Algebras Of Finite Representation Type Research Articles

Tilting theory and cluster combinatorics

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

On the degree of irreducible morphisms

We study the degree of irreducible morphisms in generalized standard convex components of the Auslander–Reiten quiver of an artin algebra with the property that paths with the same origin and end vertices have equal length. We call the components with this last property components with length. In particular, we give two criteria to determine wether the degree of such an irreducible morphism f is finite or infinite. One of them is given in terms of the compositions of f with non-zero maps between modules in the component. The other states that the left degree of an irreducible map f is finite if and only if Ker f belongs to the component. We apply our results to irreducible morphisms over artin algebras of finite representation type and over tame hereditary algebras.

Algebras with finitely many orbits

Any algebra of finite representation type has a finite number of two-sided ideals. But there are stronger finiteness conditions that should be considered here. We consider finite-dimensional K-algebras that have only a finite number of left (respectively, principal left) ideals, up to conjugacy. We then characterize K-algebras A whose Jacobson radical satisfies J( A) 2=0, and with finitely many classes of principal left ideals. Finally, we consider basic algebras with J( A) 2=0. Here we characterize such algebras with finitely many classes of left ideals.

The Hochschild cohomology ring of a selfinjective algebra of finite representation type

This paper describes the Hochschild cohomology ring of a selfinjective algebra Λ \Lambda of finite representation type over an algebraically closed field K K , showing that the quotient HH ∗ ⁡ ( Λ ) / N \operatorname {HH}^*(\Lambda )/\mathcal {N} of the Hochschild cohomology ring by the ideal N {\mathcal N} generated by all homogeneous nilpotent elements is isomorphic to either K K or K [ x ] K[x] , and is thus finitely generated as an algebra. We also consider more generally the property of a finite dimensional algebra being selfinjective, and as a consequence show that if all simple Λ \Lambda -modules are Ω \Omega -periodic, then Λ \Lambda is selfinjective.

POLYNOMIALS RELATED TO HALL NUMBERS

Let Λ be a finite dimensional algebra of finite representation type over a finite field k. For any modules A, B and Pin mod Λ with P projective, we prove that there exists a polynomial ϕ B (P)Λ over Z whose evaluation at |E| for any conservative finite field extension E of Λ is the sum of Hall numbers F B E C E A E where C E runs through isoclasses in mod Λ E and P E is the projective cover of C E . As a consequence of this result and its dual version, Hall polynomials ϕ E CA exist when C or A is semisimple. As applications of the main result, we obtain the existence of Hall polynomials for Nakayama algebras and some selfinjective algebras. *Supported by the Volkswagen Foundation.

AUSLANDER ALGEBRAS ARE 2-GORENSTEIN

Let R be a finitely dimensioanl algebra of finite representation type over an algebraically closed field k. Assume that R is basic. Let M be the sum of the representatives of the indecomposable Λ-m...

POLYNOMIALS RELATED TO HALL NUMBERS

Let Λ be a finite dimensional algebra of finite representation type over a finite field k. For any modules A, B and Pin mod Λ with P projective, we prove that there exists a polynomial ϕ B (P)Λ over Z whose evaluation at |E| for any conservative finite field extension E of Λ is the sum of Hall numbers F B E C E A E where C E runs through isoclasses in mod Λ E and P E is the projective cover of C E . As a consequence of this result and its dual version, Hall polynomials ϕ E CA exist when C or A is semisimple. As applications of the main result, we obtain the existence of Hall polynomials for Nakayama algebras and some selfinjective algebras.

Representation-Directed Diamonds

AbstractA module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

Complexity of Representations of Quantised Function Algebras and Representation Type

Let G be a simply connected, connected, semisimple algebraic group over C, B± be opposite Borel subgroups of G, and b=Lie(B+). Let Oϵ[G] be the quantised function algebra at a root of unity ϵ and let U≥0ϵ be the quantised enveloping algebra of b at a root of unity. We study the finite dimensional factor algebras Oϵ[G](g) and U≥0ϵ(b) for g∈G and b∈B−, which were introduced by De Concini, Kac, and Procesi (1992, in “Geometry and Analysis,” pp. 41–65, Tata Inst. Fund. Res., Bombay) and De Concini and Lyubashenko (1994, Adv. Math.108, 205–262). In particular we describe the complexity of these factor algebras in terms of the Weyl group of G and deduce results on their representation type. Finally we completely describe those factor algebras of finite representation type.

