Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let k be a fixed algebraically closed field of arbitrary characteristic,let Λ be a finite dimensional self-injective k-algebra,and let V be an indecomposable non-projective left Λ-module with finite dimension over k.We prove that if τΛV is the Auslander-Reiten translation of V,then the versal deformation rings R(Λ,V)and R(Λ,τΛV)(in the sense of F.M.Bleher and the second author)are isomorphic.We use this to prove that if Λ is further a cluster-tilted k-algebra,then R(Λ,V)is universal and isomorphic to k.     

Computation of the Yoneda Algebras for Algebras of Dihedral Type

We continue a series of papers in which the Yoneda algebra is computed for algebras of dihedral and semidihedral types. In this paper, the Yoneda algebra is computed for one more family of algebras, namely, for the family D(3 ) (in the classification of K. Erdmann). In order to find the minimal resolutions of simple modules, we used for the first time a C⁺⁺ program implemented by the second author. Bibliography: 10 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 101–120.     

Projective Resolutions and Yoneda Algebras for Algebras of Dihedral Type

This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson. This work was partially supported by the grant 06-01-00200 of the Russian Foundation for Basic Research.     

Endomorphism Rings of Regular Modules over Wild Hereditary Algebras

Let X be an indecomposable regular module over a connected wild hereditary path-algebra. The main result is a factorization property for maps in the radical of End _H (X) and an upper bound for its degree of nilpotency. The bound is sharp if X has elementary quasi-top. In this, case the socle of End _H (X) can also be characterized.     

Cohomology of Algebras of Dihedral Type. I

A technique which is associated with the BensonCarlson diagrammatic method is developed. Using this technique, we describe the Yoneda algebras of the algebras that constitute a series of algebras of dihedral type. We also describe the Ext-algebras of simple modules over the algebras considered. Bibliography: 6 titles.     

Quiver Generalized Weyl Algebras,Skew Category Algebras and Diskew Polynomial Rings

The aim of the paper is to introduce new large classes of algebras—quiver generalized Weyl algebras, skew category algebras, diskew polynomial rings and skew semi-Laurent polynomial rings.     

Hopf Algebras of Linear Recurring Sequences over Rings and Modules

The module of linear recurring sequences over a commutative ring R can be considered as a Hopf algebra dual to the polynomial Hopf algebra over R. Under this approach, some notions and operations from the Hopf algebra theory have an interesting interpretation in terms of linear recurring sequences. Generalizations are also considered: linear recurring bisequences, sequences over modules, and k-sequences.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 113–148, 2003.     

Simple Modules for the Restricted Cartan Type Lie Algebras

The simple modules with homogeneous characters are considered, their dimension formulas are determined.Presented by D. Passman.     

Large Free Algebras in the Ring of Fractions of Skew Polynomial Rings

It is shown that the division ring of quotients of a skew polynomialring of automorphism type, if infinite-dimensional over itscentre k and satisfying suitable hypotheses, contains the groupalgebra of a free group of large rank (usually at least |k|).The result applies, in particular, to the skew polynomial ringsconstructed from rational function fields, and affirmativelysettles the conjectures of Makar–Limanov and Lichtmanin this case.     

Cyclically Presented Modules Over Rings of Finite Type

A ring is of finite type if it has only finitely many maximal right ideals, all two-sided. In this article, we give a complete set of invariants for finite direct sums of cyclically presented modules over a ring R of finite type. More generally, our results apply to finite direct sums of direct summands of cyclically presented right R-modules (DCP modules). Using a duality, we obtain as an application a similar set of invariants for kernels of morphisms between finite direct sums of pair-wise non-isomorphic indecomposable injective modules over an arbitrary ring. This application motivates the study of DCP modules.     

q-Wedge Modules for Quantized Enveloping Algebras of Classical Type

We use the fusion construction in twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a by-product we uniformly realize all non-spin fundamental modules for quantized enveloping algebras of classical types, and show that they admit natural crystal bases as modules for the (derived) twisted quantum affine algebra. These crystal bases are parametrized in terms of the q-wedge products.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Varieties of Algebras with Superinvolution of Almost Polynomial Growth

Let A be an associative algebra with superinvolution ? over a field of characteristic zero and let \(c_{n}^{\ast }(A)\) be its sequence of corresponding ?-codimensions. In case A is finite dimensional, we prove that such sequence is polynomially bounded if and only if the variety generated by A does not contain three explicitly described algebras with superinvolution. As a consequence we find out that no intermediate growth of the ?-codimensions between polynomial and exponential is allowed.     

Standard Modules for Tabular Algebras

We introduce cell modules for the tabular algebras defined in a previous work; these modules are analogous to the representations arising from left Kazhdan–Lusztig cells. The standard modules of the title are constructed in an elementary way by suitable tensoring of the cell modules. We show how a certain extended affine Hecke algebra of type A equipped with its Kazhdan–Lusztig basis is an example of a tabular algebra, and verify that in this case our standard modules coincide with other standard modules defined in the literature.     

Modules for Yokonuma-type Hecke Algebras

This paper describes the module categories for a family of generic Hecke algebras, called Yokonuma-type Hecke algebras. Yokonuma-type Hecke algebras specialize both to the group algebras of the complex reflection groups G(r,1,n) and to the convolution algebras of (B \(^{\prime }\),B \(^{\prime }\))-double cosets in the group algebras of finite general linear groups, for certain subgroups B \(^{\prime }\) consisting of upper triangular matrices. In particular, complete sets of inequivalent, irreducible modules for semisimple specializations of Yokonuma-type Hecke algebras are constructed.     

Prime Ideals in Skew Polynomial Rings of Derivation Type

ABSTRACT

It is shown that (Plesken, 1983 Plesken , W. ( 1983 ). Group rings of finite groups over the p-adic integers . Springer Lecture Notes in Mathematics 1026 . [Google Scholar], Theorem 8.5), describes blocks of cyclic defect groups up to Morita equivalence. In particular such a block is determined by its planar embedded Brauer tree. Applying the radical idealizer process, the head order of such blocks is calculated explicitly.