Sophie Chemla
(U Sorbonne & Paris Cité)
Duality for action bialgebroids
We will extend the
smash product construction of Brzeziński-Militaru to the bialgebroid setting.
Starting with a (braided commutative) Yetter-Drinfeld algebra B over a left bialgebroid U, we construct the action left bialgebroid B#U over B.
We study the behaviour of this smash product construction with respect to the linear duality functor and with respect to Drinfeld quantum duality functors.
We characterize those bialgebroids that are smash products. Joint work with Fabio Gavarini and Niels Kowalzig.
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Francesco D'Andrea
(U Napoli Federico II)
On morphisms of Hopf-Galois extensions
The ring structure of the K-theory of a compact space can be reconstructed from the knowledge
of compact principal bundles over the space. The K-theory of a noncommutative C*-algebra has no obvious ring structure,
but one can use Hopf-Galois extensions to construct a replacement of the K-theory ring. I will present some results
about morphisms of Hopf-Galois extensions that are a preparation for the study of the "multiplicative K-theory" of a C*-algebra.
Joint work with T. Maszczyk.
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Fabio Gavarini
(U Roma Tor Vergata)
Quantum group deformations and quantum R-(co)matrices versus
quantum duality principle
I will describe the effect on quantum groups
of deformations by twist and by 2-cocycles,
showing how such deformations affect the semiclassical limit.
We also discuss how these deformation procedures can be extended,
via a formal variation of the original recipes,
using "polar twists" and "polar 2-cocycles":
these new recipes seemingly should make no sense at all,
yet we prove that they do work, thus providing more general
deformation procedures. I will then
explain the underlying motivation: this comes from the quantum duality
principle, through which every polar twist resp. 2-cocycle for a given quantum group can be seen as a standard twist resp. 2-cocycle for a
different quantum group, associated with the original
one via the appropriate Drinfeld functor.
Time permitting, I will sketch how some standard,
well-known constructions involving R-(co)matrices for Hopf algebras (and, in particular, for quantum groups) can be extended
to the case of "polar R-(co)matrices", by just stretching a bit the arguments used for polar twists and polar 2-cocycles.
As a byproduct, I will show how this also yields new symmetries for the underlying pair of dual Poisson (formal) groups
that one gets by specialization.
Joint work with G. A. García.
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Emma Lepri
(U Glasgow)
Prorepresentability of noncommutative deformation functors
Schlessinger's Theorem, which gives necessary and sufficient conditions for a commutative deformation functor to be prorepresentable, does not generalise easily to noncommutative deformations
because of the appearance of inner automorphisms in the category of Artinian rings.
This talk is based on a joint work in progress with Eugenio Landi, where we try to overcome this difficulty using bicategories.
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Hsuan-Yi Liao
(U Tsing Hua Taiwan)
Brown functors of directed graphs
In this talk, we will consider Brown functors of directed graphs — that is, contravariant functors from the
homotopy category of finite directed graphs to the category of abelian groups,
satisfying the triviality axiom, the additivity axiom, and the Mayer-Vietoris axiom.
Our main theorem states that such a functor is representable.
In addition, I will show that the first path cohomology of directed graphs provides a nontrivial example of a Brown functor.
This talk is based on joint work with Zachary McGuirk, Dang Khoa Nguyen, and Byungdo Park.
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Antonio Miti
(U Roma La Sapienza)
A canonical morphism between twisted and untwisted higher Courant Lie infinity algebras
In "Lie infinity algebras and higher analogues of Dirac structures and Courant algebroids" (arXiv:1003.1004), Marco Zambon constructs
a Lie infinity algebra associated with any higher standard Courant algebroid
(also known as a Vinogradov algebroid), and exhibits an explicit Lie infinity morphism
from the Lie algebra associated with a standard Lie algebroid twisted by a closed 2-form to the Lie 2-algebra of the standard Courant algebroid.
