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Journal articles on the topic 'Representations up to homotopy'

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Author: Grafiati

Published: 21 July 2023

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1

Trentinaglia, Giorgio, and Chenchang Zhu. "Some remarks on representations up to homotopy." International Journal of Geometric Methods in Modern Physics 13, no. 03 (March 2016): 1650024. http://dx.doi.org/10.1142/s0219887816500249.

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Motivated by the study of the interrelation between functorial and algebraic quantum field theory (AQFT), we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomology. Furthermore, we observe that the derived representation category of any compact group is equivalent to the category of ordinary (finite-dimensional) representations of the group.

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2

Porter, Tim, and Jim Stasheff. "Homotopy Coherent Representations." Symmetry 14, no. 3 (March 9, 2022): 553. http://dx.doi.org/10.3390/sym14030553.

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Homotopy coherence has a considerable history, albeit also by other names. For this volume highlighting symmetries, the appropriate use is homotopy coherence of representations, at one time known as representations up to homotopy/homotopy coherent representations. We present a brief semi-historical survey, providing some links that may not be common knowledge.

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3

Sheng, Yunhe, and Chenchang Zhu. "Semidirect products of representations up to homotopy." Pacific Journal of Mathematics 249, no. 1 (January 3, 2011): 211–36. http://dx.doi.org/10.2140/pjm.2011.249.211.

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4

Merati, S., and M. R. Farhangdoost. "Representation up to homotopy of hom-Lie algebroids." International Journal of Geometric Methods in Modern Physics 15, no. 05 (April 2, 2018): 1850074. http://dx.doi.org/10.1142/s0219887818500743.

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A hom-Lie algebroid is a vector bundle together with a Lie algebroid like structure which is twisted by a homomorphism. In this paper, we use the idea of representations up to homotopy of Lie algebroids to construct a same structure for hom-Lie algebroids and we will explain how representations up to homotopy of length 1 are related to extensions of hom-Lie algebroids.

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5

VITAGLIANO, LUCA. "Representations of Homotopy Lie–Rinehart Algebras." Mathematical Proceedings of the Cambridge Philosophical Society 158, no. 1 (December 4, 2014): 155–91. http://dx.doi.org/10.1017/s0305004114000541.

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AbstractI propose a definition of left/right connection along a strong homotopy Lie–Rinehart algebra. This allows me to generalise simultaneously representations up to homotopy of Lie algebroids and actions ofL∞algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie–Rinehart connections.

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6

Gracia-Saz, A., M. Jotz Lean, K. C. H. Mackenzie, and R. A. Mehta. "Double Lie algebroids and representations up to homotopy." Journal of Homotopy and Related Structures 13, no. 2 (July 7, 2017): 287–319. http://dx.doi.org/10.1007/s40062-017-0183-1.

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7

Mehta, Rajan Amit. "Lie algebroid modules and representations up to homotopy." Indagationes Mathematicae 25, no. 5 (October 2014): 1122–34. http://dx.doi.org/10.1016/j.indag.2014.07.013.

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8

Jotz, M. "Obstructions to representations up to homotopy and ideals." Asian Journal of Mathematics 26, no. 2 (2022): 137–66. http://dx.doi.org/10.4310/ajm.2022.v26.n2.a1.

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9

Drummond, T., M. Jotz Lean, and C. Ortiz. "VB-algebroid morphisms and representations up to homotopy." Differential Geometry and its Applications 40 (June 2015): 332–57. http://dx.doi.org/10.1016/j.difgeo.2015.03.005.

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10

Arias Abad, Camilo, and Florian Schätz. "Deformations of Lie brackets and representations up to homotopy." Indagationes Mathematicae 22, no. 1-2 (October 2011): 27–54. http://dx.doi.org/10.1016/j.indag.2011.07.003.

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11

WILKIN, GRAEME. "HOMOTOPY GROUPS OF MODULI SPACES OF STABLE QUIVER REPRESENTATIONS." International Journal of Mathematics 21, no. 09 (September 2010): 1219–38. http://dx.doi.org/10.1142/s0129167x1000646x.

