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Theorem on semisimple lie p-algebras

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Abstract

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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Literature cited

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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.

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    G. Seligman, Some remarks on classical Lie algebras, J. Math. Mech., 6, No. 4, 549–558 (1957).

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    N. Jacobson, Lie algebras [in Russian], Moscow (1964).

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    A. I. Kostrikin, Lie fields satisfying an Engel's condition, Izv. AN SSSR, ser. matem.,21, 515–540(1957).

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    A. I. Kostrikin, The Height of Simple Lie Algebras, Dokl. AN SSSR,162, No. 5, 992–994 (1965).

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    G. Seligman, Characteristic ideals and the structure of Lie algebras, Proc. Amer. Math. Soc., 8, 159–164 (1957).

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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

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Keywords

  • Bilinear Form
  • Characteristic Zero
  • Classical Type
  • Direct Factor
  • Simple Ideal