Journal articles: 'Maurer-Cartan element'

1

Ward, Benjamin. "Maurer–Cartan elements and cyclic operads." Journal of Noncommutative Geometry 10, no. 4 (2016): 1403–64. http://dx.doi.org/10.4171/jncg/263.

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Chen, Zhuo, Mathieu Stiénon, and Ping Xu. "Geometry of Maurer-Cartan Elements on Complex Manifolds." Communications in Mathematical Physics 297, no. 1 (March 31, 2010): 169–87. http://dx.doi.org/10.1007/s00220-010-1029-4.

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Das, Apurba, and Satyendra Kumar Mishra. "The L∞-deformations of associative Rota–Baxter algebras and homotopy Rota–Baxter operators." Journal of Mathematical Physics 63, no. 5 (May 1, 2022): 051703. http://dx.doi.org/10.1063/5.0076566.

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A relative Rota–Baxter algebra is a triple ( A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra ( A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra ( A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

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Buijs, Urtzi, Yves Félix, Aniceto Murillo, and Daniel Tanré. "Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes." Canadian Mathematical Bulletin 60, no. 3 (September 1, 2017): 470–77. http://dx.doi.org/10.4153/cmb-2017-003-7.

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AbstractIn a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

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Chtioui, T., A. Hajjaji, S. Mabrouk, and A. Makhlouf. "Cohomology and deformations of twisted O-operators on 3-Lie algebras." Filomat 37, no. 21 (2023): 6977–94. http://dx.doi.org/10.2298/fil2321977c.

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The purpose of this paper is to introduce and study twisted O-operators on 3-Lie algebras. We construct an L?-algebra whose Maurer-Cartan elements are twisted O-operators and define a cohomology of a twisted O-operator T as the Chevalley-Eilenberg cohomology of a certain 3-Lie algebra induced by T with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of twisted O-operators.

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Liu, Jiefeng, and Yunhe Sheng. "Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids." Journal of Noncommutative Geometry 15, no. 1 (January 21, 2021): 147–93. http://dx.doi.org/10.4171/jncg/398.

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Das, Apurba. "Cohomology and deformations of weighted Rota–Baxter operators." Journal of Mathematical Physics 63, no. 9 (September 1, 2022): 091703. http://dx.doi.org/10.1063/5.0093066.

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Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras.

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Xu, Senrong, Wei Wang, and Jia Zhao. "Twisted Rota-Baxter operators on Hom-Lie algebras." AIMS Mathematics 9, no. 2 (2023): 2619–40. http://dx.doi.org/10.3934/math.2024129.

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<abstract><p>Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit $ L_{\infty} $-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebra s. Additionally, we employ Getzler's technique of twisting $ L_\infty $-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra.</p></abstract>

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Goncharov, Alexander B. "Hodge correlators." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 748 (March 1, 2019): 1–138. http://dx.doi.org/10.1515/crelle-2016-0013.

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Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

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HAAK, G., M. SCHMIDT, and R. SCHRADER. "GROUP THEORETIC FORMULATION OF THE SEGAL-WILSON APPROACH TO INTEGRABLE SYSTEMS WITH APPLICATIONS." Reviews in Mathematical Physics 04, no. 03 (September 1992): 451–99. http://dx.doi.org/10.1142/s0129055x92000121.

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A general group theoretic formulation of integrable systems is presented. The approach generalizes the discussion of the KdV equations of Segal and Wilson based on ideas of Sato. The starting point is the construction of commuting flows on the group via left multiplication with elements from an abelian subgroup. The initial data are then coded by elements, called abstract scattering data, in a certain coset space. The resulting equations of motion are then derived from a suitably formulated Maurer-Cartan equation (zero curvature condition) given an abstract Birkhoff factorization. The resulting equations of motion are of the Zakharov-Shabat type. In the case of flows periodic in x-space, the integrals of motion have a natural group theoretic interpretation. A first example is provided by the generalized nonlinear Schrödinger equation, first studied by Fordy and Kulish with integrals of motion which may be local or nonlocal. A suitable reduction gives the mKdV equations of Drinfeld and Sokolov. On the level of abstract scattering data the generalized Miura transformation from solutions of the mKdV equations to the KdV type equations is then just the canonical map from a coset space to a double coset space. This group theoretic approach is related to the algebraic geometric discussion of integrable systems via an affine map from the abelian group describing flows restricted to a suitable set of abstract scattering data, called algebraic geometric, onto a connected component of the Picard variety.

