A characterization of the Kostrikin radical of a Lie algebra

A radical for graded Lie algebras

“…By [5] let 0 = y ∈ ad n−1 xg (L) be a homogeneous ad-nilpotent element of index 3 and let us consider the G-graded Jordan algebra L y . In L y every element is nilpotent of bounded index since y is still strongly Engel in L and we can argue as in [6,Proposition 3.2]. By [13,Lemma 17,p.…”

“…849]) and every m-sequence in L x has finite length. But then every msequence of L starting with x has also finite length by [6,Proposition 2.2]. Now if x ∈ K(L) = K β (L), x ∈ K α (L) for some α which is not a limit ordinal, and x + K α−1 (L) ∈ K 1 L/K α−1 (L) is also a Jordan element of L/K α−1 (L).…”

“…ConsiderK = π −1 K G (L/P ) , where π : L → L/P denotes the canonical projection, which is a graded ideal of L properly containing P , so there exists some c j ∈K, hence0 = c j + P ∈ K G (L/P ) ⊂ K(L/P ). By [6,Proposition 3.6] if {c i } i∈N is a generalized m-sequence of homogeneous elements, or by 4.4 if {c i } i∈N is an m-sequence of homogeneous Jordan elements, the sequence {c i + P } i∈N has finite length, so there exists some c k + P =0, i.e., c k ∈ P , a contradiction. Now we state the two most important results of the paper: if G is an arbitrary group and L is a G-graded Lie algebra over a field of characteristic zero, firstly, the Kostrikin radical of L is the intersection of all graded-strongly prime ideals of L and, secondly, nondegeneracy and graded-nondegeneracy are equivalent notions for L. Theorem 4.6.…”

Simple, locally finite dimensional Lie algebras in positive characteristic

“…A Lie algebra is nondegenerate if there are no nonzero elements x ∈ L such that ad 2 There is also a notion of m-sequence in Lie algebras which is defined in [9]: an m-sequence in a Lie algebra L is a sequence {a n } such that a n+1 = [a n , [a n , b n ]] for some b n ∈ L. Recall that for a Lie algebra L, the Kostrikin radical K(L) is the smallest ideal of L inducing a nondegenerate quotient ( [23]). If any m-sequence beginning with an element x terminates, then x ∈ K(L).…”

“…If any m-sequence beginning with an element x terminates, then x ∈ K(L). (The other direction is currently unknown and is the subject of [9]).…”