is an isomorphism then ϕ(ℋ(m;t)i) = ℋ (m′ t′) i for all i ≥ –1. Using the result, we complete the classification of odd Hamiltonian superalgebras. Finally, we determine the automorphism group of the restricted odd Hamiltonian superalgebra and give further properties.

MSC classification

Secondary: 17B50: Modular Lie (super)algebras 17B40: Automorphisms, derivations, other operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 1 , August 2005 , pp. 113 - 130
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Berezin, F. A. and Marinov, M. S., ‘Particle spin dynamics as the grassmann varlant of classical mechanics’, Ann. Physics 104 (1977), 336–362.Google Scholar
[2]Block, R. E. and Wilson, R. L., ‘Classification of the restricted simple Lie algebras’, J. Algebra 114 (1988), 115–259.Google Scholar
[3]Jacobson, N., Lie algebras (Interscience, New York, 1962).Google Scholar
[4]Jin, N., ‘Ad-nilpotent elements, quasi-nilpotent elements and invariant filtrations of infinitedimensional Lie algebras of Cartan-type’, Sci. China Ser. A 35 (1992), 1191–1200.Google Scholar
[5]Kac, V.G., ‘On the classification of the simple Lie algebras over a field with nonzero characteristic’, Math. USSR Izv. 4 (1970), 391–413.Google Scholar
[6]Kac, V.G., ‘Description of filtered Lie algebras with which graded Lie algebras of Cartan-type are associated’, Math. USSR lzv. 8 (1974), 801–835.CrossRefGoogle Scholar
[7]Kav, V.G., ‘Lie superaigebras’, Adv. Math. 26 (1977), 8–96.Google Scholar
[8]Kac, V.G., ‘Classification of infinite-dimensional simple linearly compact Lie superaigebras’, Adv. Math. 139 (1998), 1–55.Google Scholar
[9]Kochetkov, Yu., ‘Induced irreducible representations of Leites superaigebras’, in: Problem in group theory and homological algebra 139 (Yaroslav. Gos. Univ., Yaroslavl, 1983) pp. 120–123. (in Russian).Google Scholar
[10]Kostrikin, A. I. and Shafarevic, I. R., ‘Graded Lie algebras of finite characteristic’, Math. USSR lzv. 3 (1969), 237–304.CrossRefGoogle Scholar
[11]Leites, D. A., ‘New Lie superalgebras and mechanics’, Dokl. Akad. Nauk SSSR 236 (1977), 804–807.Google Scholar
English translation: Soviet Math. Dokl. (5), 18 (1977), 1277–1280.Google Scholar
[12]Leites, D. A., ‘Automorphisms and real forms of simple Lie superalgebras of formal vector fields’, in: Problem in group theory and homological algebra 139 (Yaroslav. Gos. Univ., Yaroslavl, 1983) pp. 126–128. (in Russian).Google Scholar
[13]Petrogradski, V. M., ‘Identities in the enveloping algebras for modular Lie superalgebras’, J. Algebra 145 (1992), 1–21.CrossRefGoogle Scholar
[14]Rudakov, A. N., ‘Subalgebras and automorphisms of Lie algebras of Cartan-type’, Funktsional. Anal. Prilozhen. 20 (1986), 83–84 (in Russian).Google Scholar
[15]Scheunert, M., Theory of Lie superalgebras, Lectures Notes in Math. 716 (Springer, Berlin, 1979).Google Scholar
[16]Shen, G.-Y., ‘An intrinsic property of the Lie algebra K(m, )’, Chin. Ann. Math. 2 (1981), 104–107.Google Scholar
[17]Strade, H., ‘The classification of the simple modular Lie algebras: IV. Determining of the associated graded algebra’, Ann. of Math. (2) 138 (1993), 1–59.Google Scholar
[18]Strade, H. and Farnsteiner, R., Modular Lie algebras and their representations (Marcel Dekker, 1988).Google Scholar
[19]Wilson, R. L., ‘Classification of generalized Witt algebras over algebraically closed fields’, Trans. Amer Math. Soc. 153 (1971), 191–210.CrossRefGoogle Scholar
[20]Wilson, R. L., ‘Automorphisms of graded Lie algebras of Cartan type’, Comnun. Algebra 3 (1975), 591–613.CrossRefGoogle Scholar
[21]Wilson, R. L., ‘A structural characterization of the simple Lie algebras of generalized Cartan-type over fields of prime characteristic’, J. Algebra 40 (1976), 418–465.CrossRefGoogle Scholar
[22]Wilson, R. L., ‘Simple Lie algebras of type S’, J. Algebra 62 (1980), 292–298.Google Scholar
[23]Zhang, Y.-Z., ‘Finite-dimensional Lie superalgebras of Cartan-type over fields of prime characteristic’, Chin. Sci. Bull. 42 (1997), 720–724.CrossRefGoogle Scholar
[24]Zhang, Y.-Z. and Fu, H.-C., ‘Finite-dimensional hamiltonian Lie superalgebras’, Commun. Algebra 30 (2002), 2651–2673.Google Scholar
[25]Zhang, Y.-Z. and Nan, J.-Z., ‘Finite-dimensional Lie superalgebras W(m, n, ) and S(m, n, ) of Cartan-type’, Chin. Adv. Math. 27 (1998), 240–246.Google Scholar
[26]Zhang, Y.-Z. and Shen, G.-Y., ‘The embedding theorem of Z-graded Lie superalgebras’, Sci. China Ser A 41 (1998), 1009–1016.Google Scholar