Bidifferential Operators Research Articles

Curved Versions of the Ovsienko–Redou Operators

Abstract We construct a family of conformally covariant bidifferential operators on pseudo-Riemannian manifolds. Our construction is analogous to the construction of Graham–Jenne–Mason–Sparling of conformally covariant differential operators via tangential powers of the Laplacian in the Fefferman–Graham ambient space. In fact, we completely classify the tangential bidifferential operators on the ambient space, which are expressed purely in terms of the ambient Laplacian. This gives a curved analogue of the classification, due to Ovsienko–Redou and Clerc, of conformally invariant bidifferential operators on the sphere. As an application, we construct a large class of formally self-adjoint conformally invariant differential operators.

Sobolev spaces and $$\nabla $$-differential operators on manifolds I: basic properties and weighted spaces

We study covariant Sobolev spaces and $$\nabla $$ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted $$\nabla $$ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the $$\nabla $$ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection $$\nabla $$ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our $$\nabla $$ -differential operators. We also introduce and study the notion of a $$\nabla $$ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for $$\nabla $$ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

Wick rotations in deformation quantization

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from [Formula: see text] with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space [Formula: see text] and the complex hyperbolic disc [Formula: see text]. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on [Formula: see text] and [Formula: see text]. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the [Formula: see text]-involution of pointwise complex conjugation.

Rankin–Cohen brackets on tube-type domains

A new formula is obtained for the holomorphic bi-differential operators on tube-type domains which are associated to the decomposition of the tensor product of two scalar holomorphic representations, thus generalizing the classical Rankin-Cohen brackets. The formula involves a family of polynomials of several variables which may be considered as a (weak) generalization of the classical Jacobi polynomials.

Symmetry breaking differential operators, the source operator and Rodrigues formulae

A Rodrigues type formula is obtained for the symbols of the covariant bi-differential operators on a simple Jordan algebra.

Second cohomology space of $\mathfrak{sl}(2)$ acting on the space of bilinear bidifferential operators

We consider the sl(2)-module structure on the spaces of bilinear bidifferential operators acting on the spaces of weighted densities. We compute the second cohomology group of the Lie algebra sl(2) with coefficients in the space of bilinear bidifferential operators that act on tensor densities Dλ,ν,µ.

Conformally Covariant Bi-differential Operators for Differential Forms

The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.

Conformally Covariant Bi-differential Operators on a Simple Real Jordan Algebra

Abstract For a simple real Jordan algebra V, a family of bi-differential operators from $\mathcal{C}^\infty (V\times V)$ to $\mathcal{C}^\infty (V)$ is constructed. These operators are covariant under the rational action of the conformal group of V. They generalize the classical Rankin–Cohen brackets (case $V=\mathbb{R}$).

Twist for Snyder space

We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.

Covariant bi-differential operators on matrix space

A family of bi-differential operators from C ∞ Mat(m, R)×Mat(m, R) into C ∞ Mat(m, R) which are covariant for the projective action of the group SL(2m, R) on Mat(m, R) is constructed, generalizing both the transvectants and the Rankin-Cohen brackets (case m = 1). Resume On construit une famille d' operateurs bi-differentiels de C ∞ Mat(m, R) × Mat(m, R) dans C ∞ Mat(m, R) qui sont covari-ants pour l'action projective du groupe SL(2m, R) sur Mat(m, R). Dans le cas m = 1, cette construction fournit une nouvelle approche des transvectants et des crochets de Rankin-Cohen.

Singular invariant trilinear forms and covariant (bi-)differential operators under the conformal group

The residues of the meromorphic family of conformally invariant trilinear forms on the sphere (constructed in Clerc and Ørsted, in press, [2]) are computed. Their expression involves conformally covariant differential and bidifferential operators. For the latter, new formulæ are obtained.

Rankin-Cohen Brackets for Jacobi Forms and Tensor Products of Discrete Series Representations of the Jacobi Group

Explicit formulae for Rankin–Cohen brackets for Jacobi forms are obtained by pure Lie theoretic means. The Lie algebraic nature of these holomorphic bi-differential operators, which map pairs of Jacobi forms onto another such form, is unveiled, along with their relationship with the decompositions of tensor products of two (anti-)holomorphic discrete series representations of the Jacobi group into direct sums of similar representations. These decompositions, which turn out to be nonmultiplicity-free, are a problem of an independent interest, which is explicitly solved. Some of the obtained brackets are shown to be intertwiners, projecting the appropriate tensor product onto a direct summand. The first mentioned goal is reached by characterizing all the -equivariant holomorphic bi-differential operators, through establishing a one-to-one correspondence between them and the null vectors in the tensor products of two Verma -modules, for the complexified Lie algebra of . A large class of pairs of such modules is considered, the remaining situations being relegated to a future work. A thorough analysis of the structure of the tensor products, based on Borho’s isomorphism, which provides a factorization of the universal enveloping algebra of , yields explicit formulae for the sought after null vectors; they turn out to be completely determined by the null vectors in tensor products of three Verma -modules. The bi-differential operators associated with the obtained null vectors are then shown to be Rankin–Cohen brackets for Jacobi forms. The Lie theoretic approach developed here supplies a unifying framework within which results of previous investigations undertaken by different authors, using techniques pertaining to the realm of number theory, are recovered, provided with new interpretations and generalized. In particular, Bocherer’s generic dimensions are shown to be given by the above-mentioned multiplicities.

