Internally Calabi–Yau algebras and cluster-tilting objects

Abstract

We describe what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d − 1)-cluster-tilting objects in certain stably (d − 1)-Calabi-Yau Frobenius categories, as observed by Keller-Reiten. We show that an internally d-Calabi-Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d − 1)-clustertilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger 'bimodule' internally d-Calabi-Yau condition, this Frobenius category is stably (d − 1)-Calabi-Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi-Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot's construction in the coefficient-free case.

Mutation of frozen Jacobian algebras

“…From the point of view of defining J (Q, F, W ), we could have taken F to simply be a set of arrows, rather than a subquiver, but we have opted to also record frozen vertices for better compatibility with the defining data of a cluster algebra with frozen variables (cf. also [31], in which the idempotent defined by the frozen vertices plays an important role.) Jacobian algebras are somewhat ubiquitous; it has been shown by Buan-Iyama-Reiten-Smith [11] that cluster-tilted algebras are Jacobian algebras, and Bocklandt [6] has shown that any graded 3-Calabi-Yau algebra is a (necessarily infinite-dimensional) Jacobian algebra.…”

“…4.5], there is an exact functor π : CM(B) → Sub Q k , which is a quotient by the ideal generated by an indecomposable projective B-module P n . Here Sub Q k denotes the exact category of submodules of an injective module Q k for the preprojective algebra of type A n−1 , see [18, §3], and is a Hom-finite Frobenius cluster category [31,Eg. 3.11] (in fact, it is even one of the categories C w considered in [10]; cf.…”

“…We pick out three families of Frobenius cluster categories for which some clustertilting objects have endomorphism algebra isomorphic to J (Q, F, W ) for Q without loops and 2-cycles, so we can apply Lemma 5.11. For cases (i) and (iii), proofs that the categories are indeed Frobenius cluster categories can be found in[31,. In case (ii), this is part of [30, Thm.…”

THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPEA

“…Grassmannian clusters have seen much study recently; from the representationtheoretic side, their categorifications were studied directly [35,55]: through dimer models [12]; and through Frobenius versions [78,79], as well as self-injective quivers with potential [41,76]. More general models studied in relation to Schubert cells can be found in [32,59,83].…”

Relative cluster categories and Higgs categories

“…Moreover, A is internally bimodule (n + 1)-Calabi-Yau with respect to the idempotent e = f (1 B ) in the sense of Matthew Pressland (see [42]) and restriction induces an equivalence from the Higgs category H to the category of Gorenstein projective modules over B ′ = eH 0 (A)e. More precisely, we have the following theorem. We summarize the situation in the following commutative diagram perA…”