Solvable Lie algebras with complex symplectic structures

Abstract: Let $\mathfrak g$ be a $2n$-dimensional solvable Lie algebra. A complex structure on $\mathfrak g$ is an endomorphism $J$ that satisfies $J^2=-Id$ and $N_J(X,Y)=0$, for every $X,Y\in\mathfrak g$, being $$N_J(X,Y):=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY].$$ Suppose that $\mathfrak g$ simultaneously admits a complex structure $J$ and a symplectic structure $\omega$ (i.e., a closed $2$-form $\omega\in\wedge^2\mathfrak g^*$ such that $\omega^n\neq 0$). Although $J$ and $\omega$ are initially two unrelated structures, one can ask for an additional condition involving both of them. In this sense, the pair $(J,\omega)$ is said to be a complex symplectic structure if $J$ is symmetric with respect to $\omega$, in the sense that $\omega(JX,Y)=\omega(X,JY)$, for every $X,Y\in\mathfrak g$. In this talk, we will present some methods to find certain types of solvable Lie algebras (such as nilpotent or almost Abelian) admitting complex symplectic structures.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar