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Malcev algebras with $ J\sb{2}$-potent radical


Author: Ernest L. Stitzinger
Journal: Proc. Amer. Math. Soc. 50 (1975), 1-9
MSC: Primary 17E05
DOI: https://doi.org/10.1090/S0002-9939-1975-0374224-1
MathSciNet review: 0374224
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Abstract: Let $ A$ be a Malcev algebra, $ B$ be an ideal of $ A$ and $ J_2^1(B) = J(B,A,A)$ where $ J(B,A,A)$ is the linear subspace of $ A$ spanned by all elements of the form $ J(x,y,z) = (xy)z + (yz)x + (zx)y,x \in B,y,z \in A$. For $ k \geq 1$, define $ J_2^{k + 1}(B) = J(J_2^k(B),A,A)$. Then $ B$ is called $ {J_2}$-potent if there exists an integer $ N \geq 1$ such that $ J_2^N(B) = 0$. Now let $ A$ be a Malcev algebra over a field of characteristic 0 such that the radical $ R$ of $ A$ is $ {J_2}$-potent. Then $ R$ is complemented by a semisimple subalgebra and all such complements are strictly conjugate in $ A$. The proofs follow those in the Lie algebra case.


References [Enhancements On Off] (What's this?)

  • [1] N. Jacobson, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962. MR 26 #1345. MR 0143793 (26:1345)
  • [2] E. N. Kuz'min, Mal'cev algebras and their representations, Algebra i Logika 7 (1968), no. 4, 48-69 = Algebra and Logic 7 (1968), 233-244. MR 40 #5688. MR 0252468 (40:5688)
  • [3] W. G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217-242. MR 13, 619. MR 0045702 (13:619d)
  • [4] O. Loos, Über eine Beziehung zwischen Malcev-Algebren und Lie-Tripelsystemen, Pacific J. Math. 18 (1966), 553-562. MR 33 #7385. MR 0199236 (33:7385)
  • [5] T. S. Ravisankar, On Malcev algebras, Pacific J. Math. 42 (1972), 227-234. MR 47 #1905. MR 0313350 (47:1905)
  • [6] A. A. Sagle, Malcev algebras, Trans. Amer. Math. Soc. 101 (1961), 426-458. MR 26 #1343. MR 0143791 (26:1343)

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DOI: https://doi.org/10.1090/S0002-9939-1975-0374224-1
Article copyright: © Copyright 1975 American Mathematical Society

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