Classifying bicrossed products of two Taft algebras

We classify all Hopf algebras which factor through two Taft algebras ${\mathbb{T}}_{n^2}(\stackrel{￣}{q})$ and respectively $T_{m^2}(q)$. To start with, all possible matched pairs between the two Taft algebras are described: if $\stackrel{￣}{q}\ne q^{n?1}$ then the matched pairs are in bijection with the group of d-th roots of unity in k, where $d=(m,n)$ while if $\stackrel{￣}{q}=q^{n?1}$ then besides the matched pairs above we obtain an additional family of matched pairs indexed by $k^{?}$. The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups.     

Indecomposable modules over one-dimensional Noetherian rings

In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings R. We assume there exists a positive integer N_R such that, for every indecomposable R-module M and for every minimal prime ideal P of R, the dimension of M_P, as a vector space over the field R_P, is less than or equal to N_R. If a nonzero indecomposable R-module M is such that all the localizations M_P as vector spaces over the fields R_P have the same dimension r, for every minimal prime P of R, then r=1,2,3,4 or 6. Let n be an integer ≥8. We show that if M is an R-module such that the vector space dimensions of the M_P are between n and 2n−8, then M decomposes non-trivially. For each n≥8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from n to 2n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case.     

Indecomposable modules for domestic canonical algebras

We show that-up to precisely one-each exceptional module over a domestic canonical algebra of quiver type over a field k can be represented by matrices whose entries are just 0 and 1. In the case we calculate the matrices of these representations explicitly.     

Green rings of Drinfeld doubles of Taft algebras

Abstract

In this article, we investigate the representation rings (or Green rings) of the Drinfeld doubles of the Taft algebras. It is shown that these Green rings are commutative rings generated by infinitely many elements subject to certain relations. The generators together with the subjecting relations are given. The stable Green rings of of the Drinfeld doubles of the Taft algebras are also described.     

Tame concealed algebras and cluster quivers of minimal infinite type

The well-known list of Happel-Vossieck of tame concealed algebras in terms of quivers with relations, and the list of A. Seven of minimal infinite cluster quivers are compared. There is a 1-1 correspondence between the items in these lists, and we explain how an item in one list naturally corresponds to an item in the other list. A central tool for understanding this correspondence is the theory of cluster-tilted algebras.     

Degenerations for modules over blocks of group algebras

Let B be a block of the group algebra KG of a finite Group G over an algebraically closed field K. We prove that every degeneration of finite dimensional B-modules is given by short exact sequences if and only if B is of finite representation type. Received: 7 July 1997     

Hopf subalgebras and tensor powers of generalized permutation modules

By means of a certain module V$V$ and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R$R$ of a finite-dimensional Hopf algebra H$H$ is finite. The module V$V$ is the counit representation induced from R$R$ to H$H$, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V$V$ is either semisimple with R^∗$R^{\ast }$ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R$R$, then the depth of R$R$ in H$H$ is finite. One assigns a nonnegative integer depth to V$V$, or any other H$H$-module, by comparing the truncated tensor algebras of V$V$ in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character.     

Copointed Hopf algebras over S4

We study the realizations of certain braided vector spaces of rack type as Yetter–Drinfeld modules over a cosemisimple Hopf algebra H. We apply the strategy developed in [1] to compute their liftings and use these results to obtain the classification of finite-dimensional copointed Hopf algebras over ${\mathbb{S}}_4$.     

Direct products of simple modules over Dedekind domains

Let R be a (commutative) Dedekind domain and let the R-module M be a direct product of simple R-modules. Then any homomorphism from a closed submodule K of M to M can be lifted to M. Received: 9 December 2002     

Tensor products of n-complete algebras

If A and B are n- and m-representation finite k-algebras, then their tensor product $\mathrm{\Lambda }=A{?}_{k}B$ is not in general $(n+m)$-representation finite. However, we prove that if A and B are acyclic and satisfy the weaker assumption of n- and m-completeness, then Λ is $(n+m)$-complete. This mirrors the fact that taking higher Auslander algebra does not preserve d-representation finiteness in general, but it does preserve d-completeness. As a corollary, we get the necessary condition for Λ to be $(n+m)$-representation finite which was found by Herschend and Iyama by using a certain twisted fractionally Calabi–Yau property.     

Characteristic modules and tensor products over quasi-hereditary algebras

Let A be a monomial quasi-hereditary algebra with a pure strong exact Borel subalgebra B.It is proved that the category of induced good modules over B is contained in the category of good modules over A;that the characteristic module of A is an induced module of that of B via the exact functor-(?)_B A if and only if the induced A-module of an injective B-module remains injective as a B-module.Moreover,it is shown that an exact Borel subalgebra of a basic quasi-hereditary serial algebra is right serial and that the characteristic module of a basic quasi-hereditary serial algebra is exactly the induced module of that of its exact Borel subalgebra.     

Resolutions over Koszul algebras

In this paper we show that if is a Koszul algebra with Λ₀ isomorphic to a product of copies of a field, then the minimal projective resolution of Λ₀ as a right Λ-module provides all the information necessary to construct both a minimal projective resolution of Λ₀ as a left Λ-module and a minimal projective resolution of Λ as a right module over the enveloping algebra of Λ. The main tool for this is showing that there is a comultiplicative structure on a minimal projective resolution of Λ₀ as a right Λ-module.Received: 14 September 2004     

Jordan recoverability of some subcategories of modules over gentle algebras

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.     

On the Hochschild cohomology ring of tensor products of algebras

We prove that, as Gerstenhaber algebras, the Hochschild cohomology ring of the tensor product of two algebras is isomorphic to the tensor product of the respective Hochschild cohomology rings of these two algebras, when at least one of them is finite dimensional. In case of finite dimensional symmetric algebras, this isomorphism is an isomorphism of Batalin–Vilkovisky algebras. As an application, we explain by examples how to compute the Batalin–Vilkovisky structure, in particular, the Gerstenhaber Lie bracket, over the Hochschild cohomology ring of the group algebra of a finite abelian group.     

Finite generation of some cohomology rings via twisted tensor product and Anick resolutions

Over a field of prime characteristic $p>2$, we prove that the cohomology rings of some pointed Hopf algebras of dimension $p^3$ are finitely generated. These are Hopf algebras arising in the ongoing classification of finite dimensional pointed Hopf algebras in positive characteristic. They include bosonizations of Nichols algebras of Jordan type in a general setting. When $p=3$, we also consider their Hopf algebra liftings, that is Hopf algebras whose associated graded algebra with respect to the coradical filtration is given by such a bosonization. Our proofs are based on an algebra filtration and a lemma of Friedlander and Suslin, drawing on both twisted tensor product resolutions and Anick resolutions to locate the needed permanent cocycles in May spectral sequences.     

Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.     

Graded tensor products

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z₂-graded tensor products of P.I. algebras, J. Algebra 291 (2005) 274-296]. For example, a finite-dimensional algebra is regular if it has a basis whose multiplication table satisfies some commutation relations. The matrix algebra M_n(F) over the field F is regular, which is closely related to M_n(F) being Z_n-graded. We study the polynomial identities of various types of tensor products of such algebras. In particular, using the theory of Hopf algebras, we prove a far reaching extension of the A⊗B theorem for Z₂-graded PI algebras.