On Cocommutative Hopf Algebras of Finite Representation Type

Let G be a finite algebraic group, defined over an algebraically closed field k of characteristic p>0. Such a group decomposes into a semidirect product G=G0×Gred with a constant group Gred and a normal infinitesimal subgroup G0. If the principal block B0(G) of the group algebra H(G) has finite representation type, then both constituents have the same property, with at least one of them being semisimple. We determine the structure of the infinitesimal constituent G0 up to the classification of V-uniserial groups.

The indecomposable modules of self-injective algebra and their composition factors

In this paper, a sufficient and necessary condition for the indecomposable modules of the self-injective algebra of finite-representation type to be determined by their composition factors is presented by means of the quiver with relations with the help of covering.

Strong Symmetry Defined by Twisting Modules, Applied to Quasi-Hereditary Algebras with Triangular Decomposition and Vanishing Radical Cube

For a finite dimensional algebra with triangular decomposition, a new kind of modules is defined, the twisting modules. Using the structure of these modules, for some algebras with vanishing radical cube the characteristic tilting module is described and its endomorphism ring is computed. This covers both Temperley–Lieb algebras and q-Schur algebras of finite representation type. Our basic tool is a new symmetry condition, stronger than the symmetry provided by the existence of a triangular decomposition in general.

INFINITESIMAL SCHUR ALGEBRAS OF FINITE REPRESENTATION TYPE

INFINITESIMAL SCHUR ALGEBRAS OF FINITE REPRESENTATION TYPE Get access STEPHEN R. DOTY, STEPHEN R. DOTY Search for other works by this author on: Oxford Academic Google Scholar DANIEL K. NAKANO, DANIEL K. NAKANO Search for other works by this author on: Oxford Academic Google Scholar KARL M. PETERS KARL M. PETERS Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 48, Issue 3, September 1997, Pages 323–345, https://doi.org/10.1093/qmath/48.3.323 Published: 01 September 1997 Article history Received: 13 May 1995 Revision received: 11 November 1995 Published: 01 September 1997

QF-3 Algebras of Finite Representation Type

QF-3 Algebras of Finite Representation Type

On Schur algebras and related algebras V: some quasi-hereditary algebras of finite type

In a recent paper, Erdmann determined which of the Schur algebras S( n, r) have finite representation type and described the finite type Schur algebras up to Morita equivalence. The present paper grew out of a desire to see Erdmann's results in the more general context of algebras which are quasi-hereditary in the sense of Cline et al. (1988). Weconsider here the class of quasi-hereditary algebras which have a duality fixing simples. This class includes the “generalized Schur algebras” defined and studied by the first author, and the Schur algebras themselves in particular. In the first part we describe the possible Morita types of the quasi-hereditary algebras of finite representation type over an algebraically closed field with duality fixing simples. This is then applied, in the second part, to give the block theoretic refinement of Erdmann's results.

Semigroup algebras of finite representation type

In what followsK denotes a field. An algebra means an asociative algebra, modules over algebras are right ones.

Short Chains and Short Cycles of Modules

We show that for a large class of artin algebras including the algebras of finite representation type, an indecomposable module $M$ is not the middle of a short chain if and only if there is no short cycle $M \to N \to M$ of nonzero nonisomorphisms between indecomposable modules. We apply this to get sufficient conditions for modules to be determined by their composition factors. We also show that if for an algebra of finite representation type there is a sincere indecomposable $\Lambda$-module that is not the middle of a short chain, then $\Lambda$ is a tilted algebra.

Short chains and short cycles of modules

We show that for a large class of artin algebras including the algebras of finite representation type, an indecomposable module M M is not the middle of a short chain if and only if there is no short cycle M → N → M M \to N \to M of nonzero nonisomorphisms between indecomposable modules. We apply this to get sufficient conditions for modules to be determined by their composition factors. We also show that if for an algebra of finite representation type there is a sincere indecomposable Λ \Lambda -module that is not the middle of a short chain, then Λ \Lambda is a tilted algebra.

Restricted Lie algebras with bounded cohomology and related classes of algebras

We determine the structure of restricted Lie algebras with bounded cohomology over arbitrary fields of prime characteristic. As a byproduct a classification of the serial restricted Lie algebras and the restricted Lie algebras of finite representation type is obtained. In addition, we derive complete information on the finite dimensional indecomposable restricted modules of these algebras over algebraically closed fields.

Some examples of semigroup algebras of finite representation type

The semigroup algebras over a field K of the semigroups Tn of all permutations of a set of n elements are considered. It is proved: if n≤3 and (n!)-1∈ K then the algebra KTn has a finite representation type. Also the finiteness of the representation type of the semigroup algebra KS is established, where S is the sub-semigroup of Tn (n is arbitrary) such that S=Jn∪G where Jn={x∈Tn|rank x=1}, while G is a doubly transitive subgroup of the symmetric group Sn, the order of G being invertible in K.