Abstract
Restricted algebras, or Lie p-algebras, of vector fields are finite-dimensional analogs of the corresponding classical algebras defined over fields of positive characteristic p. Our computations of p-algebras of Lie vector fields that preserve the symplectic structure (i.e., Hamiltonian and Poisson algebras) revealed important and interesting specific features of the structure of their cohomologies. Explanations of these specific features are presented.
Similar content being viewed by others
REFERENCES
Fuks, D.B., Kogomologii beskonechnomernykh algebr Li (Cohomology of Infinite-Dimensional Lie Algebras), Moscow: Nauka, 1984.
Fuks, D.B., Finite-Dimensionality of the Homology of the Lie Algebra of Hamiltonian Vector Fields on the Plane, Funk. Anal. Prilozh., 1985, vol. 19, no.4, pp. 68–73.
Jacobson, N., Restricted Lie Algebras of Characteristic p, Trans. Amer. Math. Soc., 1941, vol. 50, pp. 15–25.
Hochschild, G., Cohomology of Restricted Lie Algebras, Amer. J. Math., 1954, vol. 76, pp. 555–580.
Kostrikin, A.I. and Shafarevich, I.R., Cartan Pseudogroups and Lie p-Algebras, Dokl. Akad. Nauk SSSR, 1966, vol. 168, pp. 740–742.
Kostrikin, A.I. and Shafarevich, I.R., Graded Lie Algebras of Finite Characteristic, Izv. Akad. Nauk. SSSR: Ser. Mat., 1969, vol. 33, pp. 251–322.
Strade, H., The Classification of the Simple Lie Algebras over Fields with Positive Characteristic, Hamburger Beitrage zur Mathematik aus dem Mathematischen Seminar, vol. 31, 1997.
Strade, H., The Classification of the Simple Modular Lie Algebras: VI. Solving the Final Case, Trans. Amer. Math. Soc., 1998, vol. 250, pp. 2553–2628.
Strade, H. and Farnsteiner, R., Modular Lie Algebras and Their Representations, in Marcel Dekker Textbooks and Monographs, vol. 116, New York: Marcel Dekker, 1988.
Premet, A.A. and Strade, H., Simple Lie Algebras of Small Characteristic: I. Sandwich Elements, J. Algebra, 1997, vol. 189, pp. 419–480.
Premet, A.A. and Strade, H., Simple Lie algebras of Small Characteristic: II. Exceptional Roots, J. Algebra, 1999, vol. 216, pp. 190–301.
Premet, A.A. and Strade, H., Simple Lie Algebras of Small Characteristic: III. Toral Rank 2 Case, J. Algebra, 2001, vol. 242, pp. 236–337.
Kornyak, V.V., Modular Algorithm for Computing Cohomology: Lie Superalgebra of Special Vector Fields on (2¦?2)-dimensional Odd-Symplectic Superspace, in Computer Algebra in Scientific Computing, Ganzha, V.G., Mayr, E.W., and Vorozhtsov, E.V., Eds., München: TUM, 2003, pp. 227–240. http: //arXiv.org/abs/math.RT/0305155.
Author information
Authors and Affiliations
Additional information
Translated from Programmirovanie, Vol. 31, No. 2, 2005.
Original Russian Text Copyright © 2005 by Kornyak.
Rights and permissions
About this article
Cite this article
Kornyak, V.V. Cohomologies of restricted Lie algebras of Hamiltonian vector fields: Computer analysis. Program Comput Soft 31, 87–90 (2005). https://doi.org/10.1007/s11086-005-0018-4
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11086-005-0018-4