Homological properties of sklyanin algebras

In their recent paper [13], Tate and Van den Bergh studied certain quadratic algebras, called the “Sklyanin algebras”. They proved that these algebras have the Hilbert series of a polynomial algebra, are Noetherian and Koszul, and satisfy the Auslander-Gorenstein and Cohen-Macaulay conditions. This paper gives an alternative proof of these results, as suggested in [13], and thereby answering a question in their paper.     

Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C _{C G (F)}(h) = 1 for all nonidentity elements h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.     

Crossed Products by Twisted Partial Actions: Separability,Semisimplicity, and Frobenius Properties

In this article, we are concerned with the crossed product S = R☆_α G by a twisted partial action α of a group G on a ring R. We give necessary and sufficient (resp., sufficient) conditions for S to be a separable (resp., Frobenius, semisimple) extension of R.     

The Structure Theorems of Weak Hopf Algebras

In this article, we mainly give the structure theorem of endomorphism algebras of weak Hopf algebras, and give another structure theorem as well as some applications for weak Doi–Hopf modules.     

Group Algebras of Finite Groups as Lie Algebras

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G.     

When is the Diagonal Functor Frobenius?

Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or _R ?, the category of left modules over a ring R.     

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Radford's Formula for Bifrobenius Algebras and Applications

In a biFrobenius algebra H, in particular in the case that H is a finite dimensional Hopf algebra, the antipode 𝒮:H → H can be decomposed as 𝒮 = _tc ○ c_φ where c_φ:H → H* and _tc:H* → H are the Frobenius and coFrobenius isomorphisms. We use this decomposition to present an easy proof of Radford's formula for 𝒮⁴. Then, in the case that the map 𝒮 satisfies the additional condition that 𝒮  id = id  𝒮 = u ?, we prove the trace formula tr(𝒮²) = ?(t)φ(1). We finish by applying the above results to study the semisimplicity and cosemisimplicity of H.     

Cobalanced Extensions of a Torsion-Free Module by a Torsion Module Over an Integral Domain

Cobalanced extensions of torsion-free modules by torsion modules over domains are investigated. As in the case of Abelian Groups, these sequences will split if and only if the torsion-free module is a pure submodule of a vector module. Unlike this case, even the torsion-free modules of finite rank need not be locally completely decomposable. For 1-dimensional Noetherian domains, necessary and sufficient conditions are given for that to be true.