Embedding of a Lie algebra into Lie-admissible algebras

Author:
Hyo Chul Myung

Journal:
Proc. Amer. Math. Soc. **73** (1979), 303-307

MSC:
Primary 17A30; Secondary 17A20

DOI:
https://doi.org/10.1090/S0002-9939-1979-0518509-8

MathSciNet review:
518509

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Abstract: Let *A* be a flexible Lie-admissible algebra over a field of characteristic 2, 3. Let *S* be a finite-dimensional classical Lie subalgebra of which is complemented by an ideal *R* of . It is shown that *S* is a Lie algebra under the multiplication in *A* and is an ideal of *A* if and only if *S* contains a classical Cartan subalgebra *H* which is nil in *A* and such that and . In this case, the multiplication between *S* and *R* is determined by linear functionals on *R* which vanish on [*R, R*]. If *A* is finite-dimensional and of characteristic 0 then this can be applied to give a condition that a Levi-factor *S* of be embedded as an ideal into *A* and to determine the multiplication between *S* and the solvable radical of .

**[1]**P. J. Laufer and M. L. Tomber,*Some Lie admissible algebras*, Canad. J. Math.**14**(1962), 287–292. MR**0136636**, https://doi.org/10.4153/CJM-1962-020-9**[2]**Hyo Chul Myung,*Some classes of flexible Lie-admissible algebras*, Trans. Amer. Math. Soc.**167**(1972), 79–88. MR**0294419**, https://doi.org/10.1090/S0002-9947-1972-0294419-7**[3]**Hyo Chul Myung,*A subalgebra condition in Lie-admissible algebras*, Proc. Amer. Math. Soc.**59**(1976), no. 1, 6–8. MR**0422361**, https://doi.org/10.1090/S0002-9939-1976-0422361-6**[4]**G. B. Seligman,*Modular Lie algebras*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR**0245627****[5]**R. M. Santilli,*Lie-admissible approach to the hadronic structure*, Volumes I, II, and III, Hadronic Press, Inc., Nonantum, Mass. (to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0518509-8

Keywords:
Flexible algebra,
Lie-admissible algebra,
classical Lie algebra,
Cartan subalgebra,
Levi-factor

Article copyright:
© Copyright 1979
American Mathematical Society