The Real Jacobi Group Revisited

Abstract

The real Jacobi group $G_1^J(\mathbb{R})$, defined as the semi-direct product of the group $\mathrm{SL}(2,\mathbb{R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group $\mathrm{Sp}(2,\mathbb{R})$. The left-invariant one-forms on $G_1^J(\mathbb{R})$ and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates $(x,y,\theta ,p,q,\kappa )$, and a left-invariant metric depending of 4 parameters $(\alpha ,\beta ,\gamma ,\delta )$ is obtained. An invariant metric depending of $(\alpha ,\beta )$ in the variables $(x,y,\theta )$ on the Sasaki manifold $\mathrm{SL}(2,\mathbb{R})$ is presented. The well known Kähler balanced metric in the variables $(x,y,p,q)$ of the four-dimensional Siegel-Jacobi upper half-plane ${\mathcal{X}}_1^J=\frac{G_1^J(\mathbb{R})}{\mathrm{SO}(2)\times \mathbb{R}}\approx {\mathcal{X}}_1\times {\mathbb{R}}^2$ depending of $(\alpha ,\gamma )$ is written down as sum of the squares of four invariant one-forms, where ${\mathcal{X}}_1$ denotes the Siegel upper half-plane. The left-invariant metric in the variables $(x,y,p,q,\kappa )$ depending on $(\alpha ,\gamma ,\delta )$ of a five-dimensional manifold ${\tilde{\mathcal{X}}}_1^J=\frac{G_1^J(\mathbb{R})}{\mathrm{SO}(2)}\approx {\mathcal{X}}_1\times {\mathbb{R}}^3$ is determined.

References in corpus (9)

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• Applications of the Jacobi group to Quantum Mechanics
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Cited by in corpus (4)

• Invariant metric on the extended Siegel-Jacobi upper half space
• Hamiltonian systems on almost cosymplectic manifolds
• Remarks on the geometry of the extended Siegel--Jacobi upper half-plane
• Geodesics on the extended Siegel-Jacobi upper half-plane