Some classes of flexible Lie-admissible algebras
HTML articles powered by AMS MathViewer
- by Hyo Chul Myung PDF
- Trans. Amer. Math. Soc. 167 (1972), 79-88 Request permission
Abstract:
Let $\mathfrak {A}$ be a finite-dimensional, flexible, Lie-admissible algebra over a field of characteristic $\ne 2$. Suppose that ${\mathfrak {A}^ - }$ has a split abelian Cartan subalgebra $\mathfrak {H}$ which is nil in $\mathfrak {A}$. It is shown that if every nonzero root space of ${\mathfrak {A}^ - }$ for $\mathfrak {H}$ is one-dimensional and the center of ${\mathfrak {A}^ - }$ is 0, then $\mathfrak {A}$ is a Lie algebra isomorphic to ${\mathfrak {A}^ - }$. This generalizes the known result obtained by Laufer and Tomber for the case that ${\mathfrak {A}^ - }$ is simple over an algebraically closed field of characteristic 0 and $\mathfrak {A}$ is power-associative. We also give a condition that a Levi-factor of ${\mathfrak {A}^ - }$ be an ideal of $\mathfrak {A}$ when the solvable radical of ${\mathfrak {A}^ - }$ is nilpotent. These results yield some interesting applications to the case that ${\mathfrak {A}^ - }$ is classical or reductive.References
- A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc. 64 (1948), 552â593. MR 27750, DOI 10.1090/S0002-9947-1948-0027750-7
- Richard E. Block, Determination of $A^{+}$ for the simple flexible algebras, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 394â397. MR 235002, DOI 10.1073/pnas.61.2.394
- Claude Chevalley, Théorie des groupes de Lie. Tome II. Groupes algébriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1152, Hermann & Cie, Paris, 1951 (French). MR 0051242
- J. Dixmier, Sous-algĂšbres de Cartan et dĂ©compositions de Levi dans les algĂšbres de Lie, Trans. Roy. Soc. Canada Sect. III 50 (1956), 17â21 (French). MR 83988
- Murray Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices. II, Duke Math. J. 27 (1960), 21â31. MR 113911
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- P. J. Laufer and M. L. Tomber, Some Lie admissible algebras, Canadian J. Math. 14 (1962), 287â292. MR 136636, DOI 10.4153/CJM-1962-020-9
- Hyo Chul Myung, A remark on the proof of a theorem of Laufer and Tomber, Canadian J. Math. 23 (1971), 270. MR 269707, DOI 10.4153/CJM-1971-026-1 â, Flexible Lie-admissible algebras, Thesis, Michigan State University, East Lansing, Mich., 1970.
- Robert H. Oehmke, On flexible algebras, Ann. of Math. (2) 68 (1958), 221â230. MR 106934, DOI 10.2307/1970244
- G. B. Seligman, Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR 0245627, DOI 10.1007/978-3-642-94985-2
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 167 (1972), 79-88
- MSC: Primary 17A20
- DOI: https://doi.org/10.1090/S0002-9947-1972-0294419-7
- MathSciNet review: 0294419