相关论文: Cluster algebras I: Foundations

We introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological…

The article concerns the existence and uniqueness of quantisations of cluster algebras. We prove that cluster algebras with an initial exchange matrix of full rank admit a quantisation in the sense of Berenstein-Zelevinsky and give an…

We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and…

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…

The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras which we call factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes coordinate algebras…

We define a class of associative algebras generalizing 'clannish algebras', as introduced by the second author, but also incorporating semilinear structure, like a skew polynomial ring. Clannish algebras generalize the well known 'string…

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the appearance of…

In this note we study dual coalgebras of algebras over arbitrary (noetherian) commutative rings. We present and study a generalized notion of coreflexive comodules and use the results obtained for them to characterize the so called…

We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures). Systematic usage of…

We prove a convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool we introduce new semigroup of partial permutations. We…

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…

We study quantum cluster algebras from marked surfaces without punctures. We express the quantum cluster variables in terms of the canonical submodules. As a byproduct, we obtain the positivity for this class of quantum cluster algebra.

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…

We give a brief introduction to (upper) cluster algebras and their quantization using examples. Then we present several important families of bases for these algebras using topological models. We also discuss tropical properties of these…

We calculate the cluster modular groups of affine and doubly extended typecluster algebras in a uniform way by introducing a new family of quivers. We use this uniformdescription to construct a natural finite quotient of the cluster complex…

Dickson's commutative semifields are an important class of finite division algebras. We generalise Dickson's construction of commutative division algebras by doubling both finite field extensions and central simple algebras and not…

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a braided tensor category, we give a detailed construction of the…

In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some "perfection" property (simpleness, primitiveness, primeness or…

In the theory of generalized cluster algebras, we build the so-called cluster formula and $D$-matrix pattern. Then as applications, some fundamental conjectures of generalized cluster algebras are solved affirmatively.

We define a class of algebras describing links of binary semi-isolating formulas on a set of realizations for a family of 1-types of a complete theory. These algebras include algebras of isolating formulas considered before. We prove that a…