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On ideally finite Lie algebras which are lower semi-modular

David Towers

1985
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Proceedings of the Edinburgh Mathematical Society
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The purpose of this paper is twofold: first to correct the statement of Theorem 1 in [4] , and secondly to consider related problems in the class of ideally finite Lie algebras. Throughout, L will denote a Lie algebra over a field K, F(L) will be its Frattini subalgebra and $(L) its Frattini ideal. We will denote by X the class of Lie algebras all of whose maximal subalgebras have codimension 1 in L. The Lie algebra with basis {u-i.Uo.Uj} and multiplication U_ 1 M 0 = U_ 1 , M_ 1 M 1 = U 0 , u
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... M_ 1 M 1 = U 0 , u o u l = u l will be labelled MO). Theorem 1 of [4] claimed that a necessary and sufficient condition for L to belong to . The necessity is correct, but not the sufficiency. The only problem is that L^O) may not belong to X; when it does is described in the following lemma. Lemma 0. Lj(0)e X if and only if K has characteristic two, or Proof. If K has characteristic two, then F^^O)) is spanned by u 0 and so all maximal subalgebras of L x (0) are two dimensional. Let S(A,n,v) be the 1-dimensional subalgebra of Lj(0) spanned by l u _ ! +fiu o + vu l {l,y.,veK). Then any 1-dimensional subalgebra of L x (0) is of the form S(X,ix,Q>) or S(l,n, 1). If y/K^K then S(A,/i,0) is contained in the subalgebra spanned by u 0 and «_!, and S(J.,(i, 1) is contained in that spanned by Xu^l+fiu 0 + u l and ocu^ -w_ x where If y/K^K, let ae^/E, a$K. Then, when the characteristic of K is different from two, the subalgebra of 1^(0) spanned by (a 2 /2)u l -u_ x is maximal. Using the above lemma and the fact that L is supersoluble whenever L/(p(L) is supersoluble ([1], Theorem 6), we can correct Theorem 1 of [4] as follows. Theorem 1. Let L be a finite-dimensional Lie algebra. (i) If y/K^K and K has characteristic different from 2, then LeX if and only if L is supersoluble. (ii) Ify/K^K or K has characteristic two, then LeX if and only if L/(L) isomorphic to Lj(0), or is {0}, and R is a supersoluble ideal of L/(f>(L) (possibly {0}). 9

doi:10.1017/s0013091500003151
fatcat:lq2uejwjpjd3xobk2coraqhcnu