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality $L_\infty$-quasi-isomorphism for Hochschild chains is compatible with cap-products. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality $L_\infty$-quasi-isomorphism for Hochschild cochains. As in the cohomological situation our proof relies on a homotopy argument involving a variant of {\bf Kontsevich eye}. In particular we clarify the r\^ole played by the {\bf I-cube} introduced in \cite{CR1}. Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an $A_\infty$-algebra. In particular we prove that they naturally inherit the structure of an $A_\infty$-algebra with an $A_\infty$-(bi)module.

Universal Star Products

One defines the notion of universal deformation quantization: given any manifold M, any Poisson structure Λ on M and any torsionfree linear connection ∇ on M, a universal deformation quantization associates to this data a star product on (M, Λ) given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor Λ, the curvature tensor R and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poissoncohomology.

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Let X be a complex manifold with strongly pseudoconvex boundary M . If \psi is a defining function for M , then -\log\psi is plurisubharmonic on a neighborhood of M in X , and the (real) 2-form \sigma = i \partial \bar \partial (-\log \psi) is a symplectic structure on the complement of M in a neighborhood in X of M ; it blows up along M . The Poisson structure obtained by inverting \sigma extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure near M is completely determined up to isomorphism by the contact structure on M . In addition, when -\log\psi is plurisubharmonic throughout X , and X is compact, bidifferential operators constructed by Engliš for the Berezin–Toeplitz deformation quantization of X are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on M , along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.

The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators

The Lie theoretic nature of the Rankin–Cohen brackets is here uncovered. These bilinear operations, which, among other purposes, were devised to produce a holomorphic automorphic form from any pair of such forms, are instances of SL ( 2 , R ) -equivariant holomorphic bi-differential operators on the upper half-plane. All of the latter are here characterized and explicitly obtained, by establishing their one-to-one correspondence with singular vectors in the tensor product of two sl ( 2 , C ) Verma modules. The Rankin–Cohen brackets arise in the generic situation where the linear span of the singular vectors of a given weight is one-dimensional. The picture is completed by the special brackets which appear for the finite number of pairs of initial lowest weights for which the above space is two-dimensional. Explicit formulæ for basis vectors in both situations are obtained and universal Lie algebraic objects subsuming all of them are exhibited. A few applications of these results and Lie theoretic approach are then considered. First, a generalization of the latter yields Rankin–Cohen type brackets for Hilbert modular forms. Then, some Rankin–Cohen brackets are shown to intertwine the tensor product of two holomorphic discrete series representations of SL ( 2 , R ) with another such representation occurring in the tensor product decomposition. Finally, the sought for precise relationship between the Rankin–Cohen brackets and Gordan's transvection processes of the nineteenth century invariant theory is unveiled.

Universality of Fedosov's construction for star products of Wick type on pseudo-Kähler manifolds

In this paper we construct star products on a pseudo-Kähler manifold ( M, ω, I) using a modification of the Fedosov method based on a different fibrewise product similar to the Wick product on C n . Having fixed the used connection to be the pseudo-Kähler connection these star products shall depend on certain data given by a formal series of closed two-forms on M and a certain formal series of symmetric contravariant tensor fields on M. In a first step we show that this construction is rich enough to obtain star products of every equivalence class by computing Deligne's characteristic class of these products. Among these products we uniquely characterize the ones which have the additional property to be of Wick type which means that the bidifferential operators describing the star products only differentiate with respect to holomorphic directions in the first argument and with respect to anti-holomorphic directions in the second argument. These star products are in fact strongly related relvtar products with separation of variables introduced and studied by Karabegov. This characterization gives rise to special conditions on the data that enter the Fedosov procedure. Moreover, we compare our results that are based on an obviously coordinate independent construction to those of Karabegov that were obtained by local considerations and give an independent proof of the fact that star products of Wick type are in bijection to formal series of closed two-forms of type (1, 1) on M. Using this result we finally succeed in showing that the given Fedosov construction is universal in the sense that it yields all star products of Wick type on a pseudo-Kähler manifold. Due to this result we can make some interesting observations concerning these star products; we can show that all these star products are of Vey type and in addition we can uniquely characterize the ones that have the complex conjugation incorporated as an anti-automorphism.

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order $k$ in the deformation parameter have at most $k$ derivatives in each argument. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance and give necessary and sufficient conditions for them to yield a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.

Berezin–Toeplitz Quantization and Invariant Symbolic Calculi

It is well known that one can often construct an invariant star-product by expanding the product of two Toeplitz operators asymptotically into a series of another Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin–Toeplitzquantization. We show that on bounded symmetric domains (Hermitian symmetric spaces of noncompact type), one can in fact obtain in a similar way any invariant star-productwhich is G-equivalent to the Berezin–Toeplitz star-product, by using, instead of Toeplitz operators, other suitable assignments f↦Qf from compactly supported C∞ functions f to bounded linear operators Qf on the corresponding Hilbert spaces. (This procedureis referred to as prime quantization by some authors.) Along the way, we establish two technical results which are of interest in their own right, namely a controlled-growth parameter generalization of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, and the fact that any invariant bi-differential operator (Hochschild two-cochain) on a bounded symmetric domain automatically maps the Schwartz space into itself.

AN EXPLICIT FORMULA FOR PBW QUANTIZATION

ABSTRACT Let k be a field of characteristic zero, a k-Lie algebra, the symmetrization map. The PBW quantization is the one parameter family of associative products: where is the homogeneous component of degree of the map , . In this paper we give an explicit formula for B. As an application, we prove that for each , is a bidifferential operator of order .