Related papers: Noncommutative marked surfaces
The aim of the paper is to define noncommutative cluster structure on several algebras ${\mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a…
For an unpunctured marked surface $\Sigma$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum…
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric…
A non associative, noncommutative algebra is defined that may be interpreted as a set of vector modules over a noncommutative surface of rotation. Two of these vector modules are identified with the analogues of the tangent and cotangent…
We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each…
For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $\Sigma$ with at least two marked points and no punctures, we prove that the cluster algebra…
It was shown by Fock, Goncharov and Fomin, Shapiro, Thurston that some cluster algebras arise from triangulated orientable suraces. Subsequently Dupont and Palesi generalised this construction to include unpunctured non-orientable surfaces,…
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of…
We introduce C-Algebras of compact Riemann surfaces $\Sigma$ as non-commutative analogues of the Poisson algebra of smooth functions on $\Sigma$. Representations of these algebras give rise to sequences of matrix-algebras for which…
We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\sigma$. We give a geometric realization of such cluster algebras by…
Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of…
To a given tiling a non commutative space and the corresponding C*-algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for…
We define a non-commutative version of the $A_1$ T-system, which underlies frieze patterns of the integer plane. This system has discrete conserved quantities and has a particular reduction to the known non-commutative Q-system for $A_1$.…
We define a new family of noncommutative generalizations of cluster algebras called polygonal cluster algebras. These algebras generalize the noncommutative surfaces of Berenstein-Retakh, and are inspired by the emerging theory of…
We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…
We examine several algebraic properties of the noncommutive $z$-plane and Riemann surfaces. The starting point of our investigation is a two-dimensional noncommutative field theory, and the framework of the theory will be converted into…
We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum…
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect…
In this paper, we discover a new noncommutative algebra. We refer this algebra as the constant term algebra of type $A$, which is generated by certain constant term operators. We characterize a structural result of this algebra by…