Representation-finite Algebras Research Articles

On the interaction of the Coxeter transformation and the rowmotion bijection

Let P be a finite poset and L the associated distributive lattice of order ideals of P . Let \rho denote the rowmotion bijection of the order ideals of P viewed as a permutation matrix and C the Coxeter matrix for the incidence algebra kL of L . Then, we show the identity (\rho^{-1}C)^{2}=\mathrm{id} , as was originally conjectured by Sam Hopkins. Recently, it was noted that the rowmotion bijection is a special case of the much more general grade bijection R that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from n -representation finite algebras, we show that (R^{-1}C)^{2}=\mathrm{id} if n is even and (R^{-1}C+\mathrm{id})^{2}=0 when n is odd.

On the nilpotency index of the radical of a module category

Let A be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of QA are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of A. Precisely, we study the above mentioned problem whenever A is a monomial or a toupie algebra where their Auslander-Reiten quivers are component with length. We also study the above problem for string algebras. We give concrete formulas to compute such index for many classes of algebras such as toupie algebras, tree algebras and some classes of pullback algebras over the hereditary algebras of type An, Dn and E6. Furthermore, we determine the index taking into account the ordinary quiver of the given algebras.

Rowmotion in slow motion

Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and mod A has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.

Right n-Nakayama Algebras and their Representations

In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra Λ is representation-finite if and only if Λ is right n-Nakayama for some positive integer n. We classify hereditary right n-Nakayama algebras. We also define right n-coNakayama algebras and show that an artin algebra Λ is right n-coNakayama if and only if Λ is left n-Nakayama. We then study right 2-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right 2-Nakayama algebra. We end by classifying finite dimensional right 2-Nakayama algebras in terms of their quivers with relations.

On a new formula for the Gorenstein dimension

Let A be a finite dimensional algebra over a field K with enveloping algebra Ae=Aop⊗KA. Following [13], we call algebras A that have the property that the subcategory of Gorenstein projective modules in mod−A coincides with the subcategory {X∈mod−A|ExtAi(X,A)=0 for all i≥1} left weakly Gorenstein. The class of left weakly Gorenstein algebras is a large class that includes for example all Gorenstein algebras and all representation-finite algebras. We prove that the Gorenstein dimension of A coincides with the Gorenstein projective dimension of the regular module as an Ae-module for left weakly Gorenstein algebras A. We give three application of this result. The first generalises a formula by Happel for the global dimension of algebras. The second application generalises a criterion of Shen for an algebra to be selfinjective. As a final application we prove a stronger version of the first Tachikawa conjecture for left weakly Gorenstein algebras.

Higher Zigzag Algebras

Given a Koszul algebra of finite global dimension we define its higher zigzag algebra as a twisted trivial extension of the Koszul dual. If our original algebra is the path algebra of a tree-type quiver, this construction recovers the zigzag algebras of Huerfano-Khovanov. We study examples of higher zigzag algebras coming from Iyama's type A higher representation finite algebras, give their presentations by quivers and relations, and describe relations between spherical twists acting on their derived categories. We connect this to the McKay correspondence in higher dimensions: if G is a finite abelian subgroup of SL_{d+1} then these relations occur between spherical twists for G -equivariant sheaves on affine (d+1) -space.

STRONGLY QUASI-HEREDITARY ALGEBRAS AND REJECTIVE SUBCATEGORIES

Ringel’s right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline–Parshall–Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.

Derived equivalences and stable equivalences of Morita type, II

We consider the question of lifting stable equivalences of Morita type to derived equivalences. One motivation comes from an approach to Broué’s abelian defect group conjecture. Another motivation is a conjecture by Auslander and Reiten on stable equivalences preserving the number of non-projective simple modules. A machinery is presented to construct lifts for a large class of algebras, including Frobenius-finite algebras introduced in this paper. In particular, every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. Consequently, the Auslander–Reiten conjecture is true for stable equivalences of Morita type between Frobenius-finite algebras. Examples of Frobenius-finite algebras are abundant, including representation-finite algebras, Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we show that, for a Nakayama-stable idempotent element e in an algebra A over an algebraically closed field, each tilting complex over eAe can be extended to a tilting complex over A that induces an almost ν-stable derived equivalence studied in the first paper of this series. Moreover, the machinery is applicable to verify Broué’s abelian defect group conjecture for several cases mentioned in the literature.

The radical of a module category of string algebra

ABSTRACTWe study how to read from the ordinary quiver of a representation-finite string algebra, the minimal lower bound m ≥ 1 such that the mth power of the radical of its module category vanishes. We also show how to read the degree of any irreducible morphism in such a representation-finite string algebras considering strings.

Torsion classes, wide subcategories and localisations

For a finite dimensional algebra $A$, we establish correspondences between torsion classes and wide subcategories in $mod(A)$. In case $A$ is representation finite, we obtain an explicit bijection between these two classes of subcategories. Moreover, we translate our results to the language of ring epimorphisms and universal localisations. It turns out that universal localisations over representation finite algebras are classified by torsion classes and support $\tau$-tilting modules.