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The purpose of this paper is to describe a method for computing homotopy groups of the space of α-stable representations of a quiver with fixed dimension vector and stability parameter α. The main result is that the homotopy groups of this space are trivial up to a certain dimension, which depends on the quiver, the choice of dimension vector, and the choice of parameter. As a corollary we also compute low dimensional homotopy groups of the moduli space of α-stable representations of the quiver with fixed dimension vector, and apply the theory to the space of non-degenerate polygons in three-dimensional Euclidean space.

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12

Sheng, Yunhe, and Chenchang Zhu. "Higher extensions of Lie algebroids." Communications in Contemporary Mathematics 19, no. 03 (April 5, 2017): 1650034. http://dx.doi.org/10.1142/s0219199716500346.

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We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integrating an exact Courant algebroid.

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13

Arias Abad, Camilo, and Marius Crainic. "Representations up to homotopy and Bottʼs spectral sequence for Lie groupoids." Advances in Mathematics 248 (November 2013): 416–52. http://dx.doi.org/10.1016/j.aim.2012.12.022.

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14

Brahic, Olivier, and Cristian Ortiz. "Integration of $2$-term representations up to homotopy via $2$-functors." Transactions of the American Mathematical Society 372, no. 1 (March 19, 2019): 503–43. http://dx.doi.org/10.1090/tran/7586.

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15

Arias Abad, Camilo, and Florian Schätz. "The A∞ de Rham Theorem and Integration of Representations up to Homotopy." International Mathematics Research Notices 2013, no. 16 (July 4, 2012): 3790–855. http://dx.doi.org/10.1093/imrn/rns166.

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16

Vélez, Alexander Quintero. "Boundary Coupling of Lie Algebroid Poisson Sigma Models and Representations up to Homotopy." Letters in Mathematical Physics 102, no. 1 (March 3, 2012): 31–64. http://dx.doi.org/10.1007/s11005-012-0549-6.

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17

Puschnigg, Michael. "Finitely summable Fredholm modules over higher rank groups and lattices." Journal of K-Theory 8, no. 2 (December 23, 2010): 223–39. http://dx.doi.org/10.1017/is010011023jkt131.

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AbstractWe give a classification (up to smooth homotopy) of finitely summable Fredholm representations (Fredholm modules) over higher rank groups and lattices. Our results are a direct consequence of work of Bader, Furman, Gelander and Monod on generalizations of Kazhdan's property T for locally compact groups.

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18

Reinhold, Ben. "L∞-algebras and their cohomology." Emergent Scientist 3 (2019): 4. http://dx.doi.org/10.1051/emsci/2019003.

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We give an overview of different characterisations of L∞-structures in terms of symmetric brackets and (co)differentials on the symmetric (co)algebra. We then do the same for their representations (up to homotopy) and approach L∞-algebra cohomology using the commutator bracket on the space of coderivations of the symmetric coalgebra. This leads to abelian extensions of L∞-algebras by 2-cocycles.

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19

Merati, S., M. R. Farhangdoost, and A. R. Attari Polsangi. "Representation up to Homotopy and Hom-Lie Algebroid Modules." Journal of Dynamical Systems and Geometric Theories 18, no. 1 (January 2, 2020): 27–37. http://dx.doi.org/10.1080/1726037x.2020.1788817.

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20

Merati, S., and M. R. Farhangdoost. "Representation up to homotopy of double algebroids and their transgression classes." Journal of Dynamical Systems and Geometric Theories 16, no. 1 (January 2, 2018): 89–99. http://dx.doi.org/10.1080/1726037x.2018.1436269.

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21

Rutter, J. W. "The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 120, no. 1-2 (1992): 47–60. http://dx.doi.org/10.1017/s0308210500014979.

