Van C. Nguyen

Dominant dimension and tilting modules

We study which algebras have tilting modules that are both generated and cogenerated by projective–injective modules. Crawley–Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension 2, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting–cotilting module generated–cogenerated by projective–injective modules are precisely 1-minimal Auslander–Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

Classification of connected Hopf algebras of dimension p3 I

Abstract Let p be a prime, and k be an algebraically closed field of characteristic p. In this paper, we provide the classification of connected Hopf algebras of dimension p 3 , except for the case when the primitive space of the Hopf algebra is a two-dimensional abelian restricted Lie algebra. Each isomorphism class is presented by generators x, y, z with relations and Hopf algebra structures. Let μ be the multiplicative group of ( p 2 + p − 1 ) -th roots of unity. When the primitive space is one-dimensional and p is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by A k 1 / μ .

Primitive Deformations of Quantum p -Groups

In this paper, working over an algebraically closed field k of prime characteristic p, we introduce a concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional Hopf algebras which are Hopf deformations of restricted universal enveloping algebras. We illustrate this technique for the case when the restricted Lie algebra has dimension 3. Together with our previous classification results, we provide a complete classification of p3-dimensional connected Hopf algebras over k of characteristic p > 2.

Finite generation of the cohomology of some skew group algebras

We prove that some skew group algebras have Noetherian cohomology rings, a property inherited from their component parts. The proof is an adaptation of Evens’ proof of finite generation of group cohomology. We apply the result to a series of examples of finite-dimensional Hopf algebras in positive characteristic.

GERSTENHABER BRACKETS ON HOCHSCHILD COHOMOLOGY OF TWISTED TENSOR PRODUCTS

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.

Tate and Tate–Hochschild cohomology for finite dimensional Hopf algebras

Abstract Let A be any finite dimensional Hopf algebra over a field k . We specify the Tate and Tate–Hochschild cohomology for A and introduce cup products that make them become graded rings. We establish the relationship between these rings. In particular, the Tate–Hochschild cohomology of A is isomorphic (as algebras) to its Tate cohomology with coefficients in an adjoint module. Consequently, the Tate cohomology ring of A is a direct summand of its Tate–Hochschild cohomology ring. As an example, we explicitly compute both the Tate and Tate–Hochschild cohomology for the Sweedler algebra H 4 .