Margaret Beattie

A GENERALIZATION OF THE SMASH PRODUCT OF A GRADED RING

For any group G and G-graded ring R, there exists a ring S = R ♯ G∗, defined analogously to the smash product of R with the dual of the group ring for finite G, such that the categories of unital S-modules and G-graded R-modules are isomorphic. The category of unital S-modules is equivalent to the category of A-modules for a ring A with identify if and only if S is finitely generated as an S-S bimodule. Finally the category isomorphism is applied to obtain a characterization of the graded Jacobson radical for infinite G.

A direct sum decomposition for the Brauer group of H-module algebras

Let R be a commutative ring with unit, H a finitely generated projective, commutative and cocommutative Hopf algebra over R. In [3], Long defined a Brauer group BM(R, H) whose elements are equivalence classes of R- Azumaya H-module algebras. In this note, we show that there is a split exact sequence 1 -+ R(R) + BM(R, H) + Gal(R, H) + 1, where Gal(R, H) is the group of Galois H-objects over R, defined in [l], and B(R) is the ordinary Brauer group of R. This sequence generalizes that obtained in [5] for graded algebras, i.e., for H = GR, the dual of the group ring RG.

Lifting of Nichols algebras of typeB 2

We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan typeB2 subject to the small restriction that the diagnonal elements of the braiding matrix are primitiventh roots of 1 with oddn≠5. As well, we compute the liftings of a Nichols algebra of Cartan typeA2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered in previous work of Andruskiewitsch and Schneider. We study the problem of when the liftings of a given Nichols algebra are quasi-isomorphic. The Appendix (with I. Rutherford) contains a generalization of the quantum binomial formula. This formula was used in the computation of liftings of typeB2 but is also of interest independent of these results.

Hopf Algebras of Dimension 14

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H_1, the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. Also a Hopf algebra of dimension pq where p and q are odd primes with p<q and q less than or equal to 1 + 3p, and also less than or equal to 13, is semisimple and thus a group algebra or the dual of a group algebra. We also have some partial results in the classification problem for dimension 16.

An isomorphism theorem for ore extension hopf algebras

A complete description of isomorphisms between Ore extension Hopf algebras is given and used to enumerate all Ore extension Hopf algebras Hwith ∣G(H)∣ = 4 and skew-primitives whose square lies in H 0.

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” AT, T sumHom(H,End (A)). If T is convolution invertible, the categories of relative right Hopf modules over A and ATare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products

Duality Theorems for Rings with Actions or Coactions

graded. For S a ring on which G acts as a group of automorphisms, let A be S * G, the skew group ring; then, from our Morita context, we obtain G * (S * G) z MG(S)fi”, where MG( #” is the ring of matrices over S with rows and columns indexed by G and with only finitely many nonzero entries. Note that if S is a field, then MG(S)sn is dense in End,(S * G) with the finite-discrete topology [S-J. To obtain our second duality theorem, namely that for

A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings

Let G be a finite abelian group, and R a commutative ring. The Brauer-Long group BD(R, G) is described by an exact sequence 1 -* BDS(R, G) -* BD(R, G) Aut(G x G*)(R) where BDS (R, G) is a product of etale cohomology groups, and Im fl is a kind of orthogonal subgroup of Aut(G x G*)(R). This sequence generalizes some other well-known exact sequences, and restricts to two split exact sequences describing Galois extensions and strongly graded rings.

The picard group of a category of graded modules

For R a G-graded ring, we study Pic(R-gr), the group of isomorphism classes of autoequivalences of the category of graded left R-modules. For G infinite, this requires generalizing the classical sequences involving Pic(A), A a fc-algebra, to A a ring with local units. Then for G either finite or infinite, we characterize the inner automorphisms in some subgroups H of the automorphism group of the smash product R#PG and thus obtain some subgroups of Pic(R-gr).

On perfect graded rings

We prove that a group graded ring with finite support is right perfect if and only if the identity graded component is right perfect.