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SynopsisWe give here a group extension sequence for calculating, for a non-simply-connected space X, the group of self-homotopy-equivalence classes which induce the identity automorphism of the fundamental group, that is the kernel of the representation → aut (π1(X)). This group extension sequence gives in terms of , where Xn is the n-th stage of a Postnikov decomposition. As special cases, we calculate for non-simply-connected spaces having at most two non-trivial homotopy groups, in dimensions 1 and n, as the unit group of a semigroup structure on ; and we calculate up to extension for non-simply-connected spaces having at most three non-trivial homotopy groups. The group is, for nice spaces, isomorphic to the groups and of self-homotopy-equivalence classes of X in the categories top*M and top M, respectively, where X→M = K1(π1(X)) is a top fibration which determines an isomorphism of the fundamental group; and our results are obtained initially in topM.

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22

del Hoyo, Matias, and Cristian Ortiz. "Morita Equivalences of Vector Bundles." International Mathematics Research Notices 2020, no. 14 (June 26, 2018): 4395–432. http://dx.doi.org/10.1093/imrn/rny149.

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Abstract We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to prove the Morita invariance of VB-cohomology, with implications to deformation cohomology of Lie groupoids and of classic geometries. We discuss applications of our theory to Poisson geometry, providing a new insight over Marsden–Weinstein reduction and the integration of Dirac structures. We conclude by proving that the derived category of VB-groupoids is a Morita invariant, which leads to a notion of VB-stacks, and solves (an instance of) an open question on representations up to homotopy.

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23

Grady, Ryan, and Owen Gwilliam. "LIE ALGEBROIDS AS SPACES." Journal of the Institute of Mathematics of Jussieu 19, no. 2 (February 13, 2018): 487–535. http://dx.doi.org/10.1017/s1474748018000075.

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In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

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24

SHENG, YUNHE, and CHENCHANG ZHU. "INTEGRATION OF SEMIDIRECT PRODUCT LIE 2-ALGEBRAS." International Journal of Geometric Methods in Modern Physics 09, no. 05 (July 3, 2012): 1250043. http://dx.doi.org/10.1142/s0219887812500430.

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The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite-dimensional case.

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25

Merati, S., and M. R. Farhangdoost. "VB-Hom Algebroid Morphisms and 2-Term Representation Up to Homotopy of Hom-Lie Algebroids." Iranian Journal of Science and Technology, Transactions A: Science 45, no. 3 (March 30, 2021): 937–44. http://dx.doi.org/10.1007/s40995-020-01049-1.

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26

Shepherd, Kendrick M., Xianfeng David Gu, René R. Hiemstra, and Thomas J. R. Hughes. "Quadrilateral layout generation and optimization using equivalence classes of integral curves: theory and application to surfaces with boundaries." Journal of Mechanics 38 (2022): 128–55. http://dx.doi.org/10.1093/jom/ufac002.

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Abstract Extracting quadrilateral layouts from surface triangulations is an important step in texture mapping, semi-structured quadrilateral meshing for traditional analysis and spline reconstruction for isogeometric analysis. Current methods struggle to yield high-quality layouts with appropriate connectivity between singular nodes (known as “extraordinary points” for spline representations) without resorting to either mixed-integer optimization or manual constraint prescription. The first of these is computationally expensive and comes with no guarantees, while the second is laborious and error-prone. In this work, we rigorously characterize curves in a quadrilateral layout up to homotopy type and use this information to quickly define high-quality connectivity constraints between singular nodes. The mathematical theory is accompanied by appropriate computational algorithms. The efficacy of the proposed method is demonstrated in generating quadrilateral layouts on the United States Army’s DEVCOM Generic Hull vehicle and parts of a bilinear quadrilateral finite element mesh (with some linear triangles) of a 1996 Dodge Neon.

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27

Nagasaki, Ikumitsu. "Linearity of homotopy representations, II." Manuscripta Mathematica 82, no. 1 (December 1994): 277–92. http://dx.doi.org/10.1007/bf02567702.

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28

Anderson, Laura. "Homotopy Sphere Representations for Matroids." Annals of Combinatorics 16, no. 2 (January 26, 2012): 189–202. http://dx.doi.org/10.1007/s00026-012-0125-x.

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29

Lubawski, Wojciech, and Krzysztof Ziemiański. "Homotopy representations of the unitary groups." Algebraic & Geometric Topology 16, no. 4 (September 12, 2016): 1913–51. http://dx.doi.org/10.2140/agt.2016.16.1913.

