Develops reduction method for strong Dirac maps.
problem Generalizing Poisson momentum maps.
method General procedure for reduction along strong Dirac maps.
result Recover and introduce new Poisson, quasi-Poisson, and Dirac reduced structures.
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48 results for “quasi-Poisson theory”
We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to…
New Lie 2-algebra structure for multiplicative forms on quasi-Poisson groupoids.
problem Understanding Lie 2-algebra structures on geometric stacks.
method Construction of graded weak Lie 2-algebras from multiplicative forms and differential forms.
result Established a morphism between Lie 2-algebras and weak Lie 2-algebras of multiplicative forms.
We provide a quasi-Poisson version of the Drinfeld's correspondence between Poisson homogeneous spaces and Lagrangian subalgebras.
We prove a quasi-Poisson bracket formula for the space of representations of the fundamental groupoid of a surface with boundary, which generalizes Goldman's Poisson bracket formula. We also deduce a similar formula for quasi-Poisson cross-sections.
Deform symplectic structures using moment maps and Lie algebra elements.
problem Dealing with symplectic structure deformations on manifolds.
method Using quasi-Poisson theory and Lie algebra elements to deform symplectic structures.
result Concrete examples of symplectic structure deformations on complex projective and Grassmannian spaces.
We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and hamiltonian actions, to the realm of Dirac geometry. As an example, we show how hamiltonian quasi-Poisson manifolds fit into this framework by constructing an ``inversion'' procedur…
New constructions and examples from moduli spaces.
problem Understanding quasi-Poisson -spaces and their moment maps.
method Lifting Theorem establishing a bijective correspondence.
result Simple constructions of fusion and conjugation.
Manin pairs and moment mapsmath.DG
A Lie group G in a group pair (D,G), integrating a Lie algebra g in a Manin pair (d,g) has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, that generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are quasi-h…
Quasi-Poisson Manifoldsmath.DG
A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in $\wedge^3 \g$ associated to an invariant inner product. We introduce the concept of the fusion for such manifolds, and we relate quasi-Poisson manifolds …
Shows natural quasi-Poisson structure on multiplicative Grothendieck-Springer space.
problem Exploring new structures in algebraic geometry.
method Reduction along Dirac realizations.
result Natural quasi-Poisson structure exists on multiplicative Grothendieck-Springer space.
Develops new Poisson structures for moduli spaces.
problem Creating Poisson structures for moduli spaces.
method Introduces quasi Poisson and quasi Hamiltonian structures, novel momentum mappings.
result Bijective correspondence between quasi Poisson and quasi Hamiltonian structures.
Given an oriented surface S with base point * on the boundary, we introduce for all N>0, a canonical quasi-Poisson bracket on the space of N-dimensional linear representations of π_1(S,*). Our bracket extends the well-known Poisson bracket on GL_N-invariant functions on this space. Our main tool is a natural structure …
New Poisson structures defined on surface moduli spaces.
problem Generalizing Poisson brackets to quasi-surfaces.
method Defined quasi-Poisson brackets on quasi-surfaces.
result New Poisson structures on quasi-surfaces.
The paper constructs a noncommutative bracket on surface groups and proves it's Hamiltonian.
problem Noncommutative Hamiltonian structures on surface groups.
method Double quasi Poisson bracket construction and noncommutative -matrix formalism.
result Noncommutative Hamiltonian structures on cyclic spaces of unbased loops.
Using ideas from an article of P. Bieliavsky, M. Rooman and Ph. Spindel on BTZ black holes, I construct a family of interesting examples of quasi-Poisson actions as defined by A. Alekseev and Y. Kosmann-Schwarzbach. As an application, I obtain a genuine Poisson structure on which induces a Poisson structure o…
We prove the universal lifting theorem: for an -simply connected and -connected Lie groupoid $\gm$ with Lie algebroid , the graded Lie algebra of multi-differentials on is isomorphic to that of multiplicative multi-vector fields on $\gm$. As a consequence, we obtain the integration theorem for a quasi-Lie …
A new double quasi-Poisson bracket on surface groups.
problem Constructing a new mathematical structure on surface groups.
method Proposing and proving a double quasi-Poisson bracket on group algebras.
result The double quasi-Poisson bracket is a noncommutative generalization of the Goldman bracket.
Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.
problem Homotopy Poisson algebra models for reduced spaces.
method Cattaneo-Zambon compatibility and regularity conditions, equivariant map, homotopy Poisson algebra.
result Derivation of homotopy Poisson algebra generalizing classical BFV algebra.
We study Dirac structures associated with Manin pairs (\d,\g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, pro…
We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-u…
Reformulates Fock-Rosly Poisson structure using quasi-triangular r-matrices.
problem Defining Fock-Rosly Poisson structure on moduli spaces.
method Using Lie algebra actions and quasi-triangular r-matrices.
result Shows Fock-Rosly structure as mixed product Poisson structure.
