Abstract Given a negatively graded Calabi-Yau algebra, we regard it as DG algebra with vanishing differentials and study its cluster category. We show that this is sign-twisted realise category triangulated hull of an orbit derived the singularity finite-dimensional Iwanaga-Gorenstein algebra. Along way, give two results stand on their own. First, coherent sheaves over has natural tilting subca...

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspec...

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH(B) with coefficients in the B–B-bimodule B. If C is a tilted algebra such that B is the relation extension of C by E = ExtC(DC,C), then we prove that HH (B) is isomorphic, as a vector space, to the direct sum of HH(C) with HH(B,E). This yields homological interpretations for results of the first and the fourth au...

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field Fq and any sequence i of simple objects in C the element XV,i of the corresponding algebra PC,i of q-polynomials. We prove that if C was hereditary, then the assignments V 7→ XV,i define algebra homomorphisms from the (dual) Ha...

it is shown that every  almost linear bijection $h : arightarrow b$ of a unital $c^*$-algebra $a$ onto a unital$c^*$-algebra $b$ is a $c^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries  $u in a$, all $y in a$, and all $nin mathbb z$, andthat almost linear continuous bijection $h : a rightarrow b$ of aunital $c^*$-algebra $a$ of real rank zero onto a unital$c^*$-algebra...

We make an explicit combinatorial construction of the cluster algebra arising from a double wiring diagram. We also state a Schur non-negativity conjecture and prove it is true for small cases.

We construct and study cluster algebra structures in rings of invariants of the special linear group action on collections of 3D vectors, covectors, and matrices. The construction uses Kuperberg's calculus of webs on marked surfaces with boundary.

We introduce a multi-parameter generalization of the Lambda-determinant of Robbins and Rumsey, based on the cluster algebra with coefficients attached to a T -system recurrence. We express the result as a weighted sum over alternating sign matrices.

We describe the c-vectors and g-vectors of the Markov cluster algebra in terms of a special family of triples of rational numbers, which we call the Farey triples.