Yetter–Drinfeld Modules over the Hopf–Ore Extension of the Group Algebra of Dihedral Group

Let k be an algebraically closed field of characteristic zero, and D _n be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A(n, 0) of kD _n . We describe the structures and properties of simple Yetter-Drinfeld modules over A(n, 0), and classify all simple Yetter-Drinfeld modules over A(n, 0).     

Finite-Dimensional Irreducible Modules for the Three-Point 2 Loop Algebra

Recently Brian Hartwig and the second author found a presentation for the three-point ₂ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra ? by generators and relations, and displayed an isomorphism from ? to the three-point ₂ loop algebra. In this article, we describe the finite-dimensional irreducible ?-modules from multiple points of view.     

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over Twisted Heisenberg-Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.     

Classification of Simple Weight Modules for the Neveu–Schwarz Algebra with a Finite-Dimensional Weight Space

We show that the support of a simple weight module over the Neveu–Schwarz algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Neveu–Schwarz algebra, having a nontrivial finite-dimensional weight space, is a Harish–Chandra module (and hence is either a highest or lowest weight module, or else a module of the intermediate series). This result generalizes a theorem which was originally given on the Virasoro algebra.     

Modules over matrix near-rings and the ℐ0-radical

This paper clears up some questions concerning type 0 modules over matrix near-rings and the ₀-radical in matrix near-rings. It is shown that, unlike in the type 2 case, type 0 modules over matrix near-rings may arise in several non-isomorphic ways. As a result, we do not always have the same nice relationship between the ₀-radicals of a near-ring and the corresponding matrix near-ring, as we do for the ₂-radical. All near-rings concerned are zero-symmetric with identity element.     

Krull–Remak–Schmidt Fails for Artinian Modules over Local Rings

Let R be a ring. Any R-module M which is Artinian or Noetherian can be written as the direct sum of a finite number of indecomposable R-modules. The theorem of Krull–Remak–Schmidt asserts that in the case where M is of finite length, such a decomposition is unique up to isomorphism. On the other hand, examples of Noetherian R-modules which have essentially different decompositions have been known for a long time. The first examples of Artinian R-modules with essentially different decompositions were published only in 1995 by Facchini, Herbera, Levy and Vámos. In order to construct such examples, one needs to deal with suitable rings R. Note that for R Noetherian or commutative, all the Artinian modules have the Krull–Remak–Schmidt property. In 1998, Facchini raised the problem of whether the same is true in the case where R is a local ring. The aim of this note is to show that this is not so: we are going to present a local ring R and Artinian R-modules M with essentially different direct decompositions into indecomposables. The military importance of these results has been discussed during the NATO meeting at Constantia (August 2000) which was organized by K. W. Roggenkamp.     

Modules with the Beachy–Blair Condition

A right R-module satisfies the right Beachy–Blair condition if each of its faithful submodules is cofaithful. In this article we study the relationship between the right Beachy–Blair condition of a right R-module and its skew polynomial, skew monoid and skew generalized power series extensions. Consequently, several known results regarding rings satisfying the right Beachy–Blair condition are extended to a more general setting.     

On the Construction of d-Continuous Modules over Some Special Rings

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Geometric aspects of the Kapustin–Witten equations

This expository paper introduces the Kapustin?CWitten equations to mathematicians. We discuss the connections between the complex Yang?CMills equations and the Kapustin?CWitten equations. In addition, we show the relation between the Kapustin?CWitten equations, the moment map condition and the gradient Chern?CSimons flow. The new results in the paper correspond to estimates on the solutions to the Kapustin?CWitten equations given an estimate on the complex part of the connection. This leaves open the problem of obtaining global estimates on the complex part of the connection.     

A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings

The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module.     

On the Category of Koszul Modules of Complexity One of an Exterior Algebra

In this paper, we prove that there is a natural equivalence between the category F1（x） of Koszul modules of complexity 1 with filtration of given cyclic modules as the factor modules of an exterior algebra A = ∧V of an m-dimensional vector space, and the category of the finite-dimensional locally nilpotent modules of the polynomial algebra of m - 1 variables.     

Syzygies of Cohen–Macaulay Modules over One Dimensional Cohen–Macaulay Local Rings

Algebras and Representation Theory - We study syzygies of (maximal) Cohen–Macaulay modules over one dimensional Cohen–Macaulay local rings. We assume that rings are generically...     

Normal Forms of Modules over Admissible Algebras with Formal Two-ray Modules

The aim of the paper is to classify the indecomposable modules and describe the Auslander-Reiten sequences for the admissible algebras with formal two-ray modules.     

Yetter–Drinfeld Modules over Weak Braided Hopf Monoids and Center Categories

In this paper,we introduce several centralizer constructions in a monoidal context and establish a monoidal equivalence with the category of Yetter–Drinfeld modules over a weak braided Hopf monoid.We apply the general result to the calculus of the center in module categories.     

Symmetry Algebra of the Benjamin–Ono Equation

The Lie algebra structure for symmetries of the Benjamin–Ono equation is completely described.     

The Derivation Algebra and Automorphism Group of the Twisted Heisenberg–Virasoro Algebra

In this article, we give the derivation algebra Der ? and the automorphism group Aut ? of the twisted Heisenberg–Virasoro algebra ?.     

Auslander–Bridger Modules

Classically, the Auslander–Bridger transpose finds its best applications in the well-known setting of finitely presented modules over a semiperfect ring. We introduce a class of modules over an arbitrary ring R, which we call Auslander–Bridger modules, with the property that the Auslander–Bridger transpose induces a well-behaved bijection between isomorphism classes of Auslander–Bridger right R-modules and isomorphism classes of Auslander–Bridger left R-modules. Thus we generalize what happens for finitely presented modules over a semiperfect ring. Auslander–Bridger modules are characterized by two invariants (epi-isomorphism class and lower-isomorphism class), which are interchanged by the transpose. Via a suitable duality, we find that kernels of morphisms between injective modules of finite Goldie dimension are also characterized by two invariants (mono-isomorphism class and upper-isomorphism class).