Hereditary Algebras Research Articles

Formula omitted]-structures via recollements for piecewise hereditary algebras

Following the work of Beilinson, Bernstein and Deligne, we study restriction and induction of t -structures on triangulated categories with respect to recollements. For derived categories of piecewise hereditary algebras we give a necessary and sufficient condition for a bounded t -structure to be induced from recollements by derived categories of algebras. As a corollary we prove that for hereditary algebras of finite representation type all bounded t -structures can be obtained in this way.

TILTING MUTATION FOR m-REPLICATED ALGEBRAS

Let A be a finite-dimensional hereditary algebra over an algebraically closed field k, A(m)be the m-replicated algebra of A and [Formula: see text] be the m-cluster category of A. In this paper, we introduce the notion of mutation team in mod A(m), and prove that each faithful almost complete tilting module over A(m)has a mutation team by showing that the sequence of the complements satisfies the properties of the mutation team. We also prove that for each partial mutation team in the m-left part of mod A(m), there exists a faithful almost complete tilting module having the partial mutation team as the set of indecomposable complements. As an application, we prove that m-cluster mutation in [Formula: see text] can be realized as tilting mutation in mod A(m), and we also give the relationship between connecting sequences in mod A(m)and higher AR-angles in the m-cluster category [Formula: see text].

$n$-representation-finite algebras and $n$-APR tilting

We introduce the notion of $n$-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of $n$-APR tilting and show that it preserves $n$-representation-finiteness. We give some combinatorial description of this procedure and use this to completely describe a class of $n$-representation-finite algebras called “type A”.

From m-clusters to m-noncrossing partitions via exceptional sequences

Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural bijections between these two sets are known. For any positive integer m, both m-clusters and m-noncrossing partitions have been defined, and the cardinality of both these sets is the Fuss–Catalan number C m (W). We give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category D b(H) for any finite-dimensional hereditary algebra H.

Resolutions of mesh algebras: periodicity and Calabi–Yau dimensions

A triangulated category $${(\mathcal{T},\Sigma)}$$ is said to be Calabi–Yau of dimension d if Σ d is a Serre functor. We determine which stable module categories of self-injective algebras Λ of finite type are Calabi–Yau and compute their Calabi–Yau dimensions, correcting errors in previous work. We first show that the Calabi–Yau property of mod-Λ can be detected in the minimal projective resolution of the stable Auslander algebra Γ of Λ, over its enveloping algebra. We then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the (generalized) preprojective algebras of Dynkin graphs. For representation-finite self-injective algebras of torsion order t = 1, we obtain a complete description of their stable Calabi–Yau properties, but only partial results for those algebras of torsion order t = 2. We also obtain some new information about the periods of the representation-finite self-injective algebras of torsion order t > 1. Finally, we describe how these questions can also be approached by realizing the stable categories of representation-finite self-injective algebras as orbit categories of the bounded derived categories of hereditary algebras, and illustrate this technique with several explicit computations that our previous methods left unsettled.

Geometric characterizations of the representation type of hereditary algebras and of canonical algebras

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight θ of Q, the moduli space M ( Q , d ) θ s s of θ-semi-stable d-dimensional representations of Q is just a projective space. In order to prove this, we show that the tame quivers are precisely those whose weight spaces of semi-invariants satisfy a certain log-concavity property. Furthermore, we characterize the tame quivers as being those quivers Q with the property that for each Schur root d of Q, the field of rational invariants k ( rep ( Q , d ) ) GL ( d ) is isomorphic to k or k ( t ) . Next, we extend this latter description to canonical algebras. More precisely, we show that a canonical algebra Λ is tame if and only if for each generic root d of Λ and each indecomposable irreducible component C of rep ( Λ , d ) , the field of rational invariants k ( C ) GL ( d ) is isomorphic to k or k ( t ) . Along the way, we establish a general reduction technique for studying fields of rational invariants on Schur irreducible components of representation varieties.

Three kinds of mutation

For a finite-dimensional hereditary algebra H, we consider: exceptional sequences in mod H, silting objects in the bounded derived category D , and m-cluster tilting objects in the m-cluster category, C m = D / τ − 1 [ m ] . There are mutation operations on both the set of m-cluster tilting objects and the set of exceptional sequences. It is also possible to define a mutation operation for silting objects. We compare these three different notions of mutation.

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n + 1 -power of the radical. In the case of D b ( H ) , the bounded derived category of an Ext-finite hereditary k -category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.

Cluster Tilting and Complexity

We study the notion of positive and negative complexity of pairs of objects in cluster categories. The first main result shows that the maximal complexity occurring is either one, two or infinite, depending on the representation type of the underlying hereditary algebra. In the the second result, we study the bounded derived category of a cluster tilted algebra, and show that the maximal complexity occurring is either zero or one whenever the algebra is of finite or tame type.

Combinatorial classification of piecewise hereditary algebras

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.

Topological invariants of piecewise hereditary algebras

We investigate the Galois coverings of piecewise algebras and more particularly their behaviour under derived equivalences. Under a technical assumption which is satisfied if the algebra is derived equivalent to a hereditary algebra, we prove that there exists a universal Galois covering whose group of automorphisms is free and depends only on the derived category of the algebra. As a corollary, we prove that the algebra is simply connected if and only if its first Hochschild cohomology vanishes.

