Abstract:
We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.

Abstract:
In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Boekstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.

Abstract:
In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let $H$ be a semisimple Hopf algebra over a field $\mathbb{k}$ and let $A$ be a left partial $H$-module algebra. We study the $H$-prime and the $H$-Jacobson radicals of $A$ and its relations with the prime and the Jacobson radicals of $\underline{A\#H}$, respectively. In particular, we prove that if $A$ is $H$-semiprimitive, then $\underline{A\#H}$ is semiprimitive provided that all irreducible representations of $A$ are finite-dimensional, or $A$ is an affine PI-algebra over $\mathbb{k}$ and $\mathbb{k}$ is a perfect field, or $A$ is locally finite. Moreover, we prove that $\underline{A\#H}$ is semiprime provided that $A$ is an $H$-semiprime PI-algebra, generalizing for the setting of partial actions, the main results of [20] and [19].

Abstract:
We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. In particular we prove a formula for the crossed effect of the cup-one product operation on unstable homotopy groups of spheres which was claimed by Barratt-Jones-Mahowald.

Abstract:
In this work we study the class of algebras satisfying a duality property with respect to Hochschild homology and cohomology, as in [VdB]. More precisely, we consider the class of algebras $A$ such that there exists an invertible bimodule $U$ and an integer number $d$ with the property $H^{\bullet}(A,M)\cong H_{d-\bullet}(A,U\ot_AM)$, for all $A$-bimodules $M$. We will show that this class is closed under localization and under smash products with respect to Hopf algebras satisfying also the duality property. We also illustrate the subtlety on dualities with smash products developing in detail the example $S(V)#G$, the crossed product of the symmetric algebra on a vector space and a finite group acting linearly on $V$.

Abstract:
Voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety. We prove the conjecture for one dimensional cycles on an arbitrary product of curves. As a consequence we get that numerically trivial 1-cycles on an abelian variety are smash nilpotent.

Abstract:
We analyze some relations between quasi-Hopf smash products and certain twisted tensor products of quasialgebras. Along the way we obtain also some results of independent interest, such as a duality theorem for finite dimensional quasi-Hopf algebras and a universal property for generalized diagonal crossed products.

Abstract:
In this paper, we first generalize the theorem about the existence of an enveloping action to a partial twisted smash product. Second, we construct a Morita context between the partial twisted smash product and the twisted smash product related to the enveloping action. Furthermore, we show some results relating partial actions and partial representations over the partial twisted smash products, which generalize the results of Alves and Batista (Comm. Algebra, 38(8): 2872-2902, 2010). Finally, we present versions of the duality theorems of Blattner-Montgomery for partial twisted smash products.

Abstract:
We give some general results on the ring structure of Hochschild cohomology of smash products of algebras with Hopf algebras. We compute this ring structure explicitly for a large class of finite dimensional Hopf algebras of rank one.