Special symplectic Lie groups and hypersymplectic Lie groups

TL;DR

This paper introduces a method to deform special symplectic Lie groups into hypersymplectic Lie groups using affine structures, and explores their algebraic and extension properties.

Contribution

It develops a deformation process from special symplectic to hypersymplectic Lie groups and introduces new algebraic structures like post-left-symmetric algebras.

Findings

Deformation of Lie group structures to hypersymplectic forms

Introduction of post-left-symmetric algebra as underlying structure

Construction of double extensions of special symplectic Lie groups

Abstract

A special symplectic Lie group is a triple $(G,\omega,\nabla)$ such that $G$ is a finite-dimensional real Lie group and $\omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $\nabla$. In this paper starting from a special symplectic Lie group we show how to ``deform" the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $\nabla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.