Related papers: Contractions of quantum algebraic structures
In a recent article, the coordinate ring of the nodal cubic was given the structure of a quantum homogeneous space. Here the corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity, and is shown to…
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 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…
We discuss a procedure to determine finite sets $\mathcal{M}$ within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra $\mathfrak{g}$ such that their generators define a quadratic algebra. Although…
Given an algebra A, presented by generators and relations, i.e. as a quotient of a tensor algebra by an ideal, we construct a free algebra resolution of A, i.e. a differential graded algebra which is quasi-isomorphic to A and which is…
Algebraic framework for construction of a commuting set of operators that can be interpreted as integrals of motion of the open spin chain with boundary conditions and nearest neighbour interaction is investigated.
This is a brief review of several algebraic constructions related to generalized fermionic spectra, of the type which appear in integrable quantum spin chains and integrable quantum field theories. We discuss the connection between…
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we…
A generalized algebra of quantum observables, depending on extra dimensional constants, is considered. Some limiting forms of the algebra are investigated and their possible applications to the descriptions of interactions of fundamental…
This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using $\mathcal{W}$-algebras as our starting…
We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrable highest weight representations.
In this paper, the notion of universal enveloping algebra introduced in [A. Ardizzoni, \emph{A First Sight Towards Primitively Generated Connected Braided Bialgebras}, submitted. (arXiv:0805.3391v3)] is specialized to the case of braided…
After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a…
This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of…
This is an introduction to quantum algebra, from a geometric perspective. The classical spaces $X$, such as the Lie groups, homogeneous spaces, or more general manifolds, are described by various algebras $A$, defined over various fields…
We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.
The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture…
This talk introduces a Cartan-geometric framework for generalised geometries governed by a differential graded Lie algebra. In contrast to ordinary Cartan geometry, the tangent bundle is extended and qu both a global duality group and a…
Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external…
The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general…