Alexander Karabegov

Formal Symplectic Groupoid of a Deformation Quantization

We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid ‘with separation of variables’ over an arbitrary Kähler-Poisson manifold.

An invariant formula for a star product with separation of variables

Abstract We give an invariant formula for a star product with separation of variables on a pseudo-Kahler manifold.

Fedosov’s formal symplectic groupoids and contravariant connections

Abstract Using Fedosov’s approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kahler–Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.

A Covariant Poisson Deformation Quantization with Separation of Variables up to the Third Order

We give a simple formula for the operator C3 of the standard deformation quantization with separation of variables on a Kähler manifold M. Unlike C1 and C2, this operator cannot be expressed in terms of the Kähler–Poisson tensor on M. We modify C3 to obtain a covariant deformation quantization with separation of variables up to the third order which is expressed in terms of the Poisson tensor on M and can thus be defined on an arbitrary complex manifold endowed with a Poisson bivector field of type (1,1).

On the phase form of a deformation quantization with separation of variables

Abstract Given a star product with separation of variables on a pseudo-Kahler manifold, we obtain a new formal (1, 1)-form from its classifying form and call it the phase form of the star product. The cohomology class of a star product with separation of variables equals the class of its phase form. We show that the phase forms can be arbitrary and they bijectively parametrize the star products with separation of variables. We also describe the action of a change of the formal parameter on a star product with separation of variables, its formal Berezin transform, classifying form, phase form, and canonical trace density.

On Gammelgaard’s Formula for a Star Product with Separation of Variables

We show that Gammelgaard’s formula expressing a star product with separation of variables on a pseudo-Kähler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard’s formula which gives more insight into the question why the directed graphs in his formula have no cycles.

A mapping from the unitary to doubly stochastic matrices and symbols on a finite set

We prove that the mapping from the unitary to doubly stochastic matrices that maps a unitary matrix (ukl) to the doubly stochastic matrix (|ukl|2) is a submersion at a generic unitary matrix. The proof uses the framework of operator symbols on a finite set.

Felix Alexandrovich Berezin and His Work

This is a survey of Berezin’s work focused on three topics: representation theory, general concept of quantization, and supermathematics.

On the Inverse Mapping of the Formal Symplectic Groupoid of a Deformation Quantization

To each natural star product on a Poisson manifold M we associate an antisymplectic involutive automorphism of the formal neighborhood of the zero section of the cotangent bundle of M. If M is symplectic, this mapping is shown to be the inverse mapping of the formal symplectic groupoid of the star product. The construction of the inverse mapping involves modular automorphisms of the star product.

Deformation quantization with separation of variables of an endomorphism bundle

Abstract Given a holomorphic Hermitian vector bundle E and a star-product with separation of variables on a pseudo-Kahler manifold, we construct a star product on the sections of the endomorphism bundle of the dual bundle E ∗ which also has the appropriately generalized property of separation of variables. For this star product we prove a generalization of Gammelgaard’s graph-theoretic formula.