Related papers: Differential and integral forms on non-commutative…
This article provides a synthesis of recent advances in the study of the PI property in various classes of noncommutative algebras of polynomial type.
A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique…
A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated…
This is an overview on derived nonhomogeneous Koszul duality over a field, mostly based on the author's memoir arXiv:0905.2621. The paper is intended to serve as a pedagogical introduction and a summary of the covariant duality between…
We define a noncommutative differential calculus constructed from the inner derivation, then several relevant examples are showed. It is of interest to note that for certain $C^*$-algebra, this calculus is closely related to the classical…
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.
For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative…
We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory.
In differential geometry, the notation d^n f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher order are useful objects that can be interpreted in terms of…
This is a compilation of some well known propositions of Alain Connes concerning the use of noncommutative geometry in mathematical physics.
This work reports on the construction of a nonlinear distributional geometry (in the sense of Colombeau's special setting) and its applications to general relativity with a special focus on the distributional description of impulsive…
A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or…
Traditionally, Hodge structures are associated with complex projective varieties. In my expository lectures I discussed a non-commutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry.
We discuss the differential algebras used in Connes' approach to Yang-Mills theories with spontaneous symmetry breaking. These differential algebras generated by algebras of the form functions $\otimes$ matrix are shown to be skew…
This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems…
A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and non degenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central…
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the…
Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint…
We examine several algebraic properties of the noncommutive $z$-plane and Riemann surfaces. The starting point of our investigation is a two-dimensional noncommutative field theory, and the framework of the theory will be converted into…
Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.