Left invariant special K\"ahler structures

TL;DR

This paper constructs and characterizes left invariant special Kähler structures on Lie groups, introducing new algebraic constructions and providing explicit examples of such geometric structures.

Contribution

It introduces the twisted Cartesian product and double extension methods for special Kähler Lie algebras, enabling classification of simply connected structures with bi-invariant connections.

Findings

Construction of left invariant special Kähler structures on cotangent bundles.

Introduction of twisted Cartesian product for special Kähler Lie algebras.

Characterization of flat special Kähler structures via étale Kähler affine representations.

Abstract

We construct left invariant special K\"ahler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special K\"ahler Lie algebras according to two linear representations by infinitesimal K\"ahler transformations. We also exhibit a double extension process of a special K\"ahler Lie algebra which allows us to get all simply connected special K\"ahler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or K\"ahler) affine transformations of its Lie algebra containing a nontrivial $1$-parameter subgroup formed by central translations. We show a characterization of left invariant flat special K\"ahler structures using \'etale K\"ahler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.