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On the triviality of homogeneous algebras over an algebraically closed field


Author: Lowell Sweet
Journal: Proc. Amer. Math. Soc. 48 (1975), 321-324
MSC: Primary 17E05
DOI: https://doi.org/10.1090/S0002-9939-1975-0364382-7
MathSciNet review: 0364382
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Abstract: Let $ A$ be a finite-dimensional algebra (not necessarily associative) over a field $ K$. Then $ A$ is said to be homogeneous if $ \operatorname{Aut} (A)$ acts transitively on the one-dimensional subspaces of $ A$. If $ A$ is homogeneous and $ K$ is algebraically closed, then it is shown that either $ {A^2} = 0$ or $ \dim A = 1$.


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  • [1] J. Boen, O. Rothaus and J. Thompson, Further results on $ p$-automorphic $ p$-groups, Pacific J. Math. 12 (1962), 817-821. MR 27 #2553b. MR 0152576 (27:2553b)
  • [2] D. Ž. Djoković, Real homogeneous algebras, Proc. Amer. Math. Soc. 41 (1973), 457-462. MR 0332902 (48:11227)
  • [3] F. Gross, Finite automorphic algebras over $ {\text{GF(2)}}$, Proc. Amer. Math. Soc. 31 (1972), 10-14. MR 44 #4063. MR 0286856 (44:4063)
  • [4] A. I. Kostrikin, On homogeneous algebras, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 471-484. (Russian) MR 31 #219. MR 0175943 (31:219)
  • [5] E. E. Shult, On finite automorphic algebras, Illinois J. Math. 13 (1969), 625-653. MR 40 #1441. MR 0248187 (40:1441)
  • [6] -, On the triviality of finite automorphic algebras, Illinois J. Math. 13 (1969), 654-659. MR 40 #1442. MR 0248188 (40:1442)
  • [7] S. Swierczkowski, Homogeneous Lie algebras, Bull. Austral. Math. Soc. 4 (1971), 349-353. MR 43 #6277. MR 0280557 (43:6277)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0364382-7
Keywords: Homogeneous algebra, algebraically closed field, special nil algebra, Jordan normal form
Article copyright: © Copyright 1975 American Mathematical Society

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