The Experts below are selected from a list of 7023 Experts worldwide ranked by ideXlab platform
Das Apurba - One of the best experts on this subject based on the ideXlab platform.
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Relative Rota-Baxter systems on Leibniz Algebras
2021Co-Authors: Das Apurba, Guo ShuangjianAbstract:In this paper, we introduce relative Rota-Baxter systems on Leibniz Algebras and give some characterizations and new constructions. Then we construct a Graded Lie Algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This allows us to define a cohomology theory associated with a relative Rota-Baxter system. Finally, we study formal deformations and extendibility of finite order deformations of a relative Rota-Baxter system in terms of the cohomology theory.Comment: 25page
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(Co)homology of compatible associative Algebras
2021Co-Authors: Chtioui Taoufik, Das Apurba, Mabrouk SamiAbstract:In this paper, we define and study (co)homology theories of a compatible associative Algebra $A$. At first, we construct a new Graded Lie Algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define the cohomology of a compatible associative Algebra $A$ and as applications, we study extensions, deformations and extensibility of finite order deformations of $A$. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative Algebras.Comment: 28 pages; comments are welcom
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Deformations of associative Rota-Baxter operators
2020Co-Authors: Das ApurbaAbstract:Rota-Baxter operators and more generally $\mathcal{O}$-operators on associative Algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of Algebras. Using a method of Uchino, we construct an explicit Graded Lie Algebra whose Maurer-Cartan elements are given by $\mathcal{O}$-operators. This allows us to construct a cohomology for an $\mathcal{O}$-operator. This cohomology can also be seen as the Hochschild cohomology of a certain Algebra with coefficients in a suitable representation. Next, we study linear and formal deformations of an $\mathcal{O}$-operator which are governed by the above-defined cohomology. We introduce Nijenhuis elements associated with an $\mathcal{O}$-operator which give rise to trivial deformations. As an application, we conclude deformations of weight zero Rota-Baxter operators and associative {\bf r}-matrices.Comment: Comments are welcome; final version appear in Journal of Algebr
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Generalized Rota-Baxter systems
2020Co-Authors: Das ApurbaAbstract:Rota-Baxter systems of T. Brzezi\'{n}ski are a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant biAlgebras. In this paper, we consider Rota-Baxter systems in the presence of bimodule, which we call generalized Rota-Baxter systems. We define a Graded Lie Algebra whose Maurer-Cartan elements are generalized Rota-Baxter systems. This allows us to define a cohomology theory for a generalized Rota-Baxter system. Formal one-parameter deformations of generalized Rota-Baxter systems are discussed from cohomological points of view. We further study Rota-Baxter systems, associative Yang-Baxter pairs, covariant biAlgebras and introduce generalized averaging systems that are related to associative diAlgebras. Next, we define generalized Rota-Baxter systems in the homotopy context and find relations with homotopy dendriform Algebras. The paper ends by considering commuting Rota-Baxter systems and their relation with quadri-Algebras.Comment: 28 pages; comments are welcome
Marco Modugno - One of the best experts on this subject based on the ideXlab platform.
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Graded Lie Algebra of hermitian tangent valued forms
Journal de Mathématiques Pures et Appliquées, 2006Co-Authors: Josef Janyska, Marco ModugnoAbstract:Abstract We define the Hermitian tangent valued forms of a complex 1-dimensional line bundle equipped with a Hermitian metric. We provide a local characterization of these forms in terms of a local basis and of a local fibred chart. We show that these forms constitute a Graded Lie Algebra through the Frolicher–Nijenhuis bracket. Moreover, we provide a global characterization of this Graded Lie Algebra, via a given Hermitian connection, in terms of the tangent valued forms and forms of the base space. The bracket involves the curvature of the given Hermitian connection.
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Graded Lie Algebra of hermitian tangent valued forms
arXiv: Mathematical Physics, 2005Co-Authors: Josef Janyska, Marco ModugnoAbstract:In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie Algebra of Hermitian vector fields is naturally isomorphic to a Lie Algebra of "special functions" of the phase space. Indeed, this is the source of the covariant quantisation of the above special functions. In the original version of the theory, this result was formulated and proved in a rather involved way; now, we have achieved a more direct and simple approach to the classification of Hermitian vector fields and to their representation via special phase functions. In view of a possible covariant quantisation of a larger class of "observables" it is natural to consider the Hermitian tangent valued forms. Thus, this paper is devoted to a self--contained analysis of the Graded Lie Algebra of Hermitian tangent valued forms of a complex line bundle and to their classification in terms of tangent valued forms and forms of the base space. The local classification is obtained in coordinates. For the global classification we need a Hermitian connection: indeed, this is just the connection required in gauge theories.
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torsion and ricci tensor for non linear connections
Differential Geometry and Its Applications, 1991Co-Authors: Marco ModugnoAbstract:Abstract We study a natural generalization of the concepts of torsion and Ricci tensor for a non- linear connection on a fibred manifold, with respect to a given fibred soldering form. Our results are achieved by means of the differentials and codifferentials induced by the Frolicher-Nijenhuis Graded Lie Algebra of tangent valued forms.
Giovanni Felder - One of the best experts on this subject based on the ideXlab platform.
