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Soluble Lie algebras having finite-dimensional maximal subalgebras

  • Ian N. Stewart (a1)
Abstract

Infinite-dimensional soluble Lie algebras can possess maximal subalgebras which are finite-dimensional. We give a fairly complete description of such algebras: over a field of prime characteristic they do not exist; over a field of zero characteristic then, modulo the core of the aforesaid maximal subalgebra, they are split extensions of an abelian minimal ideal by the maximal subalgebra. If the field is algebraically closed, or if the maximal subalgebra is supersoluble, then all finite-dimensional maximal subalgebras are conjugate under the group of automorphisms generated by exponentials of inner derivations by elements of the Fitting radical. An example is given to indicate the differences encountered in the insoluble case, and the nonexistence of group-theoretic analogues is briefly discussed.

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References
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[1]Amayo, R.K. and Stewart, I.N., Infinite-dimensional, Lie algebras (Noordhoff, Leyden, to appear).
[2]Barnes, Donald W. and Newell, Martin L., “Some theorems on saturated homomorphs of soluble Lie algebras”, Math. Z. 115 (1970), 179187.
[3]Curtis, Charles W., “Noncommutative extensions of Hubert rings”, Proc. Amer. Math. Soc. 4 (1953), 945955.
[4]Hall, P., “On the finiteness of certain soluble groups”, Proc. London Math. Soc. (3) 9 (1959), 595622.
[5]Hartley, B., “Locally nilpotent ideals of a Lie algebra”, Proc. Cambridge Philos. Soc. 63 (1967), 257272.
[6]Jacobson, Nathan, Lie algebras (Interscience [John Wiley & Sons], New York, London, 1962).
[7]Roseblade, J.E., “Polycyclic group rings and the Nullstellensatz”, Conference on group theory, University of Wisconsin-Parkside 1972, 156167 (Lecture Notes in Mathematics, 319. Springer-Verlag, Berlin, Heidelberg, New York, 1973).
[8]Roseblade, J.E., “Group rings of polycyclic groups”, J. Pure Appl. Algebra 3 (1973), 307328.
[9]Stewart, I.N., “An algebraic treatment of Mal'cev's theorems concerning nilpotent Lie groups and their Lie algebras”, Compositio Math. 22 (1970), 289312.
[10]Stewart, I.N., “A property of locally finite Lie algebras”, J. London Math. Soc. (2) 3 (1971), 334340.
[11]Towers, D.A., “A Frattini theory for algebras”, PhD thesis, University of Leeds, 1972.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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