Universal deformation rings and generalized quaternion defect groups

We determine the universal deformation rings R(G,V) of certain mod 2 representations V of a finite group G which belong to a 2-modular block of G whose defect groups are isomorphic to a generalized quaternion group D. We show that for these V, a question raised by the author and Chinburg concerning the relation of R(G,V) to D has an affirmative answer. We also show that R(G,V) is a complete intersection even though R(G/N,V) need not be for certain normal subgroups N of G which act trivially on V.     

Universal deformation rings and cyclic blocks

In this paper we determine the universal deformation rings of certain modular representations of finite groups which belong to cyclic blocks. The representations we consider are those for which every endomorphism is stably equivalent to multiplication by a scalar. We then apply our results to study the counterparts for universal deformation rings of conjectures about embedding problems in Galois theory. Received July 19, 1999 / Revised May 13, 2000 / Published online October 30, 2000     

Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.     

Nakayama oriented pullbacks and stably hereditary algebras

We introduce a new class of algebras, the Nakayama oriented pullbacks, obtained from pullbacks of surjective morphisms of algebras A?C and B?C. We prove that such a pullback is tilted when A and B are hereditary. We also show that stably hereditary algebras respecting the clock condition are Nakayama oriented pullbacks, and we use results about these pullbacks to show when a stably hereditary algebra is tilted or iterated tilted.     

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 representations of cyclotomic BMW algebras

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

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.     

Cellularity of twisted semigroup algebras

In this paper, the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of cellular twisted semigroup algebras of type JH. Partition algebras, Brauer algebras and Temperley-Lieb algebras all are examples of cellular twisted semigroup algebras of type JH. Our main result shows that the twisted semigroup algebra of a regular semigroup is cellular of type JH with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular algebras. Here we do not assume that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Moreover, for a twisted semigroup algebra, we do not require that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units in the ground ring. Note that trivially twisted semigroup algebras are the usual semigroup algebras. So, our results extend not only a recent result of East, but also some results of Wilcox.     

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     

Vanishing of self-extensions over symmetric algebras

We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type Z_A∞$\mathbb{Z}{A}_{\infty }$. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander–Reiten Conjecture and the Extension Conjecture.     

Self-injective algebras with an indecomposable standardly stratifying complement

For a basic self-injective algebra A which has an indecomposable standardly stratifying complement M, we determine its quiver and some relations, and a kind of coarse structure.     

Rhombus filtrations and Rauzy algebras

Peach introduced rhombal algebras associated to quivers given by tilings of the plane by rhombi. We develop general techniques to analyze rhombal algebras, including a filtration by what we call rhombus modules. We introduce a way to relate the infinite-dimensional rhombal algebra corresponding to a complete tiling of the plane to finite-dimensional algebras corresponding to finite portions of the tiling. Throughout, we apply our general techniques to the special case of the Rauzy tiling, which is built in stages reflecting an underlying self-similarity. Exploiting this self-similar structure allows us to uncover interesting features of the associated finite-dimensional algebras, including some of the tree classes in the stable Auslander-Reiten quiver.     

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. Let T(Λ)=Λ?D(Λ) be the trivial extension of Λ by its minimal injective cogenerator D(Λ). We characterize, in terms of quivers and relations, the algebras Λ^′ such that T(Λ)?T(Λ^′).     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

Representations of finite partially ordered sets over commutative artinian uniserial rings

Categories of representations of finite partially ordered sets over commutative artinian uniserial rings arise naturally from categories of lattices over orders and abelian groups. By a series of functorial reductions and a combinatorial analysis, the representation type of a category of representations of a finite partially ordered set S over a commutative artinian uniserial ring R is characterized in terms of S and the index of nilpotency of the Jacobson radical of R. These reductions induce isomorphisms of Auslander-Reiten quivers and preserve and reflect Auslander-Reiten sequences. Included, as an application, is the completion of a partial characterization of representation type of a category of representations arising from pairs of finite rank completely decomposable abelian groups.     

Decomposition of some pointed Hopf algebras given by the canonical Nakayama automorphism

Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.