Category Of Finite-dimensional Modules Research Articles

N-representation infinite algebras

From the viewpoint of higher dimensional Auslander–Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n-regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext1-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras.Applying Minamotoʼs theory on Fano algebras in non-commutative algebraic geometry, we describe the category of n-regular modules in terms of the corresponding preprojective algebra. Then we introduce n-representation tame algebras, and show that the category of n-regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an n-representation tame algebra is at least n+2.

Bernstein–Gelfand–Gelfand Reciprocity and Indecomposable Projective Modules for Classical Algebraic Supergroups

We prove a BGG type reciprocity law for the category of finite dimensional modules over algebraic supergroups satisfying certain conditions. The equivalent of a standard module in this case is a virtual module called Euler characteristic due to its geometric interpretation. In the orthosymplectic case, we also describe indecomposable projective modules in terms of those Euler characteristics.

From triangulated categories to module categories via localisation

We show that the category of finite dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the 2 2 -Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.

From triangulated categories to module categories via localization II: calculus of fractions

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.

Serre functors for Lie algebras and superalgebras

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 𝒪 associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category 𝒪 and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category 𝒪 for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra 𝔮(2).

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid. Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C . The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford–Majid bosonization of Hopf algebras). We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedlerʼs Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).

Generalized trace and modified dimension functions on ribbon categories

In this paper, we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are nonzero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.

On selfadjoint functors satisfying polynomial relations

We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors, are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products, Serre subcategories, quotients and centralizer subalgebras.

1-Dimensional representations and parabolic induction for W-algebras

A W-algebra is an associative algebra constructed from a reductive Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of W-algebras. Under some conditions on a nilpotent element (satisfied by all rigid elements) we obtain a criterium for a finite dimensional module to have dimension 1. It is stated in terms of the Brundan–Goodwin–Kleshchev highest weight theory. This criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras. We make an explicit computation in a special case in type E8. Our second principal result is a version of a parabolic induction for W-algebras. In this case, the parabolic induction is an exact functor between the categories of finite dimensional modules for two different W-algebras. The most important feature of the functor is that it preserves dimensions. In particular, it preserves one-dimensional representations. A closely related result was obtained previously by Premet. We also establish some other properties of the parabolic induction functor.

Infinite dimensional modules for Frobenius kernels

We prove that the projectivity of an arbitrary (possibly infinite dimensional) module for a Frobenius kernel can be detected by restrictions to one-parameter subgroups. Building upon this result, we introduce the support cone of such a module, extending the construction of support variety for a finite dimensional module, and show that such support cones satisfy most of the familiar properties of support varieties. We also verify that our representation-theoretic definition of support cones admits an interpretation in terms of Rickard idempotent modules associated to thick subcategories of the stable category of finite dimensional modules.

Stably thick subcategories of modules over Hopf algebras

We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional co-commutative Hopf algebra B. Our method is based on that of Benson–Carlson–Rickard [BCR1], who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

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.

Stable equivalence preserves representation type

Given two finite dimensional algebras \(\Lambda\) and \(\Gamma\), it is shown that \(\Lambda\) is of wild representation type if and only if \( \Gamma \) is of wild representation type provided that the stable categories of finite dimensional modules over \( \Lambda \) 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 \( \Lambda \) and \( \Gamma \), 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.