Stefan Berceanu
A representation of the Jacobi algebra $\mathfrak{h}_1\rtimes \mathfrak{su}(1,1)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}\times \mathcal{D}_1$ is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with polynomials coefficients act is constructed.
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In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi-Nijenhuis structures.
This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to complex contact groupoids. A crucial tool in our proof is what we call the "homogenization scheme", which allows us to identify holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and holomorphic contac…
We present a holomorphic representation of the Jacobi algebra $\mathfrak{h}_n\rtimes \mathfrak{sp}(n,\R)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}^n\times \mathcal{D}_n$. We construct the Hilbert space of holomorphic functions on which these differential operators act.
We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.
The standard formulation of Jacobi manifolds in terms of differential operators on line bundles, although effective at capturing most of the relevant geometric features, lacks a clear algebraic interpretation similar to how Poisson algebras are understood to be the algebraic counterpart of Poisson manifolds. We propose a formalism, based on the dimensioned algebra technology recently developed by the author, to capture the algebraic counterparts of Jacobi manifolds as dimensioned Poisson algebras. Particularly, we give a generalisation of the functor $\text{C}^\infty:\textsf{Smooth}\to \textsf…
Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introduced. Matrix elements of the squeezed operators, expectation values of polynomial operators in infinitesimal generators of the Jacobi group, the squeezing region and a description of Mandel's parameter are presented.
A Kahlerian approche to the Schrodinger equation in Siegel jacobi Space of the lognormal distribution
本文基于对数正态统计流形上的 Kähler 几何，在 Siegel Jacobi 空间中构建含时薛定谔方程以描述谱曲线演化，揭示了哈密顿向量场与全纯等距生成向量场的一致性，并建立了 Kähler 几何、统计模型与量子力学之间的联系。
We prove that any $4$-dimensional almost-Kähler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is Kähler.
The vector space of holomorphic polyvector fields on any complex manifold has a natural Gerstenhaber algebra structure. In this paper, we study BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on smooth compact toric varieties. We give a necessary and sufficient condition for the existence of BV operators of the Gerstenhaber algebra of holomorphic polyvector fields on any smooth compact toric variety.
We give examples of Lie-Rinehart algebras whose enveloping algebra is not a full Hopf algebroid in the sense of Bohm and Szlachanyi. We construct these examples as quotients of a canonical Lie-Rinehart algebra over a Jacobi algebra which does admit an antipode.
Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra $A$ and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vector space $V$ a non-abelian cohomological type object ${\mathcal J}{\mathcal H}^{2} \, (V, \, A)$ is constructed: it classifies all Jacobi algebras containing $A$ as a subalgebra of codimension equal to ${\rm dim} (V)$. Representations of $A$ are used in order to give the decomposi…
We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacobi--Ricci commuting (i.e. the Ricci operator commutes with the Jacobi operator of any vector).
We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also discuss a new example of a Lie bialgebroid on Poisson manifolds.
We develop the deformations theory of a Dirac--Jacobi structure within a fixed Courant--Jacobi algebroid. Using the description of split Courant--Jacobi algebroids as degree $2$ contact $\mathbb{N} Q$ manifolds and Voronov's higher derived brackets, each Dirac--Jacobi structure is associated with a cubic $L_\infty$ algebra for any choice of a complementary almost Dirac--Jacobi structure. This $L_\infty$ algebra governs the deformations of the Dirac--Jacobi structure: there is a one-to-one correspondence between the MC elements of this $L_\infty$ algebra and the small deformations of the Dirac-…
Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von Neumann algebras as developed in an earlier paper by the authors. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic $L^2$ torsion, which shows that it is not in general ind…
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considere…
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