Pseudo-algebraic Ricci solitons on Einstein nilradicals

We develop a variational method to find pseudo-algebraic Ricci solitons on connected Lie groups. As applications, we prove that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and that any algebraic Ricci soliton on a semi-simple Lie group is Einstein. Furthermore, we construct several Lorentz algebraic Ricci solitons on the nilpotent Lie groups which have a codimension one abelian ideal.

Locally conformally balanced metrics on almost abelian Lie algebras

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.

Abelian ideals and amazing roots

Let [Formula: see text] be a simple Lie algebra with a Borel subalgebra [Formula: see text]. To any long positive root [Formula: see text], one associates two ideals of [Formula: see text]: the abelian ideal [Formula: see text] and not necessarily abelian ideal [Formula: see text]. It is known that [Formula: see text], and [Formula: see text] is said to be amazing if the equality holds. The set of amazing roots, [Formula: see text], is closed under the operation “∨” in [Formula: see text], and [Formula: see text] is said to be primitive, if it cannot be written as [Formula: see text] with incomparable amazing roots [Formula: see text]. We classify the amazing roots and notice that the number of primitive roots equals [Formula: see text]. Moreover, if [Formula: see text] (respectively, [Formula: see text]) is the set of simple (respectively, primitive) roots, then there is a natural bijection [Formula: see text]. We also study the subset [Formula: see text] of [Formula: see text].

Finite presentability of some metabelian Hopf algebras

We classify the Hopf algebras U (L)#kQ of homological type FP2 where L is a Lie algebra and Q an Abelian group such that L has an Abelian ideal A invariant under the Q-action via conjugation and U (L/A)#kQ is commutative. This generalises the classification of finitely presented metabelian Lie algebras given by J. Groves and R. Bryant.

On the cohomology of a class of nilpotent Lie algebras

Let g denote a finite dimensional nilpotent Lie algebra over ℂ containing an Abelian ideal a of codimension 1, with z ∈ g/a. We give a combinatorial description of the Betti numbers of g in terms of the Jordan decomposition induced by ad(z)|a. As an application we prove that the filiform-nilpotent Lie algebras arising in the case t = 1 have unimodal Betti numbers.

On a Theorem of D. W. Barnes

Let L be a solvable Lie algebra and A be an abelian ideal of L. For a ∊ A, let da be the (right) inner derivation of L generated by a and let exp da = 1 + da. Since A is abelian, exp da is an automorphism and exp da exp db = exp da+b for all a, b ∊ A. Let I(L, A) be the subgroup of the automorphism group of L generated by exp da for all a ∊ A.