Ado-Iwasawa extras

Abstract

Let L be a finite-dimensional Lie algebra over the field F. The Ado-Iwasawa Theorem asserts the existence of a finite-dimensional L-module which gives a faithful representation p of L. Let 5 be a subnormal subalgebra of L, let 5 be a saturated formation of soluble Lie algebras and suppose that 5 € ff. 1 show that there exists a module V with the extra property that it is 3-hypercentral as 5-module. Further, there exists a module V which has this extra property simultaneously for every such 5 and 3, along with the Hochschild extra that p(x) is nilpotent for every x € L with ad(x) nilpotent. In particular, if L is supersoluble, then it has a faithful representation by upper triangular matrices.2000 Mathematics subject classification: primary 17B30,17B56; secondary 17B50,17B55.

Character Clusters for Lie Algebra Modules Over a Field of Nonzero Characteristic

“…Let F be a saturated formation of soluble Lie algebras over F . Comparing Theorem 4.2 with [3, Lemma 1.1] suggests a further relationship between clusters and saturated formations beyond that of [3,Theorem 6.4].…”

Faithful completely reducible representations of modular Lie algebras

“…There are several extensions of this result which assert the existence of such a module V with various additional properties. See, for example, Hochschild [3], Barnes [1]. Of importance for this paper is Jacobson's Theorem, [4,Theorem 5.5.2] that every finite-dimensional Lie algebra L over a field F of characteristic p > 0 has a finitedimensional faithful completely reducible module V .…”

Schunck Classes of Soluble Leibniz Algebras

“…Every non-zero Schunck class contains all nilpotent algebras, but not every nilpotent Lie algebra is restrictable. The two theories however are linked by Theorem 6.4 of Barnes [5], which I quote here for convenience of reference.…”