The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

Derived dynkin extensions

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri].

We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions.

Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments.     

Minimal algebras of infinite representation type with preprojective component

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field k. Denote by e₁,...,e_n a complete set of primitive orthogonal idempotents in A and by A_i= A/Ae_iA. A is called a minimal algebra of infinite representation type provided A is itself of infinite representation type,whereas all A_i, 1≤i≤n,are of finite representation type. The main result gives the classification of the minimal algebras having a preprojective component in their Auslander-Reiten quiver. The classification is obtained by realizing that these algebras are essentially given by preprojective tilting modules over tame hereditary algebras.     

T(2,2,2,2)型tubular代数的合成代数

设A为有限域上的T(2,2,2,2)型tubular代数,它是tame遗传代数A0的单点扩张,也是tame遗传代数A∞的单点余扩张．记P0为预投射A0-模集合,L∞是预内射A∞-模集合,则P0,L∞(?)mod A记T=mod A＼(P0∪L∞)．本文证明A的合成代数C(A)具有形为P0·J·L∞的三角分解．     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

Finiteness of the strong global dimension of radical square zero algebras

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.     

Cartan determinants for gentle algebras

The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very helpful for derived equivalence classifications. In this paper we determine the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.Received: 29 October 2004     

Stable equivalence preserves representation type

Given two finite dimensional algebras and , it is shown that is of wild representation type if and only if is of wild representation type provided that the stable categories of finite dimensional modules over and $\Gamma$ are equivalent. The proof uses generic modules. In fact, a stable equivalence induces a bijection between the isomorphism classes of generic modules over and , and the result follows from certain additional properties of this bijection. In the second part of this paper the Auslander-Reiten translation is extended to an operation on the category of all modules. It is shown that various finiteness conditions are preserved by this operation. Moreover, the Auslander-Reiten translation induces a homeomorphism between the set of non-projective and the set of non-injective points in the Ziegler spectrum. As a consequence one obtains that for an algebra of tame representation type every generic module remains fixed under the Auslander-Reiten translation. Received: July 24, 1996     

A characterization of congruence permutable locally finite varieties

A variety of universal algebras is said to be congruence permutable if for every algebra A of and every pair of congruences α, β from A we have αβ = βα. We show that if is locally finite (i.e., every finitely generated member of is finite) then congruence permutability is equivalent to a local property of the finite members of , expressible in the language of tame congruence theory. This answers a question of R. McKenzie and D. Hobby.     

TAME TWO-POINT ALGEBRAS WITHOUT LOOPS

We determine the representation type of the algebras whose quiver has precisely two vertices and admits no loops by listing all minimal wild algebras of this form. It turns out that such an algebra A is tame if and only if A/rad³ A is tame, and in this case A degenerates to a special biserial algebra. Moreover, A is wild if and only if it is controlled wild.     

Generalised Chain Conditions,Prime Ideals,and Classes of Locally Finite Lie Algebras

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radicalof any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is qussi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.     

On the codimension growth of almost nilpotent Lie algebras

We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra L is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of L exists and is a positive integer.     

The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.     

Based algebras and standard bases for quasi-hereditary algebras

Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly full-based. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.     

Note on Co-split Lie Algebras

It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should be a weak co-split Lie algebra.A class of non-semi-simple co-split Lie algebras is given.     

A homological approach to representations of algebras II: tame hereditary algebras

We determine the pure global dimension of finite dimensional hereditary or radical-squared zero algebras over algebraically closed fields. The results are applied to algebras of dimension four and to the incidence algebras of critical ordered sets studied by Loupias. We further prove that the path algebra of an oriented cycle shares with Dedekind domains the Kulikov property (submodules of pure-projective modules are pure-projective).     

Derivations of Tensor Product Algebras

It is known that under certain finite dimensionality condition the derivation algebra of tensor product of two algebras can be obtained in terms of the derivation algebras and the centroids of the involved algebras. We extend this theorem to infinite dimensional case and as an application, we determine the derivation algebra of the fixed point algebra of the tensor product of two algebras, with respect to the tensor product of two finite order automorphisms. These provide the framework for calculating the derivations of some infinite dimensional Lie algebras.     

Representation Theory of some Infinite-dimensional Algebras Arising in Continuously Controlled Algebra and Topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Received October 2003