arXiv subjects

Sophie Chemla

Publications and source records attributed to Sophie Chemla.

10 recordsLinked to original sources

arXiv2026

Hopf bimodules for bialgebroids

Generalising a result for Hopf algebras, we not only define the four possible types of Hopf modules in the bialgebroid setting but also yield the notion of two-sided two-cosided Hopf modules, also known as Hopf bimodules or tetramodules, in this realm. By explicitly formulating a fundamental theorem for Hopf modules via the concept of Hopf-Galois comodules, we prove that the category of Hopf bimodules can be endowed with the structure of a (pre-)braided monoidal category in two different ways, which, in turn, are shown to be both braided monoidally equivalent to the category of Yetter-Drinfel'd modules, that is, to the monoidal centre of the category of left bialgebroid modules or comodules. As an illustration, we discuss relative Hopf bimodules associated to Ehresmann-Schauenburg bialgebroids.

math.QA↗

arXiv2026

Duality properties for induced and coinduced representations in positive characteristic

Let $k$ be a field of positive characteristic $p>2$. We prove a duality property concerning the kernel of coinduced representations of Lie superalgebras. This property was already proved by M. Duflo for Lie algebras in any characteristic under more restrictive finiteness conditions. It was then generalized to Lie superalgebras in characteristic 0 in previous works of the author. In a second part of the article, we study the links between coinduced representations and induced representations in the case of restricted Lie superalgebras.

math.RT↗

arXiv2025

Duality for action bialgebroids

We study the effect of linear duality on action bialgebroids (also known as smash product or scalar extension bialgebroids) and, for those bearing a quantisation nature, the effect of Drinfeld functors underlying the quantum duality principle. By means of various categorical equivalences, it is shown that any braided commutative Yetter-Drinfeld algebra over any bialgebroid is also a braided commutative Yetter-Drinfeld algebra over the respective dual bialgebroid. This implies that the action bialgebroid of the dual exists, which is then proven to be isomorphic, as a bialgebroid, to the dual of the initial action bialgebroid: in short, (linear) duality commutes with the action bialgebroid construction. Similarly, for quantum groupoids to which the Drinfeld duality functors apply and the quantum duality principle holds, these Drinfeld duality functors are shown to commute with the action bialgebroid construction as well.

math.RA↗

arXiv2023

Differential calculus over double Lie algebroids

The notion of double Lie algebroid was defined by M. Van den Bergh and was illustrated by the double quasi Poisson case. We give new examples of double Lie algebroids and develop a differential calculus in that context. We recover the non commutative de Rham complex and the double Poisson-Lichnerowicz cohomology (Pichereau-vanWeyer) as particular cases of our construction.

math.RA↗

arXiv2021

Poincaré Duality for Hopf algebroids

We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of Van Den Bergh [VDB]. We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds in [CLYZ].

math.RA↗

arXiv2020

Integral theory for left Hopf left bialgebroids

We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but, for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or $u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G. Böhm who studied integral theory for Hopf algebroids. We make use of recent results about left Hopf left bialgebroids. We apply our results to the restricted enveloping algebra of a restricted Lie Rinehart algebra.

math.RA↗

arXiv2016

Duality features of left Hopf algebroids

We explore special features of the pair (U^*, U_*) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S^* from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phung's categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations.

math.RA↗

arXiv2015

Duality functors for quantum groupoids

We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum groupoids", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of "quantum groupoids". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of "quantum groupoids". On the other hand, we develop a suitable version of a "quantum duality principle" for quantum groupoids, which extends the one for quantum groups - dealing with Hopf algebras - originally introduced by Drinfeld (cf. [V. G. Drinfeld, "Quantum groups", Proc. ICM (Berkeley, 1986), 1987, pp. 798-820], sec. 7) and later detailed in [F. Gavarini, "The quantum duality principle", Annales de l'Institut Fourier 53 (2002), 809-834].

math.QA↗

arXiv2009

Duality properties for quantum groups

Some duality properties for induced representations of enveloping algebras involve the character $Trad_{\goth g}$. We extend them to deformation Hopf algebras $A_{h}$ of a noetherian Hopf $k$-algebra $A_{0}$ satistying $Ext^{i}_{A_{0}}(k, A_{0})=\{0\}$ except for $i=d$ where it is isomorphic to $k$. These duality properties involve the character of $A_{h}$ defined by right multiplication on the one dimensional free $k[[h]]$-module $Ext^{d}_{A_{h}} (k[[h]], A_{h})$. In the case of quantized enveloping algebras, this character lifts the character $Trad_{\goth g}$. We also prove Poincar{é} duality for such deformation Hopf algebras in the case where $A_{0}$ is of finite homological dimension. We explain the relation of our construction with quantum duality.

math.QA↗

arXiv2008

Formality theorem with coefficients in a module

In this article, $X$ will denote a ${\cal C}^{\infty}$ manifold. In a very famous article, Kontsevich showed that the differential graded Lie algebra (DGLA) of polydifferential operators on $X$ is formal. Calaque extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid $E$ over $X$, he defined the DGLA of $E$-polydifferential operators, $Γ(X, ^{E}D^{*}_{poly})$, and showed that it is formal. Denote by $Γ(X, ^{E}T^{*}_{poly})$ the DGLA of $E$-polyvector fields. Considering $M$, a module over $E$, we define $Γ(X, ^{E}T_{poly}^{*}(M))$ the $Γ(X, ^{E}T^{*}_{poly})$-module of $E$-polyvector fields with values in $M$. Similarly, we define the $Γ(X, ^{E}D^{*}_{poly})$-module of $E$-polydifferential operators with values in $M$, $Γ(X, ^{E}D^{*}_{poly}(M))$. We show that there is a quasi-isomorphism of $L_{\infty}$-modules over $Γ(X, ^{E}T^{*}_{poly})$ from $Γ(X, ^{E}T^{*}_{poly}(M))$ to $Γ(X, ^{E}D^{*}_{poly}(M))$. Our result extends Calaque 's (and Kontsevich's) result.

math.QA↗