Tensor products of n-complete algebras

Abstract

If A and B are n-and m-representation finite k-algebras, then their tensor product Λ = A ⊗ k B is not in general (n + m)-representation finite. However, we prove that if A and B are acyclic and satisfy the weaker assumption of n-and m-completeness, then Λ is (n + m)-complete. This mirrors the fact that taking higher Auslander algebra does not preserve drepresentation finiteness in general, but it does preserve d-completeness. As a corollary, we get the necessary condition for Λ to be (n + m)-representation finite which was found by Herschend and Iyama by using a certain twisted fractionally Calabi-Yau property.

THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPEA

“…Since these algebras are of infinite representation type, it is common to study only particular subcategories of the module category. This is the case for finding 2-cluster-tilting subcategories as in [40,77]; or interval representations [7,14,25] and, alternatively, thin modules [10] for applications to multi-dimensional persistence homology in topological data analysis. Fomin, Pylyavskyy and Shustin conjecture that any two real morsifications of real isolated plane curve singularities are related by mutation of their associated quivers if and only if they are of the same complex topological type [27, Conjecture 5.1].…”

Tensor products of higher APR tilting modules

“…Lemma 3.19. (see [8,30]) Assume that Λ, Γ are two finite dimensional algebras over field K. If M, N ∈ mod Λ and M ′ , N ′ ∈ mod Γ, then there is a functorial isomorphism…”

“…The tensor product is a very effective research tool in representation theory of finite dimension algebras [1,8,23,24,29,30]. For n-, m-representation-finite algebras Λ, respectively Γ over perfect field K, under condition of l-homogeneity, Herschend and Iyama in [15] showed that tensor product Λ ⊗ K Γ is an (n + m)representation finite algebra which admits the (n+m)-APR tilting (Λ⊗ K Γ)-module associated with simple projective module.…”

Preprojective algebras of $d$-representation finite species with relations

“…In general, Λ 1 ⊗ K Λ 2 is not (d 1 + d 2 )representation finite if we drop the assumption that Λ 1 and Λ 2 are l-homogeneous. Relaxing the assumptions on Λ 1 and Λ 2 Pasquali proved that if Λ i is an acyclic d i -complete algebra for each i ∈ {1, 2} then Λ 1 ⊗ K Λ 2 is (d 1 + d 2 )-complete [Pas19]. In this paper we will further study the former case, that Λ i is a d i -representation finite l-homogeneous algebra, but we also assume that Λ i is a Koszul algebra for each i ∈ {1, 2} and focus on the Koszul properties of the preprojective algebra of Λ 1 ⊗ K Λ 2 .…”