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A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
To a representation of $\O_N$ (the Cuntz algebra with $N$ generators) we associate a projection valued measure and we study the case when this measure has atoms. The main technical tool are the spaces invariant for all the operators…
Let A be a finitely generated commutative algebra over a field K with a presentation A=K < X_{1}, ..., X_{n} | R >, where R is a set of monomial relations in the generators X_{1}, ..., X_{n}. So A = K[S], the semigroup algebra of the monoid…
The pair consisting of a quantum group and its corresponding coideal subalgebra, known as a quantum symmetric pair, was developed independently by M. Noumi and G. Letzter through different approaches. The purpose of this paper is threefold.…
Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms $T_0$, $T_1$ of the $q$-Onsager algebra $\mathcal O_q$, that are roughly analogous to the Lusztig automorphisms of $U_q(\widehat{\mathfrak{sl}}_2)$. We use $T_0, T_1$…
We study the representation theory of various convolution algebras attached to the $q$-deformation of $\mathrm{SL}(2,\mathbb{R})$ from an algebraic perspective and beyond the unitary case. We show that many aspects of the classical…
A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator model in the plane, whose Hamiltonian…
In this paper we study the tensor powers of the standard representation of the quantum super-algebra $U_q(sl(2|1)$, focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by $LG_n$. Their dimensions were…
We provide a generators and relation description of the deformed W_{1+\infty}-algebra introduced in previous joint work of E. Vasserot and the second author. This gives a presentation of the (spherical) cohomological Hall algebra of the…
Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance,…
We provide a generalized definition for the quantized Clifford algebra introduced by Hayashi using another parameter $k$ that we call the twist. For a field of characteristic not equal to $2$, we provide a basis for our quantized Clifford…
We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $\mathfrak{sl}_2$ at a root of unity $q$ of odd order. Our approach is diagrammatic, and it relies on…
To any strongly continuous orthogonal representation of $\R$ on a real Hilbert space $\CH_\R$, Hiai constructed $q$-deformed Araki-Woods von Neumann algebras for $-1< q< 1$, which are $W^{\ast}$-algebras arising from non tracial…
In this note, we study various relational and algebraic aspects of the bounded quasi-implication algebras introduced by Hardegree. By generalizing the constructions given by MacLaren and Goldblatt within the setting of ortholattices, we…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…
We establish a Drinfeld type new presentation for the $\imath$quantum groups arising from quantum symmetric pairs of split affine ADE type, which includes the $q$-Onsager algebra as the rank 1 case. This presentation takes a form which can…
Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting…
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The…
The procedure of double extension of vector spaces endowed with non-degenerate bilinear forms allows us to introduce the class of generalized $\mbK$-oscillator algebras over any arbitrary field $\mbK$. Starting from basic structural…
In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as…