We study Maurer-Cartan elements on homotopy Poisson manifolds of degree . They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an -term $L_\infty…
We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissoniza…
The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.
problem Integrating quasi-Poisson manifolds into a broader geometric framework.
method Develops new aspects of shifted symplectic and Poisson geometry, establishing Lie-type correspondences and systematic constructions.
result Identifies multiplicative D-valued moment maps integrating quasi-Poisson manifolds, extending known constructions.
Simplified method for algebra of Dirac structures.
problem Deformation of Dirac structures.
method Simplified method for algebra.
result Canonical -isomorphism of algebras.
The paper develops a theory linking Hamiltonian and quasi-Hamiltonian manifolds.
problem Understanding the deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds.
method Introduces a generalized Hamiltonian deformation theory and constructs a topological quantum field theory.
result Shows that the imploded cross section of the double $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$.
Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G-bundle, P\to Σ, over a compact oriented surface, Σ, carries a Poisson structure. If we trivialize P over a finite number of points on the boundary of Σ, then the moduli space carries a q…
Pseudo-Dirac Structuresmath.DG
A Dirac structure is a Lagrangian subbundle of a Courant algebroid, , which is involutive with respect to the Courant bracket. In particular, inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, $W\sub…
We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. …
In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new fi…
Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.
problem Provides a Morita invariant definition of Lie and Courant algebroids over Lie groupoids.
method Views vector fields as Maurer-Cartan elements in a differential graded Lie algebra and as functors and natural transformations.
result Obtains a unifying conceptual framework for studying various algebraic structures.
The paper explores Poisson structures on differentiable stacks, developing new mathematical tools.
problem Investigating shifted Poisson structures on differentiable stacks.
method Developed new mathematical tools including Morita equivalence of quasi-Poisson groupoids and tangent/cotangent complexes.
result Shifted Poisson structures on differentiable stacks correspond to elements of the Maurer-Cartan moduli set of a specific Lie 2-algebra.
Poisson 2-groupsmath.DG
We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-L…
This paper is a continuation of [KS]. We develop the results of [KS] principally in two directions. First, we generalize the main result of [KS], the connection between the solutions of the classical dynamical Yang-Baxter equation and Poisson homogeneous spaces of Poisson Lie groups. We hope that now we present this re…
Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Mapsmath-ph
We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both s…
Discrete Morse-Bott theory on CW complexes generalizes Forman's theory.
problem No specific problem stated; focuses on theory development.
method Derived a discrete Morse-Bott theory on CW complexes.
result Discrete Morse-Bott theory is a generalization of Forman's theory.
The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.
problem Function theory on Teichmüller space and dynamics of mapping class groups.
method Utilizes Thurston's theory and Sullivan's theory on discrete subgroups of hyperbolic space.
result Establishes connections between function theory, dynamics, and ergodic theory.
Survey of Floer theories and their connections.
problem None explicitly stated; focuses on surveying theories.
method None explicitly stated; focuses on surveying theories.
result None explicitly stated; focuses on surveying theories.
New homology theory for semi-groups with specific properties.
problem Developing a homology theory for semi-groups with self-distributivity or idempotency.
method Constructing a new homology theory and comparing it with existing theories.
result Comparison and connections with rack homology and knot theory.
New classes from 4D gauge theories.
problem Constructing characteristic classes for 4-manifold bundles.
method Using -Yang-Mills theory and Seiberg-Witten theory for families.
result Characteristic classes of 4-manifold bundles constructed.
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
New theory couples Chern-Simons to matter, topological.
problem Developing a new topological theory in 3D.
method Coupling Chern-Simons to matter, using transverse holomorphic foliation.
result The theory is equivalent to an N=2 supersymmetric Chern-Simons matter theory.
The paper defines strong emergence in field theories and proves it exists between certain theories.
problem Defining and proving the existence of strong emergence phenomena between field theories.
method Formal definition and sufficient conditions for emergence, proving existence in Euclidean background.
result Strong emergence exists between certain parameterized Lagrangian field theories.
Topological string theory derived from string geometry for non-perturbative effects.
problem Deriving non-perturbative effects in string theory.
method Formulating topological string geometry theory and deriving the partition function from fluctuations around a classical solution.
result Perturbative partition function of topological string theory derived.
New cohomology theory for Lie 2-algebras extends classical theory.
problem Classical cohomology theory limitations for Lie 2-algebras.
method Introduced a new cohomology theory for Lie 2-algebras.
result Second cohomology group classifies extensions of Lie 2-algebras.
Researchers solve M-theory's gauge enhancement problem using advanced homotopy theory.
problem Lift nonabelian gauge fields from D-branes to M-theory.
method Universal constructions in super homotopy theory, focusing on the cyclification adjunction and fiberwise stabilization.
result Gauge enhancement in M-theory is explained by lifting against the fiberwise stabilization of the unit of the cyclification adjunction.
Unified Higgs bundle vacua from M-theory on Spin(7) spaces.
problem Unifying Higgs bundle vacua from different string compactifications.
method Developed formalism for M-theory on local Spin(7) spaces and constructed explicit solutions.
result Unified 3D effective field theory from 4D M- and F-theory vacua.