Hereditary Algebras Research Articles

Generalized Green Classes

Generalized Green classes are introduced; some basic properties of members in a generalized Green class are studied. Finally, we apply the results to H(Λ), the Ringel–Hall algebra of a finite-dimensional hereditary algebra Λ over a finite field. In particular, it is proved that H(Λ) belongs to a suitable generalized Green class, and that there is direct decomposition of spaces H(Λ)=C(Λ)⊕J, where C(Λ) is the composition algebra of Λ and J is a twisted Hopf ideal of H(Λ), which is exactly the orthogonal complement of C(Λ).

Korteweg-de Vries hierarchy using the method of base equations

A base-equation method is implemented to realize the hereditary algebra of the Korteweg-de Vries (KdV) hierarchy and the N-soliton manifold is reconstructed. The novelty of our approach is that, it can in a rather natural way, predict other nonlinear evolution equations which admit local conservation laws. Significantly enough, base functions themselves are found to provide a basis to regard the KdV-like equations as higher order degenerate bi-Lagrangian systems.

Derived and Stable Equivalence Classification of Twisted Multifold Extensions of Piecewise Hereditary Algebras of Tree Type

Derived and Stable Equivalence Classification of Twisted Multifold Extensions of Piecewise Hereditary Algebras of Tree Type

THE q-ANALOGUE OF BOSONS AND HALL ALGEBRAS

ABSTRACT For the q-analogue of boson associated to a hereditary algebra over a finite field k we generalize the result of M. Kashiwara to show that the category of integrable -modules is semisimple, and the positive part of the quantized generalized Kac-Moody algebra associated to represents the unique isoclass of simple objects. Hence we obtain another proof of the result due to B. Sevenhant-M. Van den Bergh that is isomorphic to the twisted Hall algebra .

On Ringel–Hall Algebras of Tame Hereditary Algebras

In this paper, the double Ringel–Hall algebras of tame hereditary algebras are decomposed as the quantized enveloping algebras of the infinite-dimensional Lie algebras, which are the central extensions of the affine loop algebras and the infinite-dimensional Heisenberg algebras. The numbers of the generators of the Heisenberg algebras are explicitly given at each dimensional level.

ON Δ-GOOD MODULE CATEGORIES ARISING FROM HEREDITARY ALGEBRAS

Let A be a hereditary artin algebra and M a complete exceptional sequence over A. Let F(M) be the subcategory of A-mod consisting of modules with an M-filtration. A quasi-hereditary algebra is called e-quasi-hereditary provided that its Δ-good module category is equivalent to the category F(M) under an exact functor. A characterization of e-quasi-hereditary algebras is given, and a connection between the representation type of F(Δ) and the Tits form associated to it for some e-quasi-hereditary algebras is obtained in this paper.

Quasitilted One-Point Extensions of Wild Hereditary Algebras

Quasitilted One-Point Extensions of Wild Hereditary Algebras

On the construction of recursion operator and algebra of symmetries for field and lattice systems

In the paper, developing the idea of V. Sokolov ( J. Math. Phys. 40 (1999), 6473) we construct recursion operators and hereditary algebra of symmetries for many field and lattice systems.

Triangular Decomposition of Tame Non-Simply Laced Composition Algebras

Let A be a finite dimensional hereditary algebra over a finite field, and let C(A) be the composition algebra of A. Let P (resp. I) be the subalgebra of C(A) generated by the preprojective (resp. preinjective) A-modules, and let T be the subalgebra generated by rd, where rd=∑[M]∈H(A),[M] runs over the isomorphism classes of the regular A-modules with dimension vector d, and H(A) is the Ringel–Hall algebra of A. Then we prove that C(A)=P·T·I, if A is one of the types B̃n, C̃n, BC̃n, BD̃n, CD̃n, F̃41, F̃42, G̃21, G̃22, or Ã11.

Strictly wild algebras with radical square zero

It is proved that a wild algebra with radical square zero is strictly wild if and only if it has a wild hereditary algebra as its factor algebra, which, on the one hand supports one conjecture in [3], on the other hand can be used to construct many non-strictly wild algebras.

Derived-tame Tree Algebras

In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form χA is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of χA. Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.

DERIVED PICARD GROUPS OF FINITE DIMENSIONAL HEREDITARY ALGEBRAS

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of Db(mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db(mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db(mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

The spectrum of a module category

Introduction The functor category Definable subcategories Left approximations duality Ideals in the category of finitely presented modules Endofinite modules Krull-Gabriel dimension The infinite radical Functors between module categories Tame algebras Rings of definable scalars Reflective definable subcategories Sheaves Tame hereditary algebras Coherent rings Appendix A. Locally coherent Grothendieck categories Appendix B. Dimensions Appendix C. Finitely presented functors and ideals Bibliography.

