arXiv research

Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.

Study of presymplectic forms on almost abelian Lie algebras, determining moduli spaces and their finiteness.

The paper analyzes Lie symmetries in a specific geometric context.

Develops Lie algebraic approach for compact complex homogeneous manifolds.

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces i…

The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie algebra and to prove some elementary properties of $\operatorname{D}$-Lie algebras, the category of $\operatorname{D}$-Lie algebras, the category of modules on a $\operatorname{D}$-Lie algebra and extensions of $\operatorname{D}$-Lie algebras. …

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct p…

Study locally conformally balanced metrics on specific Lie algebras.

Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.

Characterizes complex structures on specific Lie groups.

Study abelian factors in Lie algebras from graph edge labels.

Paper confirms conjecture for specific Lie algebras.

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie grou…

QP perspective on Poisson-Lie T-duality topology changes.

Study on Hermitian metrics on Lie algebras with specific ideals.

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

Study pseudo-Kähler structures on almost abelian solvmanifolds.

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

Unique complex structures on specific Lie algebras.

Study coKähler structures on Lie algebras using Fino-Vezzoni correspondence.

Characterizes almost abelian Lie algebras with integrable complex structure

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g…

Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.

The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.

We study the dg-Lie algebra f_n generated by the coefficients of the universal translation invariant flat dg-connection on the n-dimensional affine space. We describe its "semiabelianization" (in particular, the universal quotient which is a crossed module of Lie algebras) in terms of closed differential forms of arbit…

The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.

Let S be a compact connected oriented surface with one boundary component. We extend each of Johnson's and Morita's homomorphisms to the Ptolemy groupoid of S. Our extensions are canonical and take values into finitely generated free abelian groups. The constructions are based on the 3-dimensional interpretation of the…

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

The paper explores geometric structures on Hom-Lie groups and algebras.

The paper explores associative structures in pseudo-Riemannian Lie algebras and their geometric implications.

It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping.

Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.

The paper classifies solitons on specific Lie groups.

Cocalibrated G_2-structures and cocalibrated G_2^*-structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin_0(3,4), respectively. In this article, we classify which seven-dimensional real Lie algebra…

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and …

We associate Hamiltonian homological evolutionary vector fields --which are the non-Abelian variational Lie algebroids' differentials-- with Lie algebra-valued zero-curvature representations for partial differential equations.

Classifies two-step solvable Lie groups with SKT structures.

Characterizes almost Abelian Lie algebras with special $H$-structures.

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson's approach to loop groups and the approach of Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebra…

Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.

The Streets-Tian conjecture is confirmed for Lie algebras with specific abelian ideals.

Symmetric spaces' connections form Lie admissible triple algebras.

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras integrates to a corresponding Lie group extension $\hat{G}$ of $G$ by $A$, where $G$ is a connected, simply connected Lie group and $A$ is a quotient of its Lie algebra by some discrete subgroup. When $G$ is non-simply connected…

Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.

We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which ve…