scholarly journals Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

2018 ◽  
pp. 681-704 ◽  
Author(s):  
Brent Pym ◽  
Travis Schedler
Keyword(s):  
Symplectic Form ◽  
Vector Fields ◽  
Structural Features ◽  
Poisson Manifolds ◽  
Deformation Spaces ◽  
Hamiltonian Vector Fields ◽  
Finite Dimensional ◽  
Degeneracy Condition ◽  
Deformation Complex ◽  
Elliptic Algebras

This chapter introduces a natural non-degeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds are privileged by the fact that their deformation spaces are as finite-dimensional as one could ever hope: the corresponding derived deformation complex is a perverse sheaf. The chapter develops some basic structural features of these manifolds, highlighting the role played by the divergence of Hamiltonian vector fields. As an application, it establishes the deformation invariance of certain families of Poisson manifolds defined by Feigin and Odesskii, along with the ‘elliptic algebras’ that quantize them.

scholarly journals Classification of Simple Lie Superalgebras in Characteristic 2

2021 ◽  
Author(s):  
Sofiane Bouarroudj ◽  
Alexei Lebedev ◽  
Dimitry Leites ◽  
Irina Shchepochkina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Simple Lie Algebra ◽  
Hamiltonian Vector Fields ◽  
Finite Dimensional ◽  
Characteristic 2

Abstract All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a simple Lie superalgebra. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures. For Lie algebras, in addition to the known “classical” restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: a $2|4$-structure, which is a direct analog of the classical restrictedness, and a novel $2|2$-structure—one more analog, a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras known only for non-alternating orthogonal and Hamiltonian series.

Symmