The Jacobian, Reflection Arrangement and Discriminant for Reflection Hopf Algebras

We study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting.

bm-Central Limit Theorems associated with non-symmetric positive cones

Analogues of the classical Central Limit Theorem are proved in the noncommutative setting of random variables which are bmindependent and indexed by elements of positive non-symmetric cones, such as the circular cone, sectors in Euclidean spaces and the Vinberg cone. The geometry of the cones is shown to play a crucial role and the related volume characteristics of the cones is shown.

Primeness in Quantales

In this paper we propose a new concept of primeness in quantales. It is proved that this concept coincide with classical definition in commutative quantales, but no longer valid in the noncommutative setting. Also, the notions of strong and uniform strong primeness are investigated.

Noncommutative Blackwell–Ross martingale inequality

We establish a noncommutative Blackwell–Ross inequality for supermartingales under a suitable condition which generalizes Khan’s work to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY

The aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $.

A noncommutative Gauss map

The aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra $\mathfrak A$ considered separately by F. Boca and D. Mundici. The center of $\mathfrak A$ is isomorphic to $C[0,1]$, so we first consider the action of the Gauss map on $C[0,1]$ and then extend the map to $\mathfrak A$ and show that the extension inherits many desirable properties.

Noncommutative Symmetric Hall-Littlewood Polynomials

International audience Noncommutative symmetric functions have many properties analogous to those of classical (commutative) symmetric functions. For instance, ribbon Schur functions (analogs of the classical Schur basis) expand positively in noncommutative monomial basis. More of the classical properties extend to noncommutative setting as I will demonstrate introducing a new family of noncommutative symmetric functions, depending on one parameter. It seems to be an appropriate noncommutative analog of the Hall-Littlewood polynomials. Les fonctions symétriques non commutatives ont de nombreuses propriétés analogues à celles des fonctions symétriques classiques (commutatives). Par exemple, les fonctions de Schur en rubans (analogues de la base de Schur classique) admettent des développements à coefficients positifs dans la base des monômes non commutatifs. La plupart des propriétés classiques s'étendent au cas non commutatif, comme je le montrerai en introduisant une nouvelle famille de fonctions symétriques non commutatives, dépendant d'un paramètre. Cette famille semble être un analogue non commutatif approprié de la famille des polynômes de Hall-Littlewood.