Four-dimensional noncommutative deformations of U(1) gauge theory and L∞ bootstrap.

We construct a family of four-dimensional noncommutative deformations of U(1) gauge theory following a general scheme, recently proposed in JHEP 08 (2020) 041 for a class of coordinate-dependent noncommutative algebras. This class includes the $$ \mathfrak{su} $$ su (2), the $$ \mathfrak{su} $$ su (1, 1) and the angular (or λ-Minkowski) noncommutative structures. We find that the presence of a fourth, commutative coordinate x0 leads to substantial novelties in the expression for the deformed field strength with respect to the corresponding three-dimensional case. The constructed field theoretical models are Poisson gauge theories, which correspond to the semi-classical limit of fully noncommutative gauge theories. Our expressions for the deformed gauge transformations, the deformed field strength and the deformed classical action exhibit flat commutative limits and they are exact in the sense that all orders in the deformation parameter are present. We review the connection of the formalism with the L∞ bootstrap and with symplectic embeddings, and derive the L∞-algebra, which underlies our model.

Center of skew PBW extensions

In this paper we compute the center of many noncommutative algebras that can be interpreted as skew [Formula: see text] extensions. We show that, under some natural assumptions on the parameters that define the extension, either the center is trivial, or, it is of polynomial type. As an application, we provided new examples of noncommutative algebras that are cancellative.

Schur indices for reality-based algebras with two nonreal basis elements

This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension [Formula: see text] with rational field of definition for which it is a division algebra. The approach has other applications, one of which shows noncommutative association scheme of rank [Formula: see text] must have at least three symmetric relations.

Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras

We review main differential-algebraic structures \ lying in background of \ analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and \ Leibniz type algebraic structures are derived, a new nonassociative "Riemann" algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. \ As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. \ Subject to these important aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we \ revisited \ the classical Poisson manifold approach, closely related to our construction of \ Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, \ we presented its natural and simple generalization allowing effectively to describe a wide class\ of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.