Poisson manifold
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Simultaneous deformations and poisson geometry Open
We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an $L_{\infty }$ -algebra, which we construct explicitly. Our machinery is based on Voronov’s derived bracket co…
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Quantum geometry of moduli spaces of local systems and representation theory Open
Let G be a split semi-simple adjoint group, and S a colored decorated surface, given by an oriented surface with punctures, special boundary points, and a specified collection of boundary intervals. We introduce a moduli space P(G,S) param…
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The κ-(A)dS quantum algebra in (3 + 1) dimensions Open
The quantum duality principle is used to obtain explicitly the Poisson\nanalogue of the kappa-(A)dS quantum algebra in (3+1) dimensions as the\ncorresponding Poisson-Lie structure on the dual solvable Lie group. The\nconstruction is fully …
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Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds Open
We give a systematic derivation of the local expressions of the NS H-flux, geometric F- as well as non-geometric Q- and R-fluxes in terms of bivector beta- and two-form B-potentials including vielbeins. They are obtained using a supergeome…
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Drinfeld double of GL <sub>n</sub> and generalized cluster structures Open
We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward …
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Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy Open
Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establis…
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4D Higher Spin Gravity with Dynamical Two-Form as a Frobenius--Chern--Simons Gauge Theory Open
We provide an off-shell formulation of four-dimensional higher spin gravity based on a covariant Hamiltonian action on an open nine-dimensional Poisson manifold whose boundary consists of the direct product of spacetime and a noncommutativ…
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Structure-preserving reduced basis methods for Poisson systems Open
We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poiss…
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Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed Open
We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure…
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Poisson-Lie duals of the η deformed symmetric space sigma model Open
Poisson-Lie dualising the η deformation of the G/H symmetric space sigma model with respect to the simple Lie group G is conjectured to give an analytic continuation of the associated λ deformed model. In this paper we investigate when the…
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Higher-spin self-dual Yang-Mills and gravity from the twistor space Open
A bstract We lift the recently proposed theories of higher-spin self-dual Yang-Mills (SDYM) and gravity (SDGR) to the twistor space. We find that the most natural room for their twistor formulation is not in the projective, but in the full…
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Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets Open
A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an ellip…
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On covariant Poisson brackets in classical field theory Open
How to give a natural geometric definition of a covariant Poisson bracket in classical field theory has for a long time been an open problem—as testified by the extensive literature on “multisymplectic Poisson brackets,” together with the …
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N-manifolds of degree 2 and metric double vector bundles Open
This paper shows the equivalence of the categories of $N$-manifolds of degree $2$ with the category of double vector bundles endowed with a linear metric. Split Poisson $N$-manifolds of degree $2$ are shown to be equivalent to self-dual re…
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Gravity theory on Poisson manifold with <i>R</i>‐flux Open
A novel gravity theory based on Poisson Generalized Geometry is investigated. A gravity theory on a Poisson manifold equipped with a Riemannian metric is constructed from a contravariant version of the Levi‐Civita connection, which is base…
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Poisson structures for canonical algebras Open
In the paper Poisson structures on canonical algebras C are described. In particular, inner Poisson algebras on C are characterized. Furthermore, we show how to construct outer Poisson structures on C.
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Noncommutative integrable systems on b-symplectic manifolds Open
In this paper we study non-commutative integrable systems on $b$-Poisson\nmanifolds. One important source of examples (and motivation) of such systems\ncomes from considering non-commutative systems on manifolds with boundary\nhaving the r…
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Star products on graded manifolds and <i>α</i>′-corrections to Courant algebroids from string theory Open
Courant algebroids, originally used to study integrability conditions for Dirac structures, have turned out to be of central importance to study the effective supergravity limit of string theory. The search for a geometric description of T…
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Regular Poisson manifolds of compact types (PMCT 2) Open
This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine…
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Cluster algebra structures on Poisson nilpotent algebras Open
Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert …
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Deformation quantisation for (-1)-shifted symplectic structures and vanishing cycles Open
We formulate a notion of $E_0$ quantisation of $(-1)$-Poisson structures on derived Artin $N$-stacks, and construct a map from $E_0$ quantisations of $(-1)$-shifted symplectic structures to power series in de Rham cohomology. For a square …
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Strict deformation quantization of the state space of Mk(ℂ) with applications to the Curie–Weiss model Open
Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text]…
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Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors? Open
From the paper "Formality Conjecture" (Ascona 1996): "I am aware of only one\nsuch a class, it corresponds to simplest good graph, the complete graph with\n$4$ vertices $($and $6$ edges$)$. This class gives a remarkable vector field on\nth…
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Geodesic motion on the symplectic leaf of $$SO(3)$$ with distorted e(3) algebra and Liouville integrability of a free rigid body Open
The solutions to the Euler–Poisson equations are geodesic lines of SO (3) manifold with the metric determined by inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We …
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Poisson manifolds of compact types (PMCT 1) Open
This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this fir…
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A note on the symplectic topology of $b$-manifolds Open
A Poisson manifold $(M^{2n},\p)$ is $b$-symplectic if $\bigwedge^n\p$ is transverse to the zero section. In this paper we apply techniques native to Symplectic Topology to address questions pertaining to $b$-symplectic manifolds. We provid…
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Quantum Riemannian geometry of phase space and nonassociativity Open
Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and …
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Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems Open
This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric m…
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Modules and representations up to homotopy of Lie n-algebroids Open
This paper studies differential graded modules and representations up to homotopy of Lie n -algebroids, for general $$n\in {\mathbb {N}}$$ . The adjoint and coadjoint modules are described, and the corresponding split versions of the …
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Pre-Calabi-Yau structures and moduli of representations Open
We establish a system of formal noncommutative calculus for differential forms and polyvector fields, which forms the foundations for the study of pre-Calabi-Yau categories. Using an explicit trace map, we show that any $n$-Calabi-Yau stru…