Related papers: Wheeled props in algebra, geometry and quantizatio…
Generalisations of geometry have emerged in various forms in the study of field theory and quantization. This mini-review focuses on the role of higher geometry in three selected physical applications. After motivating and describing some…
In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and…
This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.
A motivation of using noncommutative and nonarchimedean geometry on very short distances is given. Besides some mathematical preliminaries, we give a short introduction in adelic quantum mechanics. We also recall to basic ideas and tools…
The scope of this review is to give a pedagogical introduction to some new calculations and methods developed by the author in the context of quantum groups and their applications. The review is self- contained and serves as a "first aid…
In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques…
This course is aimed at graduate students in physics in mathematics and designed to give a comprehensive introduction to Weyl quantization and semiclassics via Egorov's theorem. Chapter 2 gives a quick overview of classical and quantum…
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…
We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra ${\mathcal V}=T(V)\otimes T(V^\star)$, where $T(V)$ is the tensor algebra on an $n$-dimensional vector…
This note, in a rather expository manner, serves as a conceptional introduction to the certain underlying mathematical structures encoding the geometric quantization formalism and the construction of Witten's quantum invariants, which is in…
We study homotopy theory of the wheeled prop controlling Poisson structures on arbitrary formal graded finite-dimensional manifolds and prove, in particular, that Grothendieck-Teichmueller group acts on that wheeled prop faithfully and…
We survey several generalizations of the Weyl algebra including generalized Weyl algebras, twisted generalized Weyl algebras, quantized Weyl algebras, and Bell-Rogalski algebras. Attention is paid to ring-theoretic properties,…
This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…
In this contribution to the proceedings of the 68eme Rencontre entre Physiciens Theoriciens et Mathematiciens on Deformation Quantization I shall report on some recent joint work with Henrique Bursztyn on the representation theory of…
This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We show the interest of such algebras for fractional supersymmetric quantum mechanics, angular momentum theory and…
Since the subject of noncommutative geometry is now entering maturity, we felt there is need for presentation of the material at an undergraduate course level. Our review is a zero order approximation to this project. Thus, the present…
Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly…
A comparison on some facts concerning the geometric quantization of symplectic manifolds is presented here. Criticism, facts and improvements on the sophisticated theory of geometric quantization are presented touching briefly, all the…
The Wheeler-DeWitt Equation represents a tool to study Quantum Gravity and Quantum Cosmology. Its solution in a very general context is, of course, impossible. To this purpose we consider some distortions of General Relativity like…
We review the definition of geometric quantization, which begins with defining a mathematical framework for the algebra of observables that holds equally well for classical and quantum mechanics. We then discuss prequantization, and go into…