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Wang, Qi, Yunhe Sheng, Chengming Bai, and Jiefeng Liu. "Nijenhuis operators on pre-Lie algebras." Communications in Contemporary Mathematics 21, no. 07 (October 10, 2019): 1850050. http://dx.doi.org/10.1142/s0219199718500505.

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First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

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12

Jackson, Christopher S., and Carlton M. Caves. "How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of SL(2,C)." Quantum 7 (August 16, 2023): 1085. http://dx.doi.org/10.22331/q-2023-08-16-1085.

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The generalized Q-function of a spin system can be considered the outcome probability distribution of a state subjected to a measurement represented by the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As fundamental as the SCS POVM is to the 2-sphere phase-space representation of spin systems, it has only recently been reported that the SCS POVM can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of curved-phase-space correspondences for quantum systems. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators of any spin j are shown to represent elements of the Lie group SL(2,C)≅Spin(3,C), a complex version of the usual unitary operators that represent elements of SU(2)≅Spin(3,R). Consequently, the associated POVM elements represent points in the symmetric space SU(2)∖SL(2,C), which can be recognized as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path integral, (Fokker-Planck) diffusion equation, and stochastic differential equations, are applied to show that the continuous isotropic POVM quickly limits to the SCS POVM, placing spherical phase space at the boundary of the fundamental Lie group SL(2,C) in an operationally meaningful way. Two basic mathematical tools are used to analyze the evolving Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan, decomposition associated with the symmetric pair SU(2) ⊂ SL(2,C). Informed by these tools, the three schochastic techniques are applied directly to the Kraus operators in a representation-independent – and therefore geometric – way (independent of any spectral information about the spin components).The Kraus-operator-centric, geometric treatment applies not just to SU(2) ⊂ SL(2,C), but also to any compact semisimple Lie group and its complexification. The POVM associated with the continuous isotropic measurement of Lie-group generators thus corresponds to a type-IV globally Riemannian symmetric space and limits to the POVM of generalized coherent states. This generalization is the focus of a sequel to this article.

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Leung, Naichung Conan, Ziming Nikolas Ma, and Matthew B. Young. "Refined Scattering Diagrams and Theta Functions From Asymptotic Analysis of Maurer–Cartan Equations." International Mathematics Research Notices, October 8, 2019. http://dx.doi.org/10.1093/imrn/rnz220.

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Abstract We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer–Cartan elements of a (dg) Lie algebra constructed from a (not necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich–Soibelman, Gross–Siebert, and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer–Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer–Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with nondegenerate skew-symmetrized Euler forms.

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Chtioui, Taoufik, Atef Hajjaji, Sami Mabrouk, and Abdenacer Makhlouf. "Cohomologies and deformations of O-operators on Lie triple systems." Journal of Mathematical Physics 64, no. 8 (August 1, 2023). http://dx.doi.org/10.1063/5.0118911.

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In this paper, first, we provide a graded Lie algebra whose Maurer–Cartan elements characterize Lie triple system structures. Then, we use it to study cohomology and deformations of O-operators on Lie triple systems by constructing a Lie 3-algebra whose Maurer–Cartan elements are O-operators. Furthermore, we define a cohomology of an O-operator T as the Lie–Yamaguti cohomology of a certain Lie triple system induced by T with coefficients in a suitable representation. Therefore, we consider infinitesimal and formal deformations of O-operators from a cohomological viewpoint. Moreover, we provide relationships between O-operators on Lie algebras and associated Lie triple systems.

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Chuang, Joseph, Julian Holstein, and Andrey Lazarev. "Maurer–Cartan Moduli and Theorems of Riemann–Hilbert Type." Applied Categorical Structures, February 6, 2021. http://dx.doi.org/10.1007/s10485-021-09631-3.

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AbstractWe study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several different contexts, Schlessinger–Stasheff type theorems comparing the notions of homotopy and gauge equivalence for Maurer–Cartan elements as well as their categorified versions. As an application, we re-prove and generalize Block–Smith’s higher Riemann–Hilbert correspondence, and develop its analogue for simplicial complexes and topological spaces.