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effective batalin vilkovisky theories equivariant configuration spaces and cyclic chains
Progress in Mathematics, 2011Co-Authors: Alberto S Cattaneo, Giovanni FelderAbstract:The celebrated Kontsevich formality theorem [M. Kontsevich, Lett. Math. Phys. 66 (2003), no. 3, 157--216; MR2062626 (2005i:53122)] states that the differential Graded Lie Algebra gG of polydifferential operators on a smooth manifold M is formal, i.e., it is quasi-isomorphic to its cohomology which is, in turn, identified with the Schouten Lie Algebra gS of polyvector fields on M. Moreover, this quasi-isomorphism is realized by a certain L∞ map from gS to gG whose components are expressed through certain correlation functions of a topological field theory on the upper half-plane as shown by the present authors [A. S. Cattaneo and G. Felder, Comm. Math. Phys. 212 (2000), no. 3, 591--611; MR1779159 (2002b:53141)]. In the paper under review the authors consider the case of a manifold M endowed with a volume form and the differential Graded Lie Algebra gS[v], where v is a formal parameter and the differential is the divergence operator times v. Via the Kontsevich formality map the complex of negative cyclic chains of the Algebra of smooth functions on M becomes an L∞-module over gS[v]. Consider also gS endowed with the divergence operator viewed as a differential and with the trivial action of gS[v]. The main result of the paper is the construction of an L∞-map between these L∞-modules. The relevant quantum field theory is a BF theory on a disc (as opposed to the upper half-plane) which is treated in the framework of the Batalin-Vilkovisky quantization. The new feature is the presence of the zero modes of the action functional. As an application the authors construct traces on Algebras of functions with star-products associated with unimodular Poisson structures.
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effective batalin vilkovisky theories equivariant configuration spaces and cyclic chains
arXiv: Mathematical Physics, 2008Co-Authors: Alberto S Cattaneo, Giovanni FelderAbstract:Kontsevich's formality theorem states that the differential Graded Lie Algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential Graded Lie Algebra of multivector fields endowed with a divergence operator. In the case of R^d with standard volume form, we obtain an L-infinity morphism of modules over this differential Graded Lie Algebra from cyclic chains of the Algebra of functions to multivector fields. As a first application we give a construction of traces on Algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin--Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.
C Wotzasek - One of the best experts on this subject based on the ideXlab platform.
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two dimensional supersymmetric harmonic oscillator carrying a representation of the gl 2 1 Graded Lie Algebra
Journal of Mathematical Physics, 1996Co-Authors: Ashok Das, C WotzasekAbstract:We study a supersymmetric two‐dimensional harmonic oscillator which carries a representation of the general Graded Lie Algebra GL(2|1), formulate it on the superspace, and discuss its physical spectrum.
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two dimensional supersymmetric harmonic oscillator carrying a representation of the gl 2 1 Graded Lie Algebra
arXiv: High Energy Physics - Theory, 1995Co-Authors: Ashok Das, C WotzasekAbstract:We study a supersymmetric 2-dimensional harmonic oscillator which carries a representation of the general Graded Lie Algebra GL(2$\vert$1), formulate it on the superspace, and discuss its physical spectrum.
Alberto S Cattaneo - One of the best experts on this subject based on the ideXlab platform.
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effective batalin vilkovisky theories equivariant configuration spaces and cyclic chains
Progress in Mathematics, 2011Co-Authors: Alberto S Cattaneo, Giovanni FelderAbstract:The celebrated Kontsevich formality theorem [M. Kontsevich, Lett. Math. Phys. 66 (2003), no. 3, 157--216; MR2062626 (2005i:53122)] states that the differential Graded Lie Algebra gG of polydifferential operators on a smooth manifold M is formal, i.e., it is quasi-isomorphic to its cohomology which is, in turn, identified with the Schouten Lie Algebra gS of polyvector fields on M. Moreover, this quasi-isomorphism is realized by a certain L∞ map from gS to gG whose components are expressed through certain correlation functions of a topological field theory on the upper half-plane as shown by the present authors [A. S. Cattaneo and G. Felder, Comm. Math. Phys. 212 (2000), no. 3, 591--611; MR1779159 (2002b:53141)]. In the paper under review the authors consider the case of a manifold M endowed with a volume form and the differential Graded Lie Algebra gS[v], where v is a formal parameter and the differential is the divergence operator times v. Via the Kontsevich formality map the complex of negative cyclic chains of the Algebra of smooth functions on M becomes an L∞-module over gS[v]. Consider also gS endowed with the divergence operator viewed as a differential and with the trivial action of gS[v]. The main result of the paper is the construction of an L∞-map between these L∞-modules. The relevant quantum field theory is a BF theory on a disc (as opposed to the upper half-plane) which is treated in the framework of the Batalin-Vilkovisky quantization. The new feature is the presence of the zero modes of the action functional. As an application the authors construct traces on Algebras of functions with star-products associated with unimodular Poisson structures.
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effective batalin vilkovisky theories equivariant configuration spaces and cyclic chains
arXiv: Mathematical Physics, 2008Co-Authors: Alberto S Cattaneo, Giovanni FelderAbstract:Kontsevich's formality theorem states that the differential Graded Lie Algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential Graded Lie Algebra of multivector fields endowed with a divergence operator. In the case of R^d with standard volume form, we obtain an L-infinity morphism of modules over this differential Graded Lie Algebra from cyclic chains of the Algebra of functions to multivector fields. As a first application we give a construction of traces on Algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin--Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.
