Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe twisting procedure
84
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homoto
rate research
Read More
Given a double cover $pi: mathcal{G} rightarrow hat{mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $tau_{pi}$ and $tau_{pi}^{ref}$, thereby associating to a Jandl $n$-gerbe $hat{lambda}$ on $hat{mathcal{G}}$ a Jandl $(n-1)$-gerbe $tau_{pi}(hat{lambda})$ on the quotient loop groupoid of $mathcal{G}$ and an ordinary $(n-1)$-gerbe $tau^{ref}_{pi}(hat{lambda})$ on the unoriented quotient loop groupoid of $mathcal{G}$. For $n =1,2$, we interpret the character theory (resp. centre) of the category of Real $hat{lambda}$-twisted $n$-vector bundles over $hat{mathcal{G}}$ in terms of flat sections of the $(n-1)$-vector bundle associated to $tau_{pi}^{ref}(hat{lambda})$ (resp. the Real $(n-1)$-vector bundle associated to $tau_{pi}(hat{lambda})$). We relate our results to Re
In this short note we describe an alternative global version of the twisting procedure used by Dolgushev to prove formality theorems. This allows us to describe the maps of Fedosov resolutions, which are key factors of the formality morphisms, in terms of a twist of the fiberwise quasi-isomorphisms induced by the local formality theorems proved by Kontsevich and Shoikhet. The key point consists in considering $L_infty$-resolutions of the Fedosov resolutions obtained by Dolgushev and an adapted notion of Maurer-Cartan element. This allows us to perform the twisting of the quasi-isomorphism intertwining them in a global manner.
We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2-cocycles J for the function algebra O(G)), which was started in [EG1,EG2], and initiate the study of the associated one-sided twisted function algebras O(G)_J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G)_J is finitely generated with center O(G/H). We then use it to study the structure of O(G)_J for connected nilpotent G. We show that in this case O(G)_J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2-cocycles for connected nilpotent G, hence of fiber functors Rep(G)to Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G)_J for arbitrary G.
We use cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}_Jmathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $mathbb{C}$. In particular, we show that ${}_Jmathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_Jmathcal{O}(G)_{J}cong U(g)$ as algebras, where $g={rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}_Jmathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $Hsubset G$ of $J$. Finally, we work out several examples to illustrate our results.
We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $mathcal{C}$ is graded over a group $Gamma$, a torus-valued 3-cocycle on $Gamma$ can be used to deform the associator of $mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Gamma$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
