The known fact on the decomposability of any semisimple Lie algebra over an algebraically closed field of characteristic zero into a direct sum of simple ideals remains true in the case of characteristic p > 0. It is also shown that a semisimple Lie p-algebra, admitting a faithful p-representation of dimension n < p−1 has such a decomposition. Its direct factors are Lie p-algebras of the classical type with a non-degenerate bilinear form of trace. The restriction n < p−1 is essential. Bibliography has 6 references.
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A. I. Kostrikin, Squares of adjoint endomorphisms in simple Lie p-algebras, Izv. AN SSSR, Ser. matem.,31, No. 2 (1967), 445–487.
G. Seligman, Some remarks on classical Lie algebras, J. Math. Mech., 6, No. 4, 549–558 (1957).
N. Jacobson, Lie algebras [in Russian], Moscow (1964).
A. I. Kostrikin, Lie fields satisfying an Engel's condition, Izv. AN SSSR, ser. matem.,21, 515–540(1957).
A. I. Kostrikin, The Height of Simple Lie Algebras, Dokl. AN SSSR,162, No. 5, 992–994 (1965).
G. Seligman, Characteristic ideals and the structure of Lie algebras, Proc. Amer. Math. Soc., 8, 159–164 (1957).
Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 465–474, November, 1967.
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Kostrikin, A.I. Theorem on semisimple lie p-algebras. Mathematical Notes of the Academy of Sciences of the USSR 2, 773–778 (1967). https://doi.org/10.1007/BF01093937
- Bilinear Form
- Characteristic Zero
- Classical Type
- Direct Factor
- Simple Ideal