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30

Hambleton, Ian, and Ergün Yalçin. "Homotopy representations over the orbit category." Homology, Homotopy and Applications 16, no. 2 (2014): 345–69. http://dx.doi.org/10.4310/hha.2014.v16.n2.a19.

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31

Guérin, Clément, Sean Lawton, and Daniel Ramras. "Bad Representations and Homotopy of Character Varieties." Annales Henri Lebesgue 5 (February 23, 2022): 93–140. http://dx.doi.org/10.5802/ahl.119.

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32

Stasheff, Jim. "Constrained Poisson algebras and strong homotopy representations." Bulletin of the American Mathematical Society 19, no. 1 (July 1, 1988): 287–91. http://dx.doi.org/10.1090/s0273-0979-1988-15645-5.

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33

Bradlow, Steven B., Oscar García-Prada, and Peter B. Gothen. "Homotopy groups of moduli spaces of representations." Topology 47, no. 4 (September 2008): 203–24. http://dx.doi.org/10.1016/j.top.2007.06.001.

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34

NAGASAKI, Ikumitsu. "On homotopy representations with the same dimension function." Journal of the Mathematical Society of Japan 40, no. 1 (January 1988): 35–51. http://dx.doi.org/10.2969/jmsj/04010035.

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35

Crawley, Timothy, and Arthur G. Palmer III. "Approximate representations of shaped pulses using the homotopy analysis method." Magnetic Resonance 2, no. 1 (April 16, 2021): 175–86. http://dx.doi.org/10.5194/mr-2-175-2021.

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Abstract. The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation or coupling interactions can be described by three Euler angles. The Euler angles, in turn, can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist only for rectangular and hyperbolic-secant pulses. The homotopy analysis method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in nuclear magnetic resonance (NMR) spectroscopy. The results of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The results are extended in a second application of the homotopy analysis method to represent relaxation as a perturbation of the magnetization trajectory calculated in the absence of relaxation. The homotopy analysis method is powerful and flexible and is likely to have other applications in magnetic resonance.

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36

Enochs, Edgar E., and Ivo Herzog. "A Homotopy of Quiver Morphisms with Applications to Representations." Canadian Journal of Mathematics 51, no. 2 (April 1, 1999): 294–308. http://dx.doi.org/10.4153/cjm-1999-015-0.

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AbstractIt is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations. Such a decomposition enables one to describe the right adjoint of the restriction of the representation functor along a morphism of quivers having this path lifting property. These right adjoint functors are used to construct injective representations of quivers. As an application, the injective representations of the cyclic quivers are classified when the base ring is left noetherian. In particular, the indecomposable injective representations are described in terms of the injective indecomposable R-modules and the injective indecomposable R[x; x−1]-modules.

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37

Leykin, Anton, and Daniel Plaumann. "Determinantal representations of hyperbolic curves via polynomial homotopy continuation." Mathematics of Computation 86, no. 308 (February 16, 2017): 2877–88. http://dx.doi.org/10.1090/mcom/3194.

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38

Bauer, Stefan. "Dimension functions of homotopy representations for compact Lie groups." Mathematische Annalen 280, no. 2 (March 1988): 247–65. http://dx.doi.org/10.1007/bf01456053.

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39

Vazquez-Leal, H., V. M. Jimenez-Fernandez, B. Benhammouda, U. Filobello-Nino, A. Sarmiento-Reyes, A. Ramirez-Pinero, A. Marin-Hernandez, and J. Huerta-Chua. "Modified Hyperspheres Algorithm to Trace Homotopy Curves of Nonlinear Circuits Composed by Piecewise Linear Modelled Devices." Scientific World Journal 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/938598.

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We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation.

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40

Anick, David J. "Hopf algebras up to homotopy." Journal of the American Mathematical Society 2, no. 3 (September 1, 1989): 417. http://dx.doi.org/10.1090/s0894-0347-1989-0991015-7.

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41

Blanc, David, and Boris Chorny. "Representability theorems, up to homotopy." Proceedings of the American Mathematical Society 148, no. 3 (November 13, 2019): 1363–72. http://dx.doi.org/10.1090/proc/14828.

