arXiv subjects
Publications and source records attributed to Markus Kleinau.
We give a new characterisation of Auslander-Gorenstein finite dimensional algebras by showing that they are exactly the finite dimensional algebras with a well-defined Auslander-Reiten bijection. This proves a conjecture of Marczinzik. We use this to give a new proof that a finite lattice with an Auslander-Gorenstein incidence algebra has to be distributive.
We give a linear algebraic classification of Auslander regular acyclic monomial algebras via the Bruhat factorisation of the Coxeter matrix. Namely, we show under mild assumptions that a monomial acyclic quiver algebra is Auslander regular if and only if its Coxeter matrix $C$ has a Bruhat factorisation $U_1 P U_2$ with $U_1$ the identity matrix. In particular, this holds without restrictions for linear Nakayama algebras and we use the Bruhat decomposition to answer a question raised by Ringel by showing that his homological permutation coincides with the permutation coming from the Bruhat factorisation of the Coxeter matrix. We also use our methods to show that general Auslander regular acyclic quiver algebras are echelon-independent, proving a conjecture of Defant-Jiang-Marczinzik-Segovia-Speyer-Thomas-Williams, and we answer another question by Ringel on the delooping level of simple modules over Nakayama algebras.
Reading constructed a Cambrian lattice $C_\Gamma$ for each oriented finite type Coxeter diagram $\Gamma$. We show that the derived category of representations of $C_\Gamma$ is fractionally Calabi-Yau for any $\Gamma$, confirming a conjecture of Chapoton. This extends a result of Rognerud for Cambrian lattices of type $A$ with linear orientation, better known as Tamari lattices. If $\Gamma$ is crystallographic, then $C_\Gamma$ is given by the lattice of torsion classes of any hereditary algebra $\Lambda$ of type $\Gamma$. In this case we introduce and study a class of intervals in $C_\Gamma$ whose combinatorics matches the combinatorics of $2$-cluster tilting objects in the 2-cluster category of $\Lambda$. This allows us to compute the Calabi-Yau dimension of $C_\Gamma$.
In a recent article, Iyama and Marczinzik showed that a lattice is distributive if and only if the incidence algebra is Auslander regular, giving a new connection between homological algebra and lattice theory. In this article we study when a distributive lattice has a pure minimal injective coresolution, a notion first introduced and studied in a work of Ajitabh, Smith and Zhang. We will see that this problem naturally leads to studying when certain antichain modules are perfect modules. We give a classification of perfect antichain modules under the assumption that their canonical antichain resolution is minimal and use this to give a completion classification in lattice theoretic terms of incidence algebras of distributive lattices with pure minimal injective coresolution. We use our results to answer a question raised by Ajitabh, Smith and Zhang by showing that there exist Auslander-Gorenstein polynomial identity rings without a pure injective coresolution.
Let $V$ and $W$ be quiver representations over $\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of $\mathrm{Hom}_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.
Aizenbud and Lapid recently introduced a binary operation on the crystal graph $B(-\infty)$ associated to a symmetric Cartan matrix. We extend their construction to symmetrizable Cartan matrices.