Multiplicative constants and maximal measurable cocycles in bounded cohomology

Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper, we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $\mathrm{PU}(m,1)$ -cocycles of complex hyperbolic lattices.

Asymptotic generalized extended uncertainty principle

We present a formalism which allows for the perturbative derivation of the Extended Uncertainty Principle (EUP) for arbitrary spatial curvature models and observers. Entering the realm of small position uncertainties, we derive a general asymptotic EUP. The leading 2nd order curvature induced correction is proportional to the Ricci scalar, while the 4th order correction features the 0th order Cartan invariant $$\Psi _2$$Ψ2 (a scalar quadratic in curvature tensors) and the curved space Laplacian of the Ricci scalar all of which are evaluated at the expectation value of the position operator i.e. the expected position when performing a measurement. This result is first verified for previously derived homogeneous space models and then applied to other non-trivial curvature related effects such as inhomogeneities, rotation and an anisotropic stress fluid leading to black hole “hair”. Our main achievement combines the method we introduce with the Generalized Uncertainty Principle (GUP) by virtue of deformed commutators to formulate a generic form of what we call the Asymptotic Generalized Extended Uncertainty Principle (AGEUP).

Cartan invariant matrices for finite monoids

International audience Let $M$ be a finite monoid. In this paper we describe how the Cartan invariant matrix of the monoid algebra of $M$ over a field $\mathbb{K}$ of characteristic zero can be expressed using characters and some simple combinatorial statistic. In particular, it can be computed efficiently from the composition factors of the left and right class modules of $M$. When $M$ is aperiodic, this approach works in any characteristic, and generalizes to $\mathbb{K}$ a principal ideal domain like $\mathbb{Z}$. When $M$ is $\mathcal{R}$-trivial, we retrieve the formerly known purely combinatorial description of the Cartan matrix. Soit $M$ un monoïde fini. Dans cet article, nous exprimons la matrice des invariants de Cartan de l'algèbre de $M$ sur un corps $\mathbb{K}$ de caractéristique zéro à l'aide de caractères et d'une statistique combinatoire simple. En particulier, elle peut être calculée efficacement à partir des facteurs de compositions des modules de classes à gauche et à droite de $M$. Lorsque $M$ est apériodique, cette approche se généralise à toute caractéristique et aux anneaux principaux comme $\mathbb{Z}$. Lorsque $M$ est $\mathcal{R}$-trivial, nous retrouvons la description combinatoire de la matrice de Cartan précédemment connue.