Related papers: Symplectic structures on $3$-Lie algebras

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra [K]. A Lie pseudoalgebra is a generalization of the…

We are interested in the classification of left-invariant symplectic structures on Lie groups. Some classifications are known, especially in low dimensions. In this paper we establish a new approach to classify (up to automorphism and…

Let $k$ be a field of any characteristic, $V$ a finite-dimensional vector space over $k$, and $S^d(V^*)$ be the $d$-th symmetric power of the dual space $V^*$. Given a linear map $\varphi$ on $V$ and an eigenvector $w$ of $\varphi$, we…

This note provides a formula for the character of the Lie algebra of the fundamental group of a surface, viewed as a module over the symplectic group.

A Hermitian metric on a complex manifold is called strong K\"ahler with torsion (SKT) if its fundamental 2-form $\omega$ is $\partial \bar \partial$-closed. We review some properties of strong KT metrics also in relation with symplectic…

A family of solvable self-dual Lie algebras is presented. There exist a few methods for the construction of non-reductive self-dual Lie algebras: an orthogonal direct product, a double-extension of an Abelian algebra, and a Wigner…

We study symplectic structures on K\"ahler surfaces with p_g = 0. We give an example of a projective surface which admits a symplectic structure which is not compatible with any K\"ahler metric.

We study left-invariant pseudo-K\"ahler and hypersymplectic structures on semidirect products $G\rtimes H$; we work at the level of the Lie algebra $\mathfrak{g}\rtimes\mathfrak{h}$. In particular we consider the structures induced on…

The double extension and the T*-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic…

We define hypersymplectic structures on Lie algebroids recovering, as particular cases, all the classical results and examples of hypersymplectic structures on manifolds. We prove a 1-1 correspondence theorem between hypersymplectic…

We study the algebraic constraints on the structure of nilpotent Lie algebra $\mathbb{g}$, which arise because of the presence of an integrable complex structure $J$. Particular attention is paid to non-abelian complex structures.…

We propose an approach to extending the concept of a Lie algebra to ternary structures based on $\omega$-symmetry, where $\omega$ is a primitive cube root of unity. We give a definition of a corresponding structure, called a ternary Lie…

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates…

Given a $2k$-dimensional symplectic space $(Z,F)$ in $N$ variables, $1 < 2k \leq N$, over a global field $K$, we prove the existence of a symplectic basis for $(Z,F)$ of bounded height. This can be viewed as a version of Siegel's lemma for…

A Lie superalgebra is attached to any finite-dimensional J-ternary algebra over an algebraically closed field of characteristic 3, using a process of semisimplification via tensor categories. Some of the exceptional simple Lie algebras,…

S-expansions of three-dimensional real Lie algebras are considered. It is shown that the expansion operation allows one to obtain a non-unimodular Lie algebra from a unimodular one. Nevertheless S-expansions define no ordering on the…

A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be…

In this paper, first we show that $(\g,[\cdot,\cdot],\alpha)$ is a hom-Lie algebra if and only if $(\Lambda \g^*,\alpha^*,d)$ is an $(\alpha^*,\alpha^*)$-differential graded commutative algebra. Then, we revisit representations of hom-Lie…

We revisit the Faulkner construction of metric 3-Leibniz algebras admitting an embedding Lie (super)algebra. In the case of positive-definite signature, we relate the various notions of simplicity: of the 3-algebra, of the representation…

It is pointed out that affine Lie algebras appear to be the natural mathematical structure underlying the notion of integrability for two-dimensional systems. Their role in the construction and classification of 2D integrable systems is…