Twisted Smash Products and L-R Smash Products for Biquasimodule Hopf Quasigroups

We consider two new algebras from an H-biquasimodule algebra A and a Hopf quasigroup H: twisted smash product A ? H and L-R smash product A?H, and find necessary and sufficient conditions for making them Hopf quasigroups. We generalize the main results in Brzeziński and Jiao [5] and Klim and Majid [9]. Moreover, if H is a cocommutative Hopf quasigroup, we prove that A ? H is isomorphic to A?H as Hopf quasigroups.     

On Smash Products of Transitive Module Algebras

Let H be a semisimple Hopf algebra over a field of characteristic 0, and A a finite-dimensional transitive H-module algebra with a l-dimensional ideal. It is proved that the smash product A#H is isomorphic to a full matrix algebra over some right coideal subalgebra N of H. The correspondence between A and such N, and the special case A = k（X） of function algebra on a finite set X are considered.     

Smash Products of H-Simple Module Algebras

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End _A′(V ? H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Elementary Bialgebra Properties of Group Rings and Enveloping Rings: An Introduction to Hopf Algebras

This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside's theorem for Hopf algebras. It was never published, but may still be of interest to students and beginning researchers.

Let K be a field, and let A be an algebra over K. Then the tensor product A ? A = A ? _K A is also a K-algebra, and it is quite possible that there exists an algebra homomorphism Δ: A → A ? A. Such a map Δ is called a comultiplication, and the seemingly innocuous assumption on its existence provides A with a good deal of additional structure. For example, using Δ, one can define a tensor product on the collection of A-modules, and when A and Δ satisfy some rather mild axioms, then A is called a bialgebra. Classical examples of bialgebras include group rings K[G] and Lie algebra enveloping rings U(L). Indeed, most of this paper is devoted to a relatively self-contained study of some elementary bialgebra properties of these examples. Furthermore, Δ determines a convolution product on Hom _K (A, A), and this leads quite naturally to the definition of a Hopf algebra.     

On twisted smash products for bimodule algebras and the drinfeld double 1

In this paper we construct a new algebra AHof an H- bimodule algebra Aby a Hopf algebra Hand study some of its properties. The smash product, the Drinfel'd double D(H) and the Doi-Takeuchi's algebra B?,H, are all special cases of AH. Moreover,we find a necessary and sufficient condition for A Hto be a Hopf algebra and also consider the dual situation     

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras,Weak Hopf Algebras,and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object B^co(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product B^co(H)#H, which is an H-bicomodule algebra, isomorphic to B.     

Faithfully Flat Hopf Bi-Galois Extensions

This article is devoted to faithfully flat Hopf bi-Galois extensions defined by Fischman, Montgomery, and Schneider. Let H be a Hopf algebra with bijective antipode. Given a faithfully flat right H-Galois extension A/R and a right H-comodule subalgebra C ? A such that A is faithfully flat over C, we provide necessary and sufficient conditions for the existence of a Hopf algebra W so that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra. As a consequence, we prove that if R = k, there is a Hopf algebra W such that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra if and only if C is an H-submodule of A with respect to the Miyashita–Ulbrich action.     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

Weak Global Dimension of Coherent Rings

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

On the Dot Product Graph of a Commutative Ring

Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R?{(0, 0,…, 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)* = Z(R)?{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZD(R) is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to ?₂ × ?₂ × ?₂.     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

Maschke’s theorem for smash products of quasitriangular weak Hopf algebras

The paper is concerned with the semisimplicity of smash products of quasitriangular weak Hopf algebras. Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k and A any semisimple and quantum commutative weak H-module algebra. Based on the work of Nikshych et al. (Topol. Appl. 127(1–2):91–123, 2003), we give Maschke’s theorem for smash products of quasitriangular weak Hopf algebras, stating that A#H is semisimple if and only if A is a projective left A#H-module, which extends the Theorem 3.2 given in Yang and Wang (Commun. Algebra 27(3):1165–1170, 1999).     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

Doi–Hopf Modules Associated to Comodule Coalgebras

To any right comodule coalgebra C over a Hopf algebra H we associate a left H-comodule algebra A. Under certain conditions, in particular in the case where H has nonzero integrals, we show that the category of right C, H-comodules is isomorphic to a certain subcategory of the category of Doi–Hopf modules associated to A. As an application, we investigate the connection between C and the smash coproduct C ? H being right semiperfect.     

On the Commutative Twisted Group Algebras

Let G be an abelian group and let R be a commutative ring with identity. Denote by R _t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ?(R) and ?(R _t G) the nil radicals of R and R _t G, respectively, by G _p the p-component of G and by G ₀ the torsion subgroup of G. We prove that:

1. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ?(R) = 0, then there exists a twisted group algebra R _{t 1} (G/G _p ) such that R _t G/?(R _t G) ? R _{t 1} (G/G _p ) as R-algebras;

2. If R is a ring of prime characterisitic p and R* is p-divisible, then ?(R _t G) = 0 if and only if ?(R) = 0 and G _p  = 1; and

3. If B(R) = 0, the orders of the elements of G ₀ are not zero divisors in R, H is any group and the commutative twisted group algebra R _t G is isomorphic as R-algebra to some twisted group algebra R _{t 1} H, then R _t G ₀ ? R _{t 1} H ₀ as R-algebras.