Publications | Andrea Pasquali's webpage

Orbifold diagrams

Sun, 01 Jan 2023 00:00:00 +0000

We study alternating strand diagrams on the disk with an orbifold point. These are quotients by rotation of Postnikov diagrams on the disk, and we call them orbifold diagrams. We associate a quiver with potential to each orbifold diagram, in such a way that its Jacobian algebra and the one associated to the covering Postnikov diagram are related by a skew-group algebra construction. We moreover realise this Jacobian algebra as the endomorphism algebra of a certain explicit cluster-tilting object. This is similar to (and relies on) a result by Baur-King-Marsh for Postnikov diagrams on the disk.

Existence of symmetric maximal noncrossing collections of k-element sets

Mon, 01 Jun 2020 00:00:00 +0000

We investigate the existence of maximal collections of mutually noncrossing $k$-element subsets of {$ 1, \dots, n $} that are invariant under adding $k\pmod n$ to all indices. Our main result is that such a collection exists if and only if $k$ is congruent to $0, 1$ or $-1$ modulo $n/\operatorname{GCD}(k,n)$. Moreover, we present some algebraic consequences of our result related to self-injective Jacobian algebras.

Quivers with potentials and actions of finite abelian groups

Wed, 25 Dec 2019 00:00:00 +0000

Let $G$ be a finite abelian group acting on a path algebra $kQ$ by permuting the vertices and preserving the arrowspans. Let $W$ be a potential on the quiver $Q$ which is fixed by the action. We study the skew group dg algebra $\Gamma_{Q, W}G$ of the Ginzburg dg algebra of $(Q, W)$. It is known that $\Gamma_{Q, W}G$ is Morita equivalent to another Ginzburg dg algebra $\Gamma_{Q_G, W_G}$, whose quiver $Q_G$ was constructed by Demonet. In this article we give an explicit construction of the potential $W_G$ as a linear combination of cycles in $Q_G$, and write the Morita equivalence explicitly. As a corollary, we obtain functors between the cluster categories corresponding to the two quivers with potentials.

Self-Injective Jacobian Algebras from Postnikov Diagrams

Mon, 01 Apr 2019 00:00:00 +0000

We study a finite-dimensional algebra $\Lambda$ constructed from a Postnikov diagram $D$ in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus, $\Lambda$ is isomorphic to the stable endomorphism algebra of a cluster tilting module $T\in\operatorname{CM}(B)$ introduced by Jensen-King-Su in order to categorify the cluster algebra structure of $\mathbb{C}[\operatorname{Gr}_{k}(\mathbb{C}^n)]$. We show that $\Lambda$ is self-injective if and only if $D$ has a certain rotational symmetry. In this case, $\Lambda$ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense ofHerschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras.

Skew group algebras of Jacobian algebras

Tue, 01 Jan 2019 00:00:00 +0000

For a quiver with potential $(Q,W)$ with an action of a finite cyclic group $G$, we study the skew group algebra $\Lambda G$ of the Jacobian algebra $\Lambda = \mathcal P(Q, W)$. By a result of Reiten and Riedtmann, the quiver $Q_G$ of a basic algebra $\eta( \Lambda G) \eta$ Morita equivalent to $\Lambda G$ is known. Under some assumptions on the action of $G$, we explicitly construct a potential $W_G$ on $Q_G$ such that $\eta(\Lambda G) \eta\cong \mathcal P(Q_G , W_G)$. The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of $G$. If $\Lambda$ is self-injective, then $\Lambda G$ is as well, and we investigate this case. Motivated by Herschend and Iyama’s characterisation of 2-representation finite algebras, we study how cuts on $(Q,W)$ behave with respect to our construction.

Tensor products of n-complete algebras

Tue, 01 Jan 2019 00:00:00 +0000

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

Tensor products of higher almost split sequences

Sun, 01 Jan 2017 00:00:00 +0000

We investigate how the higher almost split sequences over a tensor product of algebras are related to those over each factor. Herschend and Iyama give a criterion for when the tensor product of an $n$-representation finite algebra and an $m$-representation finite algebra is $(n+m)$-representation finite. In this case we give a complete description of the higher almost split sequences over the tensor product by expressing every higher almost split sequence as the mapping cone of a suitable chain map and using a natural notion of tensor product for chain maps.