arXiv subjects
Publications and source records attributed to Xiang Tang.
We construct a universal deformation formula for Connes-Moscovici's Hopf algebra without any projective structure using Fedosov's quantization of symplectic diffeomorphisms.
In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author \cite{t1:def-gpd}, we reconstruct a universal deformation formula of the Hopf algebra $\calh_1$ associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket $RC_1$ defines a noncommutative Poisson structure for an arbitrary $\calh_1$ action.
Let $G$ be a Poisson Lie group and $\g$ its Lie bialgebra. Suppose that $\g$ is a group Lie bialgebra. This means that there is an action of a discrete group $Γ$ on $G$ deforming the Poisson structure into coboundary equivalent ones. Starting from this we construct a non-trivial stack of Hopf-Poisson algebras and prove the existence of associated deformation quantizations. This non-trivial stack is a stack of functions on the formal Poisson group, dual of the starting $Γ$ Poisson-Lie group. To quantize this non-trivial stack we use quantization of a $Γ$ Lie bialgebra which is the infinitesimal of a $Γ$ Poisson-Lie group (cf \cite{MS} for simple Lie groups and $Γ$ a covering of the Weyl group and \cite{EH} for quantization in the general case).
Inspired by the group structure on $S^1/ \bbZ$, we introduce a weak hopfish structure on an irrational rotation algebra $A$ of finite Fourier series. We consider a class of simple $A$-modules defined by invertible elements, and we compute the tensor product between these modules defined by the hopfish structure. This class of simple modules turns out to generate an interesting commutative unital ring.
We study wave functions of B-model on a Calabi-Yau threefold in various polarizations.
Let $Γ$ be a finite group acting linearly on a vector space $V$. We compute the Lie algebra cohomology of the Lie algebra of $Γ$-invariant formal vector fields on $V$. We use this computation to define characteristic classes for foliations on orbifolds.
We introduce a notion of "hopfish algebra" structure on an associative algebra, allowing the structure morphisms (coproduct, counit, antipode) to be bimodules rather than algebra homomorphisms. We prove that quasi-Hopf algebras are examples of hopfish algebras. We find that a hopfish structure on the commutative algebra of functions on a finite set G is closely related to a "hypergroupoid" structure on G. The Morita theory of hopfish algebras is also discussed.
Let X be a complex manifold with strongly pseudoconvex boundary M. If u is a defining function for M, then -log u is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form s = i \del \delbar(-log u) 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 s 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 u is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Englis 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.
In this note, we study Schwarz's conjecture on application of Q-algebras to strict quantization. We prove that in the case of a torus with a constant Poisson structure, Schwarz's formalism gives the same star product as Rieffel \cite{rif:quantization}. We construct twisted Fock modules as examples of quantization dg-modules in the case of a compact Kähler manifold. In particular, we relate this construction on $\complex\mathbb{P}^1$ to representations of a fuzzy sphere.
Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann--Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.
In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper etale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.
In this note, we will show one example of hamiltonian Lie algebra action which has no invariant star product.
We introduce a new kind of groupoid--a pseudo étale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the idea that symplectic and Poisson geometries are the semiclassical limits of the corresponding quantum geometries, we quantize these noncommutative Poisson manifolds in the framework of deformation quantization.
We define a C*-algebraic quantization of constant Dirac structures on tori, which extends the standard quantization of Poisson structures. We prove that Dirac structures in the same orbit of a natural action of O(n,n|Z) give rise to Morita equivalent algebras, completing and extending a theorem of Rieffel and Schwarz.
The solution to nonlinear Fokker-Planck equation is constructed in terms of the minimal Markov semigroup generated by the equation. The semigroup is obtained by a purely functional analytical method via Hille-Yosida theorem. The existence of the positive invariant measure with density is established and a weak form of Foguel alternative proven. We show the equivalence among self-adjoint of the elliptic operator, time-reversibility, and zero entropy production rate of the stationary diffusion process. A thermodynamic theory for diffusion processes emerges.