Related papers: Topological model for q-deformed rational number a…
We describe the relationships between the notion of $q$-deformed rational numbers, introduced in our previous work with Sophie Morier-Genoud, and the theory of dimer models. We show that $q$-deformed rationals can be calculated in terms of…
Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal{R}_{\frac{r}s}(q),~ \mathcal{S}_{\frac{r}s}(q)…
We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the…
We consider the $q$-deformation of rational numbers introduced recently by Morier-Genoud and Ovsienko. We propose three enumerative interpretations of these $q$-rationals: in terms of a new version of Ostrowski's numeration system for…
We describe a family of compactifications of the space of Bridgeland stability conditions of any triangulated category following earlier work by Bapat, Deopurkar, and Licata. We particularly consider the case of the 2-Calabi--Yau category…
The left and right $q$-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and they gave a homological interpretation for left and right $q$-deformed rational numbers. In the present paper,…
The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial…
We consider a natural $q$-deformation of the classical Markov numbers. This $q$-deformation is closely related to $q$-deformed rational numbers recently introduced by two of us. Both notions, those of $q$-rationals and $q$-Markov numbers,…
Let $A = \Bbbk Q / I$ be the path algebra of any finite quiver $Q$ modulo any two-sided ideal $I$ of relations and let $R$ be any reduction system satisfying the diamond condition for $I$. We introduce an intrinsic notion of deformation of…
We associate a deformation of Heisenberg algebra to the suitably normalized Yang $R$-matrix and we investigate its properties. Moreover, we construct new examples of quantum vertex algebras which possess the same representation theory as…
We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket…
We establish duality between real forms of the quantum deformation of the 4-dimensional orthogonal group studied by Fioresi et al. and the classification work made by Borowiec et al.. Classically these real forms are the isometry groups of…
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. For the fractional harmonic oscillator, the corresponding q-number is derived.…
We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to…
We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show…
In this paper, we generalize the Markov triples in two different directions. One is generalization in direction of using the $q$-deformation of rational number introduced by \cite{MO} in connection with cluster algebras, quantum topology…
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. A new class of fractional q-deformed Lie algebras is proposed, which for the…
We establish a link between the new theory of $q$-deformed rational numbers and the classical Burau representation of the braid group $\mathrm{B}_3$. We apply this link to the open problem of classification of faithful complex…
Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated…
The braided approach to q-deformation (due to the author and collaborators) gives natural algebras $R_{21}u_1Ru_2=u_2R_{21}u_1R$ and $R_{21}x_1x_2=x_2x_1R$ for q-Minkowski and q-Euclidean spaces respectively. These algebras are covariant…