Lie algebras admitting a hypercentrally regular automorphism

Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable

“…that the ground field is algebraically closed. Then, by (1) we see that C L (L σ ) ∩ L ξ i (σ ) Z(L) for every 1 i < n. From this (2) follows. 2 A subalgebra S of a Lie algebra L is said to be toral if ad L x is semisimple for every x ∈ S. Now let L be reductive.…”

“…(b) follows from (a) and from the fact that L σ L τ for every τ ∈ σ . In order to prove (2), assume that σ has prime order. From Proposition 2.1 (2) it follows that C L (L σ ) L σ .…”

“…In order to prove (2), assume that σ has prime order. From Proposition 2.1 (2) it follows that C L (L σ ) L σ . This yields that L σ = H and therefore σ ∈ Aut 0 (L), see [10, p. 278].…”

“…In the literature there are many generalizations of regularity. For cyclic groups G, there are considered the cases when L G is contained in the center of L [1], in the hypercenter of L [2] and when dim L G < ∞ see [12]. We are concerned with finite (not necessarily cyclic) groups G. The earlier cited result cannot be extended to non-cyclic groups, since the simple Lie algebra A p−1 of matrices p × p of zero trace, p a prime, admits a regular group of automorphisms which is an elementary abelian p-group of type (p, p) (see [3]).…”

Almost regular automorphisms

“…(Of course, the characteristic of L maybe assumed to be p, since otherwise G is nilpotent of class h(p)+ 1 by Theorem 1.17.) A positive answer is known only for p = 2, 3; in addition, A. I. Belov [5] proved that L is soluble if it is of finite dimension (but with no bound for the derived length).…”