Related papers: On Function Theory in Quantum Disc: A q-Analogue o…
This work considers a formal deformation of the quantum disc (it is developed via an application of the Berezin method) and presents an explicit formula for this deformation.
We use Berezin's quantization procedure to obtain a formal $U_q su_{1,1}$-invariant deformation of the quantum disc. Explicit formulae for the associated q-bidifferential operators are produced.
This work presents proofs of the main results of (math.QA/9808015), except those on q-Berezin transform to appear in a subsequent work. The notation and the results of (math.QA/9808037) and (math.QA/9808047) are used.
In our earlier work math.QA/9808015 some results on integral representations of functions in quantum disc were announced. It was then shown in math.QA/9808037 that the validity of those results is related to the invariance of kernels of…
We introduce and study, in the framework of a theory of quantum Cartan domains, a q-analogue of the Berezin transform on the unit ball. We construct q-analogues of weighted Bergman spaces, Toeplitz operators and covariant symbol calculus.…
Our previous work (math.QA/9808015) introduces the basic notions and announces some results on function theory in the quantum disc. The present paper establishes a relationship between those results and the quantum groups theory.
A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a…
A new approach to deformation quantization on the cylinder considered as phase space is presented. The method is based on the standard Moyal formalism for R^2 adapted to (S^1 x R) by the Weil--Brezin--Zak transformation. The results are…
The present work considers one of the simplest homogeneous spaces of the quantum group SU(1,1), the q-analogue of the unit disc in ${\Bbb C}$. We state without proofs q-analogues of Cauchy-Green formulae, integral representations of…
The aim of this paper is to give a basic overview of Deformation Quantization (DQ) to physicists. A summary is given here of some of the key developments over the past thirty years in the context of physics, from quantum mechanics to…
This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs, and extend it to a more general class of noncommutative polydomains. An important role in our study is played by…
By a transfer principle Pascal's Theorem is equivalent to a theorem about point pairs on the real line. It appears that Pascal's Theorem is equivalent to the vanishing of a common invariant of six quadratic forms. Using the q-deformed…
Let $\ds dA=\frac{dxdy}\pi$ denote the normalized Lebesgue area measure on the unit disk $\disk$ and $u$, a summable function on $\disk$. $$B(u)(z)=\int_\disk u(\zeta)\frac{(1-|z|^2)^2}{|1-\zeta\oln z|^4}dA(\zeta)$$ is called the Berezin…
This work is devoted to the study of the foundations of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth projective variety X, analogous to…
We described a $q$-deformation of a quantum dynamics in one dimension. We prove that there exists only one essential deforamtion of quantum dynamics.
We present a quantum deformation theory of the Airy curve and use it to establish a version of mirror symmetry of a point.
We present an approach to Berezin quantization (a variant of quantization in the spirit of Berezin) on para-Hermitian symmetric spaces using the notion of an "overgroup". This approach gives covariant and contravariant symbols and the…
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both,…
Over the last 25 years, the notion of "fuzzy spaces" has become ubiquitous in the high-energy physics literature. These are finite dimensional noncommutative approximations of the algebra of functions on a classical space. The most well…
We study the theory of representations of a multiparameter deformation of the function algebra of a simple algebraic group (as defined by Reshetikhin) when the quantum parameter is a root of unity. We extend the technics of De…