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de Kleijn, Niek, and Felix Wierstra. "On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$-algebra." Journal of Homotopy and Related Structures, September 25, 2021. http://dx.doi.org/10.1007/s40062-021-00290-8.

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AbstractIn this paper, we develop the $$A_\infty $$ A ∞ -analog of the Maurer-Cartan simplicial set associated to an $$L_\infty $$ L ∞ -algebra and show how we can use this to study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of $$A_\infty $$ A ∞ -algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) $$A_\infty $$ A ∞ -algebras to simplicial sets, which sends a complete curved $$A_\infty $$ A ∞ -algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads.

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Liu, Meijun, Jiefeng Liu, and Yunhe Sheng. "Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures." Symmetry, Integrability and Geometry: Methods and Applications, July 13, 2022. http://dx.doi.org/10.3842/sigma.2022.054.

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Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order n deformations to order n+1 deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota-Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota-Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul-Vinberg structures on left-symmetric algebroids.

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Hu, Meiyan, Shuai Hou, Lina Song, and Yanqiu Zhou. "3-pre-Leibniz Algebras, Deformations and Cohomologies of Relative Rota-Baxter Operators on 3-Leibniz Algebras." Journal of Nonlinear Mathematical Physics 31, no. 1 (June 11, 2024). http://dx.doi.org/10.1007/s44198-024-00198-w.

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AbstractIn this paper, first we introduce the notions of 3-pre-Leibniz algebras and relative Rota-Baxter operators on 3-Leibniz algebras. We show that a 3-pre-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the identity map is a relative Rota-Baxter operator. Conversely, a relative Rota-Baxter operator naturally induces a 3-pre-Leibniz algebra. Then we construct a Lie 3-algebra, and characterize relative Rota-Baxter operators as its Maurer-Cartan elements. Consequently, we obtain the $$L_\infty$$ L ∞ -algebra that controls deformations of relative Rota-Baxter operators on 3-Leibniz algebras. Next we define the cohomology of relative Rota-Baxter operators on 3-Leibniz algebras and show that infinitesimal deformations of a relative Rota-Baxter operator are classified by the second cohomology group. Finally, we construct an $$L_\infty$$ L ∞ -algebra whose Maurer-Cartan elements are relative Rota-Baxter 3-Leibniz algebra structures, and define the cohomology of relative Rota-Baxter 3-Leibniz algebras.

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Willwacher, Thomas. "Pre-Lie Pairs and Triviality of the Lie Bracket on the Twisted Hairy Graph Complexes." International Mathematics Research Notices, August 9, 2021. http://dx.doi.org/10.1093/imrn/rnab178.

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Abstract We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the little disks operads. We show that twisting with certain Maurer–Cartan elements trivializes their Lie algebra structure. The result can be used to understand the rational homotopy type of many connected components of these mapping spaces.

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20

Das, Apurba, Shuangjian Guo, and Yufei Qin. "L ∞ -structures and cohomology theory of compatible O-operators and compatible dendriform algebras." Journal of Mathematical Physics 65, no. 3 (March 1, 2024). http://dx.doi.org/10.1063/5.0161898.

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The notion of O-operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O-operators.

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Bonechi, Francesco, Nicola Ciccoli, Camille Laurent-Gengoux, and Ping Xu. "Shifted Poisson Structures on Differentiable Stacks." International Mathematics Research Notices, December 11, 2020. http://dx.doi.org/10.1093/imrn/rnaa293.

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Abstract The purpose of this paper is to investigate $(+1)$-shifted Poisson structures in the context of differential geometry. The relevant notion is that of $(+1)$-shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of $(+1)$-shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (1) We introduce a ${\mathbb{Z}}$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack ${\mathfrak{X}}$. It turns out that $(+1)$-shifted Poisson structures on ${\mathfrak{X}}$ correspond exactly to elements of the Maurer–Cartan moduli set of the corresponding dgla. (2) We introduce the notion of the tangent complex $T_{\mathfrak{X}}$ and the cotangent complex $L_{\mathfrak{X}}$ of a differentiable stack ${\mathfrak{X}}$ in terms of any Lie groupoid $\Gamma{\rightrightarrows } M$ representing ${\mathfrak{X}}$. They correspond to a homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^{\vee } M\rightarrow A^{\vee }[-1]$, respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak{X}}$ defines a morphism ${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$.

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Journal articles on the topic 'Maurer-Cartan element'

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