Frattini subalgebras of finitely generated soluble Lie algebras

Amayo, Ralph K.

doi:10.1090/S0002-9947-1978-0498733-7

Frattini subalgebras of finitely generated soluble Lie algebras
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by Ralph K. Amayo PDF

Trans. Amer. Math. Soc. 236 (1978), 297-306 Request permission

Abstract:

This paper is motivated by a recent one of Stewart and Towers [8] investigating Lie algebras with “good Frattini structure” (definition below). One consequence of our investigations is to prove that any finitely generated metanilpotent Lie algebra has good Frattini structure, thereby answering a question of Stewart and Towers and providing a complete Lie theoretic analogue of the corresponding group theoretic result of Phillip Hall. It will also be shown that in prime characteristic, finitely generated nilpotent-by-finite-dimensional Lie algebras have good Frattini structure.

References

Ralph K. Amayo, Engel Lie rings with chain conditions, Pacific J. Math. 54 (1974), 1–12. MR 360727
Ralph K. Amayo and Ian Stewart, Finitely generated Lie algebras, J. London Math. Soc. (2) 5 (1972), 697–703. MR 323850, DOI 10.1112/jlms/s2-5.4.697

Infinite-dimensional Lie algebras

Donald W. Barnes and Martin L. Newell, Some theorems on saturated homomorphs of soluble Lie algebras, Math. Z. 115 (1970), 179–187. MR 266969, DOI 10.1007/BF01109856
Charles W. Curtis, Noncommutative extensions of Hilbert rings, Proc. Amer. Math. Soc. 4 (1953), 945–955. MR 59254, DOI 10.1090/S0002-9939-1953-0059254-7
Nathan Divinsky, Rings and radicals, Mathematical Expositions, No. 14, University of Toronto Press, Toronto, Ont., 1965. MR 0197489
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
David Towers and Ian Stewart, The Frattini subalgebras of certain infinite-dimensional soluble Lie algebras, J. London Math. Soc. (2) 11 (1975), no. 2, 207–215. MR 396710, DOI 10.1112/jlms/s2-11.2.207