Nilpotent Lie algebras with 2-dimensional commutator ideals

Abstract

We classify all (finitely dimensional nilpotent Lie k-algebras h with 2-dimensional commutator ideals h', extending a known result to the case where h' is non-central and k is an arbitrary field. It turns\ud out that, while the structure of h depends on the field k if h' is central, it is independent of k if h' is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h' and dimk h < 12

Periodic derivations and prederivations of Lie algebras

“…. . , x 8 of g together with the pairs (x ℓ , x m ) for (ℓ, m) = (1, 2), (1, 3), (2, 3), (1,4), (2,4), (3,4), (5,6), (5,7), (5,8), (6,7), (6,8), (7,8). Proof.…”

“…It is well-known that a nilpotent Lie algebra with 1-dimensional commutator subalgebra is isomorphic to the direct sum of the Heisenberg Lie algebra h m and some abelian Lie algebra C k . Since both summands admit a periodic derivation, so does g. For the case that dim([g, g]) = 2 we can refer to theorem 1 in [1]. It says that g admits a hexagonal grading, and hence also a periodic derivation.…”

The trascendence of Kac-Moody algebras

“…Moreover, this author has recently also obtained the classification of smalldimensional solvable Lie algebras by using a simple method which allowed him to classify three-and four-dimensional solvable Lie algebras, over fields of any characteristic (recall that nilpotent Lie algebras are a subset of solvable Lie algebras) [113]. C. Bartolone, A. di Bartolo and G. Falcone (see [16]) published a paper devoted to the classification of nilpotent Lie algebras over an arbitrary field with a 2-dimensional commutator ideal. They extended some results by M.A.…”

A class of nilpotent Lie algebras admitting a compact subgroup of automorphisms

“…• If dim g = 2 and dim z = 1, then g is a uniquely determined by its dimension (cf. [2]); the (2n + 4)-and (2n + 5)-dimensional groups turn out to have maximal compact subgroup U(n) = Sp(2n, R) ∩ O(2n). Note again that if instead dim z = + 1, then we simply have a trivial Abelian extension of the case already discussed.…”

“…We mention that also nilpotent Lie algebras of type {n, 1, 1} can be explicitly described (cf. [2]), and derivations of a nilpotent Lie algebra of type {2n, 1, 1} are determined in [1].…”