On the Geometry of Siegel–jacobi Domains

Berceanu, Stefan; Gheorghe, A.

doi:10.1142/s0219887811005920

Cited by 13 publications

(20 citation statements)

References 25 publications

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“…If M ∈ M (2n, R) has the property (4.28), let 29) and the correspondence M → M of (4.28) with (4.29) is a group isomorphism…”

Section: The Starting Point In the Coherent States Approachmentioning

confidence: 99%

The Real Jacobi Group Revisited

Berceanu

2019

SIGMA

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The real Jacobi group G J 1 (R), defined as the semi-direct product of the group SL(2, R) with the Heisenberg group H 1 , is embedded in a 4×4 matrix realisation of the group Sp(2, R). The left-invariant one-forms on G J 1 (R) and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates (x, y, θ, p, q, κ), and a left-invariant metric depending of 4 parameters (α, β, γ, δ) is obtained. An invariant metric depending of (α, β) in the variables (x, y, θ) on the Sasaki manifold SL(2, R) is presented. The well known Kähler balanced metric in the variables (x, y, p, q) of the four-dimensional Siegel-Jacobi upper half-is written down as sum of the squares of four invariant one-forms, where X 1 denotes the Siegel upper half-plane. The left-invariant metric in the variables (x, y, p, q, κ) depending on (α, γ, δ) of a five-dimensional manifoldFC −1 : η = z +zw 1 − |w| 2 , and FC : A(w, z) → dη − wdη.

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“…If M ∈ M (2n, R) has the property (4.28), let 29) and the correspondence M → M of (4.28) with (4.29) is a group isomorphism…”

Section: The Starting Point In the Coherent States Approachmentioning

confidence: 99%

The Real Jacobi Group Revisited

Berceanu

2019

SIGMA

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“…We have the correspondence 15) and V ∈ X n , u ∈ C n ≡ R 2n . Let g ∈ Sp(n, R) C be of the form (1.1), (2.14) and α, z ∈ C n .…”

Section: The Cayley Transformmentioning

confidence: 99%

“…In (3.5) e 0 = e H 0 ⊗ e K 0 , where e H 0 is the minimum weight vector (vacuum) for the Heisenberg group H n , while e K 0 is the extremal weight vector for Sp(n, R) C corresponding to the weight k in (3.5), and µ parametrizes the Heisenberg group [7,9,15]. Holomorphic irreducible representations of the Jacobi group based on Siegel-Jacobi domains have been studied in mathematics [22,23,72,73,74].…”

Section: The Balanced Metricmentioning

confidence: 99%

“…Holomorphic irreducible representations of the Jacobi group based on Siegel-Jacobi domains have been studied in mathematics [22,23,72,73,74]. In [5,9,15] we have compared our results on the geometry of D J n and holomorphic representations of G J n obtained using CS with those of the mentioned mathematicians. The scalar product K : D J n × D J n → C, K(x,V ; y, W ) = (e x,V , e y,W ) kµ is [7,15]:…”

Section: The Balanced Metricmentioning

confidence: 99%

“…The Jacobi groups are semidirect products of appropriate semisimple real algebraic groups of Hermitian type with Heisenberg groups [23,33,49,74]. As unimodular, nonreductive, algebraic groups of Harish-Chandra type [68], the Jacobi groups are intensively studied in mathematics [15,22,72,73,74,76,78,79]. The Siegel-Jacobi (partially bounded [78,79]) domains are homogeneous manifolds associated to the Jacobi groups by the generalized Harish-Chandra embedding.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

Berceanu¹

2016

SIGMA

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A Useful Parametrization of Siegel–Jacobi Manifolds

Berceanu

2013

Geometric Methods in Physics

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The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following subjects are discussed via the coherent states: the geodesics, the conjugate locus and the cut locus; the divisors; the Calabi's diastasis and its domain of definition; the Euler-Poincaré characteristic of the manifold, the number of Borel-Morse cells, Kodaira embeding theorem....

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On the Geometry of Siegel–jacobi Domains

Cited by 13 publications

References 25 publications

The Real Jacobi Group Revisited

The Real Jacobi Group Revisited

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

A Useful Parametrization of Siegel–Jacobi Manifolds

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