arXiv subjects

Andrea Rivezzi

Publications and source records attributed to Andrea Rivezzi.

6 recordsLinked to original sources

arXiv2026

An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence

We describe infinitesimal deformations of post-Lie algebras and post-Hopf algebras and prove that the adjunction given by the universal enveloping algebra and primitive elements functors is compatible with the infinitesimal structure. When restricted to connected and cocommutative infinitesimal post-Hopf algebras, this becomes an equivalence of categories, which constitutes an extension of the Cartier--Milnor--Moore theorem. We classify infinitesimal post-Lie structures on $\mathfrak{sl}(2)$, and discuss a class of infinitesimal post-Lie algebras emerging from flat connections with covariantly-constant torsion. Moreover, we classify infinitesimal post-Hopf structures on Sweedler's Hopf algebra. Cocommutative infinitesimal post-Hopf algebras induce a Hochschild 2-cocycle on the associated subadjacent Hopf algebra. Finally, we prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, by using a filtered distributive law between the operads of Lie algebras and bi-magmas.

math.QA↗

arXiv2026

On the Quantization-Dequantization Correspondence for (co)Poisson Hopf Algebras

In this paper, we construct a functorial quantization of (co)Poisson Hopf algebras within a broad categorical framework. We further introduce categories naturally associated with (co)Poisson Hopf algebras, namely Drinfeld-Yetter modules. These categories provide a canonical setting in which we define explicit dequantization functors that are inverse to the quantization functors. Using this framework, we also establish functorial (de)quantization results for the corresponding module categories. Finally, we recover the classical results of Etingof and Kazhdan as special cases of our construction and discuss applications to deformation quantization \`a la Tamarkin.

math.QA↗

arXiv2025

Hopf categories associated to comonoidal functors

We provide an explicit construction of Hopf categories associated to comonoidal functors, generalizing \v{S}evera's construction of Hopf monoids through M-adapted functors. We discuss the example of the Hopf category whose underlying class is the set of twists of a Lie bialgebra. Finally, we apply the result to the setting of deformed categories.

math.CT↗

arXiv2025

Quantization of infinitesimal braidings and pre-Cartier quasi-bialgebras

In this paper we extend Cartier's deformation theorem of braided monoidal categories admitting an infinitesimal braiding to the non-symmetric case. The algebraic counterpart of these categories is the notion of a pre-Cartier quasi-bialgebra, which extends the well-known notion of quasitriangular quasi-bialgebra given by Drinfeld. Our result implies that one can quantize the infinitesimal $\mathcal{R}$-matrix of any Cartier quasi-bialgebra. We further discuss the emerging concepts of infinitesimal quantum Yang-Baxter equation and Cartier ring, the latter containing braid groups with additional generators that correspond to infinitesimal braidings. Explicit deformations of the representation categories of the gauge deformed quasitriangular quasi-bialgebras $E(n)$ are provided.

math.QA↗

arXiv2024

On the universal Drinfeld-Yetter algebra

The universal Drinfeld-Yetter algebra is an associative algebra whose co-Hochschild cohomology controls the existence of quantization functors of Lie bialgebras, such as the renowned one due to Etingof and Kazhdan. It was initially introduced by Enriquez and later re-interpreted by Appel and Toledano Laredo as an algebra of endomorphisms in the colored PROP of a Drinfeld-Yetter module over a Lie bialgebra. In this paper, we provide an explicit formula for its structure constants in terms of certain diagrams, which we term Drinfeld-Yetter looms.

math.CO↗

arXiv2023

A gentle introduction to Drinfel'd associators

In this note we give an introduction to Drinfel'd's associator coming from the Knizhnik-Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential geometry: we rather use limits of concrete parallel transports.

math.QA↗