Related papers: Universal quantum semigroupoids

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang-Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Gorenstein and Koszul of global…

We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the…

Quantum algebras (also called quantum groups) are deformed versions of the usual Lie algebras, to which they reduce when the deformation parameter q is set equal to unity. From the mathematical point of view they are Hopf algebras. Their…

For any $n$-ary associative algebra we construct a $\Z_{n-1}$ graded algebra, which is a universal object containing the $n$-ary algebra as a subspace of elements of degree 1. Similar construction is carried out for semigroups.

The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical)…

A large-scalable quantum computer model, whose qubits are represented by the subspace subtended by the ground state and the single exciton state on semiconductor quantum dots, is proposed. A universal set of quantum gates in this system may…

Basic concepts of quantum integrable systems (QIS) are presented stressing on the unifying structures underlying such diverse models. Variety of ultralocal and nonultralocal models is shown to be described by a few basic relations defining…

We introduce a new top down approach to canonical quantum gravity, called Algebraic Quantum Gravity (AQG):The quantum kinematics of AQG is determined by an abstract $*-$algebra generated by a countable set of elementary operators labelled…

Let $\mathfrak{g}=\mathfrak{gl}_{M|N}(\mathbb{k})$ be the general linear Lie superalgebra over an algebraically closed field $\mathbb{k}$ of characteristic zero. Fix an arbitrary even nilpotent element $e$ in $\mathfrak{g}$ and let…

We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If $K$ is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra $K^n$:…

A description of the quantum superalgebra $U_q[sl(n+1|m)]$ and in particular of the special linear superalgebra $sl(n+1|m)$ via creation and annihilation generators (CAGs) is given. It provides an alternative to the canonical description of…

The algebraic formulation of the quantum group covariant noncommutative geometry in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider structure groups taking values in the quantum groups and…

For a given inverse semigroup, one can associate an \'etale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated \'etale groupoids. In this paper, we focus on…

Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This…

Quantum groups and quantum homogeneous spaces - developed by several authors since the 80's - provide a large class of examples of algebras which for many reasons we interpret as `coordinate algebras' over noncommutative spaces. This…

Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$…

We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $\Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $\mathcal…

We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a…

For every element w in the Weyl group of a simple Lie algebra g, De Concini, Kac, and Procesi defined a subalgebra U_q^w of the quantized universal enveloping algebra U_q(g). The algebra U_q^w is a deformation of the universal enveloping…

We introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study properties. Our approach is slightly different from the one introduced in the thesis of S.~Roy and, unlike the investigations…