Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map $$\phi$$
 : A $$\times$$
 A $$\rightarrow$$
 F, we define a Lie algebra $${\cal L}$$
 = $${\cal L}$$
 (A, $$\phi$$
 ) over F with basis {ex | x $$\in$$
 A/{0}} and Lie product [ex,ey] = $$\phi$$
 (x,y)ex+y. We show that $${\cal L}$$
 is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der $${\cal L}$$
 of $${\cal L}$$
 is a complete Lie algebra. We describe the double extension D(
 $${\cal L}$$
 , T) of $${\cal L}$$
 by T, where T is spanned by the locally finite derivations of $${\cal L}$$
 , and determine the second cohomology group H2(D(
 $${\cal L}$$
 , T),F) using anti-derivations related to the form on D(
 $${\cal L}$$
 , T). Finally, we compute the second Leibniz cohomology groups HL2(
 $${\cal L}$$
 , F) and HL2(D(
 $${\cal L}$$
 , T), F).

Structure of Degenerate Block Algebras

Let L be a Lie algebra, and Der(L) and IDer(L) be the set of all derivations and inner derivations of L, respectively. Also, let Derc(L) denote the set of all derivations α ∈ Der(L) for which α(x) ∈ Imadx for all x ∈ L. We give necessary and sufficient conditions under which Derc(L) = Derz(L), where Derz(L) is the set of all derivations of L whose images lie in the center of L. Moreover, it is shown that any two isoclinic Lie algebras L1 and L2 satisfy Derc(L1) ≅ Derc(L2).

On special subalgebras of derivations of Lie algebras

Some 2-step nilpotent Lie algebras I☆

A remark on complete Lie algebras is given. Since the theory of complete Lie algebras is still developing, this remark cannot be complete.

Complete Lie algebras

It is proved that the derivation algebra of a centerless perfect Lie algebra of arbitrary dimension over any field of arbitrary characteristic is complete and that the holomorph of a centerless perfect Lie algebra is complete if and only if its outer derivation algebra is centerless.

Derivation Algebras of Centerless Perfect Lie Algebras Are Complete

In this paper, we will discuss the properties of solvable complete Lie algebra, describe the structures of the root spaces of solvable complete Lie algebra, prove that solvable Lie algebras of maximal rank are com-plete, and construct some new complete Lie algebras from Kac-Moody algebras.

Zhu Lin Sheng

Papers

Solvable complete lie algebras. I