Related papers: Operadic Twisting -- with an application to Delign…
An algebraic left Kan extension is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of…
We introduce the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others. We prove that, under some mild conditions, a connected generalized bialgebra is completely determined by its…
We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres in Connes and Landi [8] and in Connes and Dubois Violette [7], by using the differential and integral calculus on these spaces that is…
We endow twisted tensor products with a natural notion of counit and comultiplication, and we provide sufficient and necessary conditions making the twisted tensor product a counital coassociative coalgebra. We then characterize when the…
Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and…
In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and…
From the `cofree' cooperad $T'(A[-1])$ on a collection $A$ together with a differential, we construct an $L_\infty$-algebra structure on the total space $\bigoplus_nA(n)$ that descends to coinvariants. We use this construction to define an…
The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In…
We introduce an operad which acts on the Gerstenhaber-Schack complex of a prestack as defined by Dinh Van and Lowen, and which in particular allows us to endow this complex with an underlying $L_{\infty}$-structure. We make use of the…
We define, for a somewhat standard forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operad" functors, the right inverse and the left adjoint of the forgetful functor. Those…
The purpose of this exposition is to compare the constructions of classical nonsymmetric operads (and their algebras) to that of the globular operads of Leinster and Batanin. It is hoped that, through this comparison, understanding algebras…
In this paper, we investigate a series of W-type differential operators, which appear naturally in the symmetry algebras of KP and BKP hierarchies. In particular, they include all operators in the W-constraints for tau functions of higher…
In this paper, we prove two Kastler-Kalau-Walze type theorems for conformal perturbations of twisted Dirac operators and conformal perturbations of twisted signature operators on four-dimensional manifolds with (resp. without) boundary.
The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The $\kappa$-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are…
Algebraic operads provide a powerful tool to understand the homotopy theory of the types of (co)algebras they encode. So far, the principal results and methods that this theory provides were only available in characteristic zero. The reason…
We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of…
A d-contraction is a d-tuple $(T_1,...,T_d)$ of mutually commuting operators acting on a common Hilbert space H such that $ \|T_1\xi_1+T_2\xi_2+... +T_d\xi_d\|^2\leq \|\xi_1\|^2+\|\xi_2\|^2+...+\|\xi_d\|^2 $ for all…
In the companion paper arXiv:2110.05298, we developed the deformation theory of symplectic foliations, focusing on geometric aspects. Here, we address some algebraic questions that arose naturally. We show that the $L_{\infty}$-algebra…
This contribution studies a specific deformation of algebras with anti-involution. Starting with the observation that twisting the multiplication of such an algebra by its anti-involution generates a Hom-associative algebra of type II, it…