A New Invariant For G-Invariant Star Products

The purpose of this Letter is to propose an invariant for a G-invariant star product on a G-transitive symplectic manifold which remains invariant under the G-equivalence maps. This invariant is defined by using a quantum moment map which is a quantum analogue of the moment map on a Hamiltonian G-space. On S ² regarded as an SO(3) coadjoint orbit in , we give an example of this invariant for the canonical G-invariant star product. In this example, there arises a nonclassical term which depends only on a class of G-invariant star products.     

Oscillatory Integral Formulae for Left-invariant Star Products on a Class of Lie Groups

We use star representation geometric methods to obtain explicit oscillatory integral formulae for strongly invariant star products on Iwasawa subgroups AN of SU(1,n)     

Morita Equivalence of Fedosov Star Products and Deformed Hermitian Vector Bundles

Based on the usual Fedosov construction of star products for a symplectic manifold M, we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over M starting with a symplectic connection on M and a connection for E. In the case of a line bundle, this gives a Morita equivalence bimodule, and the relation between the characteristic classes of the Morita equivalent star products can be found very easily within this framework. Moreover, we also discuss the case of a Hermitian vector bundle and give a Fedosov construction of the deformation of the Hermitian fiber metric.     

Invariant Star Products on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> and the Canonical Trace

We calculate the canonical trace and use the Fedosov–Nest–Tsygan index theorem to obtain the characteristic class for a star product on S ². We show how, for this simple example, it is possible to extract the relevant information needed to use the Fedosov–Nest–Tsygan index theorem from a local calculation.This revised version was published online in March 2005 with corrections to the cover date.     

A Remark on Nonequivalent Star Products via Reduction for CPn

In this Letter, we construct nonequivalent star products on CPⁿ by phase space reduction. It turns out that the nonequivalent star products occur very natural in the context of phase space reduction by deforming the momentum map of the U(1)-action on Cⁿ⁺¹\{0}; into a quantum momentum map and the corresponding momentum value into a quantum momentum value such that the level set, i.e. the constraint surface, of the quantum momentum map coincides with the classical one. All equivalence classes of star products on CPⁿ are obtained by this construction.     

Convergent Star Product Algebras on ‘ax+b’

We define nontempered (exponential growth) function spaces on the Lie group ax+b which are stable under some left-invariant (convergent) star product. The techniques used to achieved the latter come from symmetric spaces geometry and star representation theory.     

Cohomology of Good Graphs and Kontsevich Linear State Products

Following the ideas presented in q-alg/9709040, we give the definition of Kontsevich star products for linear Poisson structures on ^d. We prove that all these structures are equivalent and can be defined by integral formulae. Finally, we characterize, among these star products, the Gutt and Duflo star products.     

Nonlocality of Equivariant Star Products on T*( RPn)

Lecomte and Ovsienko constructed SL _n+1(R)-equivariant quantization maps Q for symbols of differential operators on -densities on RP ⁿ . We derive some formulas for the associated graded equivariant star products on the symbol algebra Pol(T* RP ⁿ ). These give some measure of the failure of locality. Our main result expresses (for n odd) the coefficients C _p (·,·) of when = in terms of some new SL _n+1(C)-invariant algebraic bidifferential operators Z _p (·,·) on T* CP ⁿ and the operators (E + n/2 ± s)^–1 where E is the fiberwise Euler vector field and s {1, 2, ..., [p/2]}.     

Complex Star Algebras

We describe a classification of star algebras on the cotangent bundle of a complex manifold, locally isomorphic to the algebra of pseudo-differential operators; this requires a slight extension of the usual definition of star algebras. We show that in dimension 3 these are essentially trivial and come from algebras of differential operators on X; in dimension 1 and 2 there are many more, which we describe.     

Fedosov Star Products and One-Differentiable Deformations

We show that every star product on a symplectic manifold defines uniquely a 1-differentiable deformation of the Poisson bracket. Explicit formulas are given. As a corollary we can identify the characteristic class of any star product as a part of its explicit (Fedosov) expression.     

\mathfrak{g}-Relative Star Products

We consider Kontsevich star products on the duals of Lie algebras. Such a star product is relative if, for any Lie algebra, its restriction to invariant polynomial functions is the usual pointwise product. Let be a fixed Lie algebra. We shall say that a Kontsevich star product is -relative if, on ^*, its restriction to invariant polynomial functions is the usual pointwise product. We prove that, if is a semi-simple Lie algebra, the only strict Kontsevich -relative star products are the relative (for every Lie algebras) Kontsevich star products.     

Hermitian Star Products are Completely Positive Deformations

Let M be a Poisson manifold equipped with a Hermitian star product. We show that any positive linear functional on C^∞(M) can be deformed into a positive linear functional with respect to the star product.     

Deformation Quantization with Traces

In this Letter we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a constant volume form and a Poisson bivector field on ^d such that div=0, the Kontsevich star product with the harmonic angle function is cyclic, i.e. (f*g)·h·= (g*h)·f· for any three functions f,g,h on (for which the integrals make sense). We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and an arbitrary volume form, and prove a generalization of the Connes–Flato–Sternheimer conjecture on closed star products in the Poisson case.     

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence, and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.     

On the Existence of Star Products on Quotient Spaces of Linear Hamiltonian Torus Actions

We discuss BFV deformation quantization (Bordemann et al. in A homological approach to singular reduction in deformation quantization, singularity theory, pp. 443–461. World Scientific, Hackensack, 2007) in the special case of a linear Hamiltonian torus action. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms and Gotay (Adv Math 79(1):43–103, 1990) for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.      

Tangential Star Products

We etablish a necessary and sufficient condition under which there exists a tangential and well graded star product, differential or not, on the dual of a nilpotent Lie algebra . We also give enlightening examples with explicit computations.     

Star products and quantization of poisson-lie groups

We prove some theorems by Drinfeld about solutions of the triangular quantum Yang-Baxter equation and corresponding quantum groups. These theorems are to be understood in the natural setting of invariant star products on a Lie group. We also set out and prove another theorem about the invariant Hochschild cohomological meaning of the quantum Yang-Baxter equation, which underlies the others.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C (M) and ( (E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C ^*-algebras. We also discuss the semi-classical geometry arising from these deformations.     

Noncommutative Line Bundle and Morita Equivalence

Global properties of Abelian noncommutative gauge theories based on -products which are deformation quantizations of arbitrary Poisson structures are studied. The consistency condition for finite noncommutative gauge transformations and its explicit solution in the Abelian case are given. It is shown that the local existence of invertible covariantizing maps (which are closely related to the Seiberg–Witten map) leads naturally to the notion of a noncommutative line bundle with noncommutative transition functions. We introduce the space of sections of such a line bundle and explicitly show that it is a projective module. The local covariantizing maps define a new star product which is shown to be Morita equivalent to .