Symplectic symmetric pairs of contragredient Lie superalgebras

Abstract

Let L be a contragredient Lie superalgebra. A symmetric pair of L is a pair $({\mathfrak {g}},{\mathfrak {g}}^\sigma )$, where ${\mathfrak {g}}$ is a real form of L, and $\sigma$ is a ${\mathfrak {g}}$-involution with invariant subalgebra ${\mathfrak {g}}^\sigma$. We show that a symmetric pair carries invariant symplectic forms if and only if ${\mathfrak {g}}^\sigma$ has a 1-dimensional center. Furthermore, the symplectic form is pseudo-Kähler if and only if the center of ${\mathfrak {g}}^\sigma$ is compact. As an application, we classify the symplectic symmetric pairs, as well as the pseudo-Kähler symmetric pairs.