The Half-period Addition Formulae for Genus Two Hyperelliptic $\wp$ Functions and the Sp(4,$\mathbb{R}$) Lie Group Structure

TL;DR

This paper establishes that the half-period addition formulae for genus two hyperelliptic functions encode the order two Sp(4,ℝ) Lie group structure, linking complex function theory with Lie group symmetries.

Contribution

It directly demonstrates the Sp(4,ℝ) Lie group structure from differential equations of genus two hyperelliptic functions, without relying on integrable models.

Findings

Half-period addition formulae encode Sp(4,ℝ) group structure

Differential equations of hyperelliptic functions reveal Lie group symmetries

Connects hyperelliptic functions with Lie algebra and Lie group theory

Abstract

In the previous study, by using the two-flows Kowalevski top, we have demonstrated that the genus two hyperelliptic functions provide the Sp(4,$\mathbb{R}$)/$Z_2$ $\cong$ SO(3,2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic $\wp$ functions instead of using integrable models, we demonstrate that the half-period addition formula for the genus two hyperelliptic functions provides the order two Sp(4,$\mathbb{R}$) Lie group structure.