Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

Morita Theory for Comodules Over Corings

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring 𝒞. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring 𝒞 or the comodule Σ is finitely generated and projective as an A-module. That is, we obtain relations between the category of 𝒞-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End ^𝒞(Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)     

Azumaya Extensions and Galois Correspondence

For H a finite-dimensional Hopf algebra over a field k, we study H*-Galois Azumaya extensions A, i.e., A is an H-module algebra which is H*-Galois with A/A^H separable and A^H Azumaya. We prove that there is a Galois correspondence between a set of separable subalgebras of A and a set of separable subalgebras of C_A(A^H), thus generalizing the work of Alfaro and Szeto for H a group algebra. We also study Galois bases and Hirata systems.1991 Mathematics Subject Classification: 16W30, 16H05     

Hopf algebra actions

Finite groups acting on rings by automorphisms, and group-graded rings are instances of Hopf algebras H acting on H-module algebras A. We study such actions. Let A^H = {a ε A¦h · A = (h) a, all h ε H}, the ring of H-invariants, and form the smash product A # H. We study the ring extensions A^H A A # H. We prove a Maschke-type theorem for A # H-modules. We form an associated Morita context [A^H, A, A, A # H] and use these to get connections between the various rings.     

Actions of Ore extensions and growth of polynomial H-identities

We show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexp^H(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.     

Extensions of the endomorphism algebra of weak comodule algebras

Let H be a weak Hopf algebra, and let A/B be a weak right H-Galois extension. In this paper, we mainly discuss the extension of the endomorphism algebra of a module over A. A necessary and sufficient condition for such an extension of the endomorphism algebra to be weak H-Galois is obtained by using Hopf-Galois theory and Morita theory.     

THE DIMENSION OF IRREDUCIBLE MODULES FOR TRANSITIVE MODULE ALGEBRAS

We prove that for a semisimple Hopf algebra H, if A is a transitive H-module algebra and M is an irreducible A-module, then dim(A) divides dim(M)²dim(H).

     

Actions of hopf algebras on fully bounded noetherian rings

Let H be a finite dimensional Hopf algebra acting on a right noetherian algebra A and assume that the trace function [tcirc] : A → A ^H is surjective. Then A is right PBN if and only if so is A ^H . This extends the result of García-del Río for group actions which answered a question of Fisher-Osterburg, and the result of Nǎstǎsescu-Dǎscǎlescu for group graded algebras.     

Central invariants of H-module algebras

Let H be a Hopf algebra over a field k:, and A an H-module algebra, with subalgebra of H-invariants denoted by A^H . When (H, R) is quasitriangular and A is quantum commutative with respect to (H,R), (e.g. quantum planes, graded commutative superalgebras), then A^H ? center of A = Z(A). In this paper we are mainly concerned with actions of H for which A^H ? Z(A). We show that under this hypothesis there exists strong relations between the ideal structures of A^H A and A#H.

We demonstrate the theorems by constructing an example of a quantum commutative A, so that A/A^H is H ^?-Galois. This is done by giving (C G)^? G = Z_n × Z_n , a nontrivial quasitriangular structure and defining an action of it on a localization of the quantum plane.     

DOI-KOPPINEN HOPF BIMODULES ARE MODULES

I construct a generalized twisted smash product A ^H B, which gives an abstract structure of Cibils-Rosso's algebra X associated to a finite-dimensional Hopf algebra H, for the H-bimodule algebra A and H-bicomodule algebra B. I show that the Doi-Koppinen Hopf (H, B, D)-bimodules are modules over a certain algebra which is of this type. Moreover, if D is finitely generated projective as a k-module, there exists a k-module-preserving equivalence of categories between the category of Doi-Koppinen (H, B, D)-Hopf bimodules and the category of left (D ^*op ? D ^*) ^{H?H op} (B ? B ^op )-modules.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

A class of non-nilpotent 2-solvable n-Lie algebras

This article concerns a class of finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebras. It is shown that if L is a finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebra, then L can be decomposed into a semi-direct of an ideal A and an (n ? 1)-dimensional subalgebra H ₀ of L. Furthermore, H ₀ acts irreducibly on A/A ¹, and H ₀ + A ¹ is a self-normalizing maximal subalgebra of L with the core A ¹, the derived algebra of A.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. 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 n₄-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 irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

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     

Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)^* has the weak uniform A(G × H)^** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC ₂(G × H)^* is equal to the Fourier–Stieltjes algebra B(G × H).     

Hilbert's Theorem 90 for Hopf Galois Extensions

For a faithfully flat extension A/B and a right A-module M, we give a new characterization of the set of descent data on M. Assuming that B is a simple Artinian ring and A/B is H-Galois, for a certain finite dimensional Hopf algebra H, we prove that Sweedler's noncommutative cohomology H ¹(H?, A) is trivial as a pointed set.     

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.     

Structure of <Emphasis Type="Italic">H</Emphasis>-Semiprime Artinian Algebras

Let H be a Hopf algebra over a field. It is proved that every H-semiprime right artinian left H-module algebra A is quasi-Frobenius and H-semisimple. If H grows slower than exponentially, then all H-equivariant A-modules are shown to be A-projective. With the additional assumption that H is cosemisimple it is proved that the Jacobson radical of any right artinian left H-module algebra is stable under the action of H.