Indecomposable decomposition of tensor products of modules over Drinfeld doubles of Taft algebras

In this paper, we study the tensor product structure of the category of finite dimensional modules over Drinfeld doubles of Taft Hopf algebras. Tensor product decomposition rules for all finite dimensional indecomposable modules are explicitly given.     

Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.     

The Carlitz algebras

The Carlitz F_q-algebra C=C_ν, ν∈N, is generated over an algebraically closed field K (which contains a non-discrete locally compact field of positive characteristic p>0, i.e. K?F_q[[x,x⁻¹]], q=p^ν), by the (power of the) Frobenius map X=X_ν:f?f^q, and by the Carlitz derivativeY=Y_ν. It is proved that the Krull and global dimensions of C are 2, classifications of simple C-modules and ideals are given, there are only countably many ideals, they commute (IJ=JI), and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple C-module is a sum of eigenspaces of the element YX (the set of eigenvalues for YX is given explicitly for each simple C-module). This fact is crucial in finding the group of F_q-algebra automorphisms of C and in proving that any two distinct Carlitz rings are not isomorphic (C_ν?C_μ if ν≠μ). The centre of C is found explicitly, it is a UFD that contains countably many elements.     

Iyama’s finiteness theorem via strongly quasi-hereditary algebras

Let Λ be an artin algebra and X a finitely generated Λ-module. Iyama has shown that there exists a module Y such that the endomorphism ring Γ of X⊕Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Γ is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length n must have global dimension at most 2n−2. We want to show that Iyama’s better bound is related to the fact that the ring Γ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1.     

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.     

On a separation of orbits in the module variety for string special biserial algebras

Given a pair M,M^′ of finite-dimensional modules over a string special biserial algebra Λ, a fully verifiable criterion, expressed in terms of a finite set of simple linear algebra invariants, deciding if M and M^′ lie in the same orbit in module variety, equivalently, if M and M^′ are isomorphic, is formulated and proved.     

Irreducible Specht modules for Hecke algebras of type A

Let F be a field, n a non-negative integer, λ a partition of n and S^λ the corresponding Specht module for the Iwahori-Hecke algebra H_F,q(S_n). James and Mathas conjecture a necessary and sufficient condition on λ for S^λ to be irreducible. We prove the sufficiency of this condition in the case where F has infinite characteristic and also in the case where q=1.     

Comparison of admissibility conditions for cyclotomic Birman-Wenzl-Murakami algebras

We show the equivalence of admissibility conditions proposed by Wilcox and Yu (in press) [11] and by Rui and Xu (2009) [9] for the parameters of cyclotomic BMW algebras.     

The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras

With an aim of exploring homological algebra for weak Hopf modules, this paper investigates the HOM-functor and presents the structure theorem for endomorphism algebras of weak two-sided (A,H)-Hopf modules, and gives the duality theorem for weak “big” smash products.     

The representations of cyclotomic BMW algebras

In this paper, we classify the irreducible ?_r,n-modules over a field.     

Type A Hecke algebras and related algebras

Let be the Hecke algebra of the symmetric group over a field K of characteristic and a primitive -th root of one in K. We show that an -module is projective if and only if its restrictions to any -parabolic subalgebra of is projective. Moreover, we give a new construction of blocks of -parabolic subalgebras, in terms of skew group algebras over local commutative algebras. Received: 30 June 2003     

On the structure of cyclotomic Nazarov-Wenzl algebras

Ariki, Mathas and Rui [S. Ariki, A. Mathas, H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006) 47-134 (special volume in honor of Professor G. Lusztig)] introduced a class of finite dimensional algebras W_r,n, called the cyclotomic Nazarov-Wenzl algebras which are associative algebras over a commutative ring R generated by {S_i,E_i,X_j∣1≤i<n and 1≤j≤n} satisfying the defining relations given in this paper. In particular, for any positive integers i≤n−1. Note that are quotients of affine Wenzl algebras in [M. Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra 182 (1996) 664-693]. It has been proved in the first cited reference above that W_r,n is cellular in the sense of [J.J. Graham, G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996) 1-34]. Using the representation theory of cellular algebras, Ariki, Mathas and Rui have classified the irreducible W_r,n-modules under the assumption ω₀≠0 in their above-cited work. In this paper, we are going to classify the irreducible W_r,n-modules under the assumption ω₀=0. We will compute the Gram determinant associated to each cell module for W_r,n no matter whether ω₀ is zero or not. At the end of this paper, we use our formulae for Gram determinants to determine the semisimplicity of W_r,n for arbitrary parameters over an arbitrary field F with charF≠2.     

Rota-Baxter algebras and dendriform algebras

In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter algebras.     

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.     

Constructing cell data for diagram algebras

We show how the treatment of cellularity in families of algebras arising from diagram calculi, such as Jones’ Temperley-Lieb wreaths, variants on Brauer’s centralizer algebras, and the contour algebras of Cox et al. (of which many algebras are special cases), may be unified using the theory of tabular algebras. This improves an earlier result of the first author (whose hypotheses covered only the Brauer algebra from among these families).     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

Parabolic decomposition for properly stratified algebras

We generalize the machinery of exact Borel subalgebras of quasi-hereditary algebras on properly stratified algebras.     

On Lie algebras associated with representation-directed algebras

Let B be a representation-finite C-algebra. The Z-Lie algebra L(B) associated with B has been defined by Riedtmann in [Ch. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994) 526-546]. If B is representation-directed, there is another Z-Lie algebra associated with B defined by Ringel in [C.M. Ringel, Hall Algebras, vol. 26, Banach Center Publications, Warsaw, 1990, pp. 433-447] and denoted by K(B).We prove that the Lie algebras L(B) and K(B) are isomorphic for any representation-directed C-algebra B.