Wild Algebra Research Articles

THE HOMOGENISED ENVELOPING ALGEBRA OF THE LIE ALGEBRA sℓ(2,ℂ)

AbstractIn this paper, we study the homogenised algebra B of the enveloping algebra U of the Lie algebra sℓ(2,ℂ). We look first to connections between the category of graded left B-modules and the category of U-modules, then we prove B is Koszul and Artin–Schelter regular of global dimension four, hence its Yoneda algebra B! is self-injective of radical five zeros, and the structure of B! is given. We describe next the category of homogenised Verma modules, which correspond to the lifting to B of the usual Verma modules over U, and prove that such modules are Koszul of projective dimension two. It was proved in Martínez-Villa and Zacharia (Approximations with modules having linear resolutions, J. Algebra266(2) (2003), 671–697)] that all graded stable components of a self-injective Koszul algebra are of type ZA∞. Here, we characterise the graded B!-modules corresponding to the Koszul duality to homogenised Verma modules, and prove that these are located at the mouth of a regular component. In this way we obtain a family of components over a wild algebra indexed by ℂ.

Wild algebras have one-point extensions of representation dimension at least four

We show that any wild algebra has a one-point extension of representation dimension at least four, and more generally that it has an n -point extension of representation dimension at least n + 3 . We give two explicit constructions, and obtain new examples of small algebras of representation dimension four.

Semi-Continuity for Derived Categories

We prove that the number of parameters defining a complex of projective modules over an algebra is upper semi-continuous in families of algebras. The proof follows the pattern of the paper by Drozd and Greuel and rests upon universal families with projective bases. Supposing that every algebra is either derived tame or derived wild, we get that a degeneration of a derived wild algebra is also derived wild. We also discuss an apparent counter-example to the last assertion by Th. Brüstle.

Is tame open?

Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the Basic-Wild-Rank Conjecture. A covering criterion on the rank of a basic wild algebra is given, which can be effectively applied to verify the Basic-Wild-Rank Conjecture for concrete algebras. It makes all conjectures much reliable.

NON-STRICTLY WILD ALGEBRAS

Finite-dimensional algebras over an algebraically closed field are divided into two disjoint classes, called tame and wild respectively, by Drozd's tame and wild dichotomy (see [5] and [2]). A tame algebra, roughly speaking, has its n-dimensional indecomposable modules parametrized by finitely many one-parameter families, for all natural numbers n, but a wild algebra has more indecomposable modules and it is considered hopeless to classify them. In [2], Crawley-Boevey showed that all but finitely many n-dimensional indecomposable modules over a tame algebra are τ-invariant, for all natural numbers n, and conjectured that the converse would be true, where τ := DTr is the Auslander–Reiten translation (see [1]) and we call an indecomposable module X τ-invariant if X ≅ τX.

Representation theory of the system quiver

Let A be a three-point algebra with Gabriel quiver the system quiver Q. Then, up to isomorphism and duality, A is tame if and only if A is or degenerates to a factor of a tame algebra in the tame Table T if and only if A/rad 5 A is tame, and A is wild if and only if A has a wild algebra in the wild Table W as a factor if and only if A is controlled wild.

Controlled Wild Algebras

The controlled wild algebra is introduced, a covering criterion for a finite-dimensional algebra to be controlled wild is given, and this criterion is applied to the algebras with radical square zero, algebras with zero relations, local algebras and finite p-group algebras. 2000 Mathematical Subject Classification: 16G20, 16G60.

THE FINITE LENGTH MODULES FOR THIN -GRADED RINGS

A -graded ring R = is called thin, provided R 0 = K is a skew field and each K–K-bimodule Ri has left and right dimension at most 1. Such rings occur, for example, as fiber products of skew polynomial rings, as numerical skew monoid rings or as factor rings or localizations of generalized Weyl algebras. We show that in case R is (graded) affine with no homogeneous units in nonzero degree, the representation type (finite, “twisted tame infinite,” “twisted wild”) can be read off from the combinatorial structure, that is, from the vanishing of products of homogeneous elements of certain degrees. Those rings R which are of twisted wild representation type admit by our definition a representation embedding from the category of finite dimensional modules over some twisted wild algebra T(K, σ K ⊕τ K) into the category mod R of finite length R-modules. The rings which we call here “twisted tame” come close to twisted versions of string algebras, for which we obtain the classification of the indecomposable finite length modules as a version of the Butler–Ringel theorem. Finally, for rings which are either not affine or which have homogeneous units in nonzero degree, the representation type does not depend on the combinatorial structure alone, as examples show.

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.

On perpendicular categories of stones over quiver algebras

Let H = kzt be the path algebra of a quiver A, where k denotes an arbitrary field. J will always be assumed to be finite without oriented cycles. We denote by mod H the category of finite dimensional left modules over H. An indecomposable H-module X is called a stone provided Ext}~(X, X) = 0. Note that for a stone X the endomorphism ring End X = k (see for example part 1). For a stone X we denote by X l the full subcategory of mod H of the modules Y with Homn(X, Y) = 0 = Ext,(X, Y). Dually, • consists of those H-modules Y satisfying Homn(Y, X) = 0 = Ext,(Y, X). The categories X • and Xare called the right respectively left perpendicular category of X. They are equivalent to mod Ar respectively mod At, where A r and At are again finite dimensional, hereditary k-algebras, and both have one simple module less than H (compare [GL] and [S1]). We denote by Ar respectively zt t the quivers with kAr = Ar and kJz = A,. Since their introduction to Geigle, Lenzing and Schofield, perpendicular categories have become a powerful tool in the representation theory of finite dimensional algebras. They inherit essential properties from mod H, furnish a reduction procedure and open the possibility for proofs by induction. In general, it is difficult problem to determine the (right or left) perpendicular category of a given stone X. We briefly recall what is known. It is quite easy to see (Proposition 2.1) that the quiver Ar differs from a full subquiver of zl by an admissible change of orientation if and only if X is preprojective or preinjective over H. Moreover, if H is tame, then there are only finitely many regular stones X over H, and the structure of its perpendicular categories is contained in [Rill. Hence we may restrict to the context where H is wild and X is a regular stone. Baer and Straul3 proved [B2], [St] that in this situation the perpendicular categories are wild again, but combining results of Xi IX] and Kerner [K3] it follows that certain wild module categories, namely of those algebras classified in [X], do not arise as perpendicular categories of regular stones over wild algebras. On the other hand, investigations of growth numbers in [K3] prove, that for a fixed hereditary, connected, wild algebra with at least three simple modules, there are infinitely many pairwise non isomorphic algebras Ai such that mod Ai is equivalent to X • for a regular stone X over H.