Cluster tilting for higher Auslander algebras

The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The n-Auslander–Reiten translation functor τn plays an important role in the study of n-cluster tilting subcategories. We study the category Mn of preinjective-like modules obtained by applying τn to injective modules repeatedly. We call a finite-dimensional algebra Λ n-complete if Mn=addM for an n-cluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n+1)-complete. This gives an inductive construction of n-complete algebras. For example, any representation-finite hereditary algebra Λ(1) is 1-complete. Hence the Auslander algebra Λ(2) of Λ(1) is 2-complete. Moreover, for any n⩾1, we have an n-complete algebra Λ(n) which has an n-cluster tilting object M(n) such that Λ(n+1)=EndΛ(n)(M(n)). We give the presentation of Λ(n) by a quiver with relations. We apply our results to construct n-cluster tilting subcategories of derived categories of n-complete algebras.

Ringel–Hall algebras and two-parameter quantized enveloping algebras

Let g be a finite-dimensional complex simple Lie algebra and A be the finite-dimensional hereditary algebra associated to g. Let U + r,s (g) (respectively U ≥0 r,s (g)) denote the two-parameter quantized enveloping algebra of the positive maximal nilpotent (respectively Borel) Lie subalgebra of g. We study the two-parameter quantized enveloping algebras U + r,s (g) and U ≥0 r,s (g) using the approach of Ringel-Hall algebras. First of all, we show that U + r,s (g) is isomorphic to a certain two-parameter twisted Ringel-Hall algebra H r,s (A), which generalizes a result of Reineke. Based on detailed computations in H r,s (Λ), we show that H r,s (Λ) can be presented as an iterated skew polynomial ring. As an result, we obtain a PBW-basis for H r,s (A), which can be further used to construct a PBW-basis for the two-parameter quantized enveloping algebra U r,s (g). We also show that all prime ideals of U + r,s (g) are completely prime under some mild conditions on the parameters r, s. Second, we study the two-parameter extended Ringel-Hall algebra H r,s (Λ). In particular, we define a Hopf algebra structure on H r,s (Λ); and we prove that U ≥0 r,s (g) is isomorphic as a Hopf algebra to the two-parameter extended Ringel-Hall algebra H r,s (Λ).

Cluster-Tilted Algebras of Type D n

Let H be a hereditary algebra of Dynkin type D n over a field k and 𝒞 H be the cluster category of H. Assume that n ≥ 5 and that T and T′ are tilting objects in 𝒞 H . We prove that the cluster-tilted algebra Γ = End𝒞 H (T)op is isomorphic to Γ′ = End𝒞 H (T′)op if and only if T = τ i T′ or T = στ j T′ for some integers i and j, where τ is the Auslander–Reiten translation and σ is the automorphism of 𝒞 H defined in Section 4.

The Gabriel-Roiter submodules of simple homogeneous modules

Let Λ \Lambda be a connected tame hereditary algebra over an algebraically closed field. We show that if Λ = k Q \Lambda =kQ is of type A ~ n \widetilde {\mathbb {A}}_n , D ~ n \widetilde {\mathbb {D}}_n , E ~ 6 \widetilde {\mathbb {E}}_6 or E ~ 7 \widetilde {\mathbb {E}}_7 , then every Gabriel-Roiter submodule of a quasi-simple module of rank 1 1 (i.e. a simple homogeneous module) has defect − 1 -1 . In particular, any Gabriel-Roiter submodule of a simple homogeneous module yields a Kronecker pair, and thus induces a full exact embedding of the category mod ⁡ k A ~ 1 \operatorname {mod} k\widetilde {\mathbb {A}}_1 into mod ⁡ Λ \operatorname {mod}\Lambda , where A ~ 1 \widetilde {\mathbb {A}}_1 is the Kronecker quiver. Consequently, we obtain that all quasi-simple modules are Gabriel-Roiter factor modules.

Generating relations for Grothendieck groups of trivial extension algebras

In this paper we describe explicitly the generating relations for the Grothendieck groups of trivial extension algebras of tame hereditary algebras.

Representation dimension of m-replicated algebras

Let A be a finite-dimensional hereditary algebra over an algebraically closed field and A(m) be the m-replicated algebra of A. We prove that the representation dimension of A(m) is at most 3, and that the dominant dimension of A(m) is at least m.

AR quivers, exceptional sequences and algorithms in derived Hall algebras

Consider the canonical isomorphism between the positive part U + of the quantum group Uq(g) and the Hall algebra H(Λ), where the semisimple Lie algebra g and the finite-dimensional hereditary algebra Λ share a Dynkin diagram. Chen and Xiao have given two algorithms to decompose the root vectors into linear combinations of monomials of Chevalley generators of U + , respectively induced by the braid group action on the exceptional sequences of Λ-modules and the structure of the Auslander-Reiten quiver of Λ. In this paper, we obtain the corresponding algorithms for the derived Hall algebra DH(Λ), which was introduced by Toen. We show that both algorithms are applicable to the lattice algebra and Heisenberg double in the sense of Kapranov. All the new recursive formulae have the same flavor with the quantum Serre relations.

Large tilting modules and representation type

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in Angeleri Hugel, L., Herbera, D., Trlifaj, J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010) for tame hereditary algebras.

Exceptional sequences and clusters

We show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse Coxeter element in the Weyl group. We use this result to give a new combinatorial characterization of clusters tilting sets in the cluster category in the case where the hereditary algebra is of finite type.