Related papers: Symplectic Structures on Free Nilpotent Lie algebr…
In this work we introduce an obstruction for the existence of symplectic structures on nilpotent Lie algebras. Indeed, a necessary condition is presented in terms of the cohomology of the Lie algebra. Using this obstruction we obtain both…
We study symplectic structures on nilpotent Lie algebras. Since the classification of nilpotent Lie algebras in any dimension seems to be a crazy dream, we approach this study in case of 2-step nilpotent Lie algebras (in this sub-case also,…
We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space $H^2(\Lg,k)$ for certain Lie algebras $\Lg$. Among these Lie algebras are filiform CNLAs of dimension $n\le 14$. It turns…
We develop a structure theory for nilpotent symplectic alternating algebras.
We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$.…
We analyze symplectic forms on six dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable…
In this paper, we are interested in solvable complete Lie algebras, over the field $\K=\R$ or $\mathbb{C}$, which admit a symplectic structure. Specifically, important classes are studied, and a description of complete Lie Algebra with the…
We study left invariant contact forms and left invariant symplectic forms on Lie groups. We give the classification of all symplectic structures on nilpotent Lie algebras up the dimension 6.
We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T*-extension of a…
In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described…
A study is made of real Lie algebras admitting a hypersymplectic structure, and we provide a method to construct such hypersymplectic Lie algebras. We use this method in order to obtain the classification of all hypersymplectic structures…
We study symplectic structures on filiform Lie algebras -- nilpotent Lie algebras of the maximal length of the descending central sequence. In the present article we classify the Lie algebras with the structure relations of the following…
In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.
We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie…
Every symplectic Lie algebra with degenerate (including non-abelian nilpotent symplectic Lie algebras) has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding…
The present work studies deeply quadratic symplectic Lie superalgebras, obtaining, in particular, that they are all nilpotent. Consequently, we provide classifications in low dimensions and identify the double extensions that maintain…
In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them to similar results for the Lie algebra of (exact) divergence free vector fields. In…
In this work we deal with left invariant complex and symplectic structures on simply connected four dimensional solvable real Lie groups. We search the general form of such structures, when they exist and we make use of this information to…
We develop a structure theory for nilpotent symplectic alternating algebras. We then give a classification of all nilpotent symplectic alternating algebras of dimension up to 10 over any field. The study reveals a new subclasses of powerful…
The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie…