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3 papers
The Real Jacobi Group Revisited
Stefan Berceanu National Institute for Physics and Nuclear Engineering, Department of Theoretical Physics, PO BOX MG-6, Bucharest-Magurele, Romania
Abstract:
The real Jacobi group
$G^J_1(\mathbb{R})$, defined as the semi-direct product of the group
${\rm SL}(2,\mathbb{R})$ with the Heisenberg group
$H_1$, is embedded in a
$4\times 4$ matrix realisation of the group
${\rm Sp}(2,\mathbb{R})$. The left-invariant one-forms on
$G^J_1(\mathbb{R})$ and their dual orthogonal left-invariant vector fields are calculated in the
$\mathrm{S}$-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
${\rm SL}(2,\mathbb{R})$ is presented. The well known Kähler balanced metric in the variables
$(x,y,p,q)$ of the four-dimensional Siegel–Jacobi upper half-plane $\mathcal{X}^J_1=\frac{G^J_1(\mathbb{R})}{{\rm SO}(2) \times\mathbb{R}} \approx\mathcal{X}_1 \times\mathbb{R}^2$ 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 $\tilde{\mathcal{X}}^J_1= \frac{G^J_1(\mathbb{R})}{{\rm SO}(2)}\approx\mathcal{X}_1\times\mathbb{R}^3$ is determined.
Keywords:
Jacobi group, invariant metric, Siegel–Jacobi upper half-plane, balanced metric, extended Siegel–Jacobi upper half-plane, naturally reductive manifold.
MSC: 32F45,
32Q15,
53C25,
53C22 Received: May 9, 2019; in final form
November 25, 2019; Published online
December 7, 2019
Language: English
DOI:
10.3842/SIGMA.2019.096
Fulltext:
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