He poses the question of whether analogous canonical morphisms
exist in higher degrees
—
namely, for any standard higher Courant algebroids twisted by closed (n+1)-forms.
In this talk, we present a general framework that naturally yields such canonical
Lie infinity morphisms for arbitrary n, clarifying the geometric and homotopical structures
underlying these constructions.
Time permitting, we also discuss how this framework accommodates the canonical morphism between the observable Lie infinity algebra of a pre-n-plectic manifold and the higher Courant algebra we described in "Observables on multisymplectic manifolds and higher Courant algebroids" (arXiv:2209.05836).
Joint work with Domenico Fiorenza.
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Francesca Pratali
(U Sorbonne Paris Nord)
The root functor
Given a topological operad P and a set W of unary operations of P, the localization of P at W is another topological operad where the morphisms in W have been formally inverted (and which is initial with this property). The ideal situation to study localization is when P has discrete spaces of operations, so one does not need to deal with homotopy coherences.
A well-known of this phenomenom is the little disks operad: it is a result of Lurie that this topological operad is weakly equivalent to the discrete operad of disks localized at isotopy equivalences.
In this talk, I will show that, provided we are willing to switch to the formalism of ∞-operads, the same phenomenon happens for every topological operad, that is,
that every topological operad is homotopy equivalent the localization of a discrete one. As an application, we deduce a characterization of algebras over an ∞-operad as that of locally constant algebras over its discrete resolution.
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Mathieu Stiénon
(Penn State U)
Formal exponential maps and the Atiyah class of dg manifolds
Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The
infinite jets of these classical exponential maps are related to Poincaré-Birkhoff-Witt
isomorphisms and
the complete symbols of differential operators. It turns out that these formal exponential maps can be
extended to the context of graded manifolds. For dg manifolds, the formal exponential maps need not be
compatible with the homological vector field and the incompatibility is captured by a cohomology class
reminiscent of the Atiyah class of holomorphic vector bundles. Indeed, the space of vector fields on a dg
manifold carries a natural L∞-algebra
structure whose binary bracket is a cocycle representative of the
Atiyah class of the dg manifold. In particular, the de Rham complex associated with a foliation carries
an L∞-algebra structure akin to the L∞-algebra
structure on the Dolbeault complex of a Kähler manifold
discovered by Kapranov in his work on Rozansky-Witten invariants.
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Thomas Weber
(Charles U Prague)
Quantization of infinitesimal braidings
There is a well-known correspondence between braided monoidal categories and quasitriangular bialgebras: the representation category of a
quasitriangular bialgebra is braided monoidal and, conversely, for every 'nice enough' braided monoidal category, one obtains a quasitriangular bialgebra
via Tannaka-Krein reconstruction.
In this talk, we consider first-order deformations of braided monoidal categories, meaning we introduce infinitesimal braidings,
and discuss the emerging quantization problem. This is done parallelly to the algebraic picture, where infinitesimal
R-matrices and so-called pre-Cartier bialgebras appear. It turns out that the quantization problem admits a solution in terms of Drinfeld associators,
resulting in quasitriangular quasi-bialgebras. As an explicit example, the deformation of the pointed E(n) Hopf algebras is discussed.
This is based on a collaboration with Chiara Esposito, Jonas Schnitzer, and Andrea Rivezzi.
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Ping Xu
(Penn State U)
Kapranov L-infinity algebras
In his study of Rozansky-Witten invariants, Kapranov discovered a natural L ∞[1]-algebra structure on the Dolbeault complex Ω0,•(TX0,1)
of an arbitrary Kähler manifold X, where all multibrackets are Ω0,•(X)-multilinear
except for the unary bracket. Motivated by this example, we introduce
an abstract notion of Kapranov L-infinity algebras,
and prove that associated to any dg Lie algebroid, there is
a natural Kapranov L-infinity algebra. We also discuss the linearization
problem. Work in progress with Ruggero Bandiera,
Seokbong Seol, and Mathieu Stiénon.
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