Characteristic tilting modules and Ringel duals

The characteristic tilting modules of quasi-hereditary algebras which are dual extensions of directed monomial algebras are explicitly constructed; and it is shown that the Ringel dual of the dual extension of an arbitrary hereditary algebra has triangular decomposition and bipartite quiver.

Filtration-Closed Auslander-Reiten Components for Wild Hereditary Algebras

Let H=kQ be a finite-dimensional connected wild hereditary path algebra, over some field k. Denote by H-reg the category of finite-dimensional regular H-modules, that is, the category of modules M with τ H − m ( τ H m M ) ≅ M for all integers m, where τH denotes the Auslander–Reiten translation. Call a filtration ( ∗ ) M = M 0 ⊃ M 1 ⊃ … ⊃ M r ⊃ M r + 1 = 0 of a regular H-module M a regular filtration if all subquotients Mi/Mi+1 are regular. Call a regular filtration (*) a regular composition series if it is strictly decreasing and has no proper refinement. A regular component C in the Auslander–Reiten quiver Γ (H) of H-mod is called filtration closed if, for each M∈ add C, the additive closure of C, and each regular filtration (*) of M, all the subquotients Mi/Mi+1 are also in add C. We show that most wild hereditary algebras have filtration-closed Auslander–Reiten components. Moreover, we deduce from this that there are also almost serial components, that is regular components C, such that any indecomposable X∈C has a unique regular composition series. This composition series coincides with the Auslander–Reiten filtration of X, given by the maximal chain of irreducible monos ending at X. 1991 Mathematics Subject Classification: 16G70, 16G20, 16G60, 16E30.

Piecewise Hereditary One-Point Extensions

It is shown that a one-point extension of a wild hereditary algebra with a regular module is piecewise hereditary if and only if the one-point extension with some Auslander–Reiten translate of the module is quasitilted. The Auslander–Reiten structure of such piecewise hereditary one-point extensions is investigated.

Derived-tame tree algebras of type E

Let A be an algebra whose quiver is a tree and denote by wA the quadratic Euler form associated with A. Suppose A contains a convex subcategory which is derived equivalent to a hereditary algebra of type Ep; ~ EpO pa 6; 7; 8U or to a tubular algebra. Then we show that the

Root Vectors Arising from Auslander–Reiten Quivers

Consider the canonical isomorphism between the Ringel–Hall algebra H(Λ) and the positive part U+ of the Drinfield–Jimbo quantum group Uq(g), where the finite dimensional hereditary algebra Λ and the semisimple Lie algebra g enjoy a common Dynkin diagram. We get an algorithm to decompose the root vectors into combinations of monomials of the canonical generators Ei in the quantum group depending only on the structure of the Auslander–Reiten quiver of Λ. The algorithm is deduced by using a similar method as we did in X. Chen and J. Xiao [1999, Compositio Math.117, No. 2, 161–187], particularly, the derivation defined there.

The Tits and Euler forms of a tame algebra

Since the introduction by Gabriel in 1972 of the Tits quadratic form associated to finite dimensional hereditary algebras, quadratic forms have played an important role in the representation theory of algebras and related categories. There have been quadratic forms associated to finite dimensional algebras, partially ordered sets and other objects in order to determine their representation type and the classes of indecomposable representations in the corresponding Grothendieck group (see for example [3,9,14,21]). The class of finite dimensional algebras (associative, with in identity) over an algebraically closed field k is divided into two disjoint classes. One class consists of tame algebras for which the indecomposable modules occur, in each dimension, in a finite number of discrete and one-parameter families. The second class consist of wild algebras whose representation theory is as complicated as the classification of pairs of matrices up to simultaneous conjugation. In this paper we establish properties of the Tits and Euler forms associated to tame simply connected algebras. Let A be a basic connected finite dimensional algebra over k. Then A has a presentation A = kQA/I , where QA = (Q0, Q1) is the ordinary quiver of A with set of vertices Q0 and set of arrows Q1 and I is an admissible ideal of the path algebra kQA (see [9]). We shall assume that QA has no oriented cycles (and hence g`dim A < ∞). Then the Euler form χA : ZQ0 → Z is defined by

On the Coxeter polynomials of wild stars

The spectral radius of a Coxeter transformation which plays an important role in the representation theory of hereditary algebras [see V. Dlab, C.M. Ringel, Eigenvalues of Coxeter transformations and the Gelfand–Kirillov dimension of the preprojective algebras, Proc. AMS 83 (1990) 228–232] is its important invariant. This paper provides both upper and lower bounds for the spectral radii of the Coxeter transformations of wild stars (i.e. trees that have a single branching point and are neither of Dynkin nor of Euclidean type). In addition, the paper determines limit of the spectral radii of particular infinite sequences of wild stars and shows different classes of graphs with the same limit. The basic idea is to reduce the study of spectral radii of trees to the spectral radii of particular valued graphs with indefinite type of associated generalized Cartan matrix.