Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

In this paper, we investigate the tensor structure of the category of finite- dimensional weight modules over the Hopf–Ore extensions kG(χ^?1,a,0) of group algebras kG. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that k is an algebraically closed field of characteristic zero, and the orders of χ and χ(a) are the same.     

Yetter–Drinfeld modules over bosonizations of dually paired Hopf algebras

Let (R^∨,R)$(R^{\vee },R)$ be a dual pair of Hopf algebras in the category of Yetter–Drinfeld modules over a Hopf algebra H$H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter–Drinfeld modules over the bosonizations of R$R$ and of R^∨$R^{\vee }$, respectively. As an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of Nichols algebras of semi-simple Yetter–Drinfeld modules over H$H$.     

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).     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

Tilting modules over path algebras of Dynkin type and complete slice modules

Let T be a tilting A-module over a path algebra of Dynkin type.We prove thatif the indecomposable direct summands of T are in the different (?)-orbits of the AR-quiverof A,then T is a complete slice module.     

Monomial Hopf algebras over fields of positive characteristic

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n) are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.     

Diophantine definability over non-finitely generated non-degenerate modules of algebraic extensions of ℚ

We investigate the issues of Diophantine definability over the non-finitely generated version of non-degenerate modules contained in the infinite algebraic extensions of the rational numbers. In particular, we show the following. Let k be a number field and let K _inf be a normal algebraic, possibly infinite, extension of k such that k has a normal extension L linearly disjoint from K _inf over k. Assume L is totally real and K _inf is totally complex. Let M _inf be a non-degenerate O _k -module, possibly non-finitely generated and contained in O _Kinf . Then M _inf contains a submodule Mˉ _inf such that M _inf /Mˉ _inf is torsion and O _k has a Diophantine definition over Mˉ _inf . Received: 4 May 1999 / Published online: 21 March 2001     

Irreducible weight modules over the twisted Schrödinger-Virasoro algebra

It is shown that the support of an irreducible weight module over the SchrSdinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.     

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A{\mathcal {A}} . We determine explicitly all the A{\mathcal {A}} -module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A{\mathcal {A}} -module automorphisms of the (reduced) Dickson–Mùi algebras. The algebra of all A{\mathcal {A}} -module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A{\mathcal {A}} -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mùi algebra is an indecomposable A{\mathcal {A}} -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

Representations of étale Lie groupoids and modules over Hopf algebroids

The classical Serre-Swan’s theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.     

Induced modules of split Iwahori–Hecke algebras are reducible

We show that the modules of an Iwahori–Hecke algebra obtained via induction from a proper parabolic subalgebra are always reducible.     

∑–pure—injective modules over a commutative prüfer ring

We give the complete classification of ∑-pure-injective modules over a commutative Prüfer ring.     

Homological determinacy of representations of a group of type P∞ over the ring of p-adic integers

It is proved that homologically equivalent p-adic representations of a group of type p are isomorphic to within imbedding, under which the factor module is homologically trivial.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 164–171, 1983.     

Poitou–Tate duality over extensions of global fields

In this paper, we are interested in the Poitou–Tate duality in Galois cohomology. We will formulate and prove a theorem for a nice class of modules (with a continuous Galois action) over a pro-p ring. The theorem will comprise of the Tate local duality, Poitou–Tate duality and the Poitou–Tate?s exact sequence.