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42

Bruce, Andrew James, and Alfonso Giuseppe Tortorella. "Kirillov structures up to homotopy." Differential Geometry and its Applications 48 (October 2016): 72–86. http://dx.doi.org/10.1016/j.difgeo.2016.06.005.

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43

NOTBOHM, D. "HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS." Journal of the London Mathematical Society 64, no. 2 (October 2001): 472–88. http://dx.doi.org/10.1112/s0024610701002459.

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For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

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44

BRIGHTWELL, MARK, and PAUL TURNER. "REPRESENTATIONS OF THE HOMOTOPY SURFACE CATEGORY OF A SIMPLY CONNECTED SPACE." Journal of Knot Theory and Its Ramifications 09, no. 07 (November 2000): 855–64. http://dx.doi.org/10.1142/s0218216500000487.

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We discuss the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.

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45

Asadollahi, J., H. Eshraghi, R. Hafezi, and Sh Salarian. "On the homotopy categories of projective and injective representations of quivers." Journal of Algebra 346, no. 1 (November 2011): 101–15. http://dx.doi.org/10.1016/j.jalgebra.2011.08.028.

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46

Das, Apurba. "Hom-associative algebras up to homotopy." Journal of Algebra 556 (August 2020): 836–78. http://dx.doi.org/10.1016/j.jalgebra.2020.03.020.

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47

MELLOR, BLAKE. "FINITE TYPE LINK HOMOTOPY INVARIANTS." Journal of Knot Theory and Its Ramifications 08, no. 06 (September 1999): 773–87. http://dx.doi.org/10.1142/s0218216599000481.

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In [2], Bar-Natan used unitrivalent diagrams to show that finite type invariants classify string links up to homotopy. In this paper, I will construct the correct spaces of chord diagrams and unitrivalent diagrams for links up to homotopy. I will use these spaces to show that, far from classifying links up to homotopy, the only rational finite type invariants of link homotopy are the linking numbers of the components.

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48

Kanenobu, Taizo, and Toshio Sumi. "Suciu’s ribbon 2-knots with isomorphic group." Journal of Knot Theory and Its Ramifications 29, no. 07 (June 2020): 2050053. http://dx.doi.org/10.1142/s0218216520500534.

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Suciu constructed infinitely many ribbon 2-knots in [Formula: see text] whose knot groups are isomorphic to the trefoil knot group. They are distinguished by the second homotopy groups. We classify these knots by using [Formula: see text]-representations of the fundamental groups of the 2-fold branched covering spaces.

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49

Merkulov, Sergei, and Thomas Willwacher. "Classification of universal formality maps for quantizations of Lie bialgebras." Compositio Mathematica 156, no. 10 (October 2020): 2111–48. http://dx.doi.org/10.1112/s0010437x20007381.

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We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \] satisfying certain boundary conditions, where $\mathcal {A}\textit{ssb}_\infty$ is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator $\mathfrak{A}$ there is an associated ${\mathcal {L}} ie_\infty$ quasi-isomorphism between the ${\mathcal {L}} ie_\infty$ algebras $\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$ and $\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$ controlling, respectively, deformations of the standard bialgebra structure in $\odot V$ and deformations of any given Lie bialgebra structure in $V$. We study the deformation complex of an arbitrary universal formality morphism $\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$ and prove that it is quasi-isomorphic to the full (i.e. not necessary connected) version of the graph complex introduced Maxim Kontsevich in the context of the theory of deformation quantizations of Poisson manifolds. This result gives a complete classification of the set $\{F_\mathfrak{A}\}$ of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck–Teichmüller group $GRT=GRT_1\rtimes {\mathbb {K}}^*$ and can hence can be identified with the set $\{\mathfrak{A}\}$ of Drinfeld associators.

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50

Traczyk, Paweł. "The cancellation problem for homotopy equivalent representations of finite groups: a survey." Banach Center Publications 18, no. 1 (1986): 205–13. http://dx.doi.org/10.4064/-18-1-205-213.

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