Characterizing 𝜏-tilting finite algebras with radical square zero

In this paper, we give a characterization of τ \tau -tilting finite algebras with radical square zero in terms of the separated quivers, which is an analog of a famous characterization of representation-finite algebras with radical square zero due to Gabriel.

On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective–injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.

Torsion Classes and t-Structures in Higher Homological Algebra

Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalised by Jasso in the theory of $n$-abelian categories. There is also a derived version of $n$-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of $( n+2 )$-angulated categories (the reason for the shift from $n$ to $n+2$ is that angulated categories have triangulated categories as the base case). We introduce torsion classes and t-structures into the theory of $n$-abelian and $( n+2 )$-angulated categories, and prove several results to motivate the definitions. Most of the results concern the $n$-abelian and $( n+2 )$-angulated categories ${\mathcal M}( \Lambda )$ and ${\mathcal C}( \Lambda )$ associated to an $n$-representation finite algebra $\Lambda$, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in ${\mathcal M}( \Lambda )$ and intermediate t-structures in ${\mathcal C}( \Lambda )$ which is a category one can reasonably view as the $n$-derived category of ${\mathcal M}( \Lambda )$. We hint at the link to $n$-homological tilting theory.

On Lie algebras associated with representation-finite algebras

Let Λ be a representation-finite C-algebra which has Hall polynomials, with the universal cover Λ˜ which is a locally bounded directed C-algebra. In this paper we prove that the Z-Lie algebra L(Λ) associated with Λ which is defined by Riedtmann in [17] and the Z-Lie algebra K(Λ) associated with Λ which is defined by Ringel in [19] are isomorphic. As an application we show that if Λ is a representation-finite (generalized) cluster-tilted algebra or representation-finite trivial extension algebra, then K(Λ)≅L(Λ).

Hochschild cohomology rings of Temperley-Lieb algebras

The authors first construct an explicit minimal projective bimodule resolution (ℙ, δ) of the Temperley-Lieb algebra A, and then apply it to calculate the Hochschild cohomology groups and the cup product of the Hochschild cohomology ring of A based on a comultiplicative map Δ: ℙ → ℙ ⊗ A ℙ. As a consequence, the authors determine the multiplicative structure of Hochschild cohomology rings of both Temperley-Lieb algebras and representation-finite q-Schur algebras under the cup product by giving an explicit presentation by generators and relations.

Higher preprojective algebras and stably Calabi-Yau properties

In this paper, we give sufficient properties for a finite dimensional graded algebra to be a higher preprojective algebra. These properties are of homological nature, they use Gorensteiness and bimodule isomorphisms in the stable category of Cohen-Macaulay modules. We prove that these properties are also necessary for $3$-preprojective algebras using \cite{Kel11} and for preprojective algebras of higher representation finite algebras using \cite{Dugas}.

Module Varieties and Representation Type of Finite-Dimensional Algebras

In this paper we seek geometric and invariant-theoretic characterizations of (Schur-)representation finite algebras. To this end, we introduce two classes of finite-dimensional algebras: those with the dense-orbit property and those with the multiplicity-free property. We show first that when a connected algebra A admits a preprojective component, each of these properties is equivalent to A being representation-finite. Next, we give an example of an algebra which is not representation-finite but still has the dense-orbit property. We also show that the string algebras with the dense orbit-property are precisely the representation-finite ones. Finally, we show that a tame algebra has the multiplicity-free property if and only if it is Schur-representation-finite.

A Gabriel-type theorem for cluster tilting

We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we show that cluster-indecomposable modules are uniquely determined by their dimension vectors. This is a generalization of Gabriel's theorem by cluster tilting theory. We call the above roots cluster-roots and investigate their properties. Furthermore, we provide the description of quivers with relations of $n$-APR tilts. Using this, we provide a generalization of BGP reflection functors.

Hall Polynomials and Composition Algebra of Representation Finite Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k such that the indecomposable \(\mathcal{A}\)-modules are determined by their dimension vectors and for each \(M, L \in ind(\mathcal{A})\) and \(N\in mod(\mathcal{A})\), either \(F^{M}_{N L}=0\) or \(F^{M}_{L N}=0\). We show that \(\mathcal{A}\) has Hall polynomials and the rational extension of its Ringel–Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if \(\mathcal{A}\) is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in \(mod(\mathcal{A})\), or representation finite cluster tilted algebra, then \(\mathcal{A}\) has Hall polynomials and \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\).

Indecomposables live in all smaller lengths

We show that there are no gaps in the lengths of the indecomposable objects in an abelian k k -linear category over a field k k provided all simples are absolutely simple. To derive this natural result we prove that any distributive minimal representation-infinite k k -category is isomorphic to the linearization of the associated ray category which is shown to have an interval-finite universal cover with a free fundamental group so that the well-known theory of representation-finite algebras applies.