We show that the existence of a left-invariant pluriclosed Hermitian metric on unimodular Lie group with abelian complex structure forces to be 2-step nilpotent. Moreover, we prove flow starting from nilpotent preserves Strominger Kähler–like condition.

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on R which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex struc...

Our work is concerned with the problem on limit cycle bifurcation for a class of Z3-equivariant Lyapunov system of five degrees with three third-order nilpotent critical points which lie in a Z3-equivariant vector field. With the help of computer algebra system-MATHEMATICA, the first 5 quasiLyapunov constants are deduced. The fact of existing 12 small amplitude limit cycles created from the thr...

which is defined only when I is nilpotent and Q ⊂ A. Goodwillie showed that (1.2) is an isomorphism whenever it is defined, and that it coincides with (1.1) when I = 0. Using this and a 5-Lemma argument, he deduced that (1.1) is an isomorphism for any nilpotent I. The question of whether ch∗ and ch ′ ∗ agree for general nilpotent ideals I was left open in [4], and announced without proof in [7,...

The exterior algebra over the centre of a Lie algebra acts on the cohomology of the Lie algebra in a natural way. Focusing on nilpotent Lie algebras, we explore the module structure afforded by this action. We show that for all two-step nilpotent Lie algebras, this module structure is non-trivial, which partially answers a conjecture of Cairns and Jessup [4]. The presence of free submodules ind...

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove, that the set-valued algebraic theory obtained by applying π0 is the theory of ordinary n-nilpotent groups and that ...

We consider the following problem: What are possible sizes of Jordan blocks for a pair of commuting nilpotent matrices? Or equivalently, for which pairs of nilpotent orbits of matrices (under similarity) there exists a pair of matrices, one from each orbit, that commute. The answer to the question could be considered as a generalization of Gerstenhaber– Hesselink theorem on the partial order of...

We organize the nilpotent orbits in the exceptional complex Lie algebras into series and show that within each series the dimension of the orbit is a linear function of the natural parameter a = 1, 2, 4, 8, respectively for f4, e6, e7, e8. We observe similar regularities for the centralizers of nilpotent elements in a series and graded components in the associated grading of the ambient Lie alg...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

Conditions are given for a class 2 nilpotent group to have no central extensions of class 3. This is related to Betti numbers and to the problem of representing a class 2 nilpotent group as the fundamental group of a smooth projective variety. Surveys of the work on the characterization of the fundamental groups of smooth projective varieties and Kähler manifolds (see [1],[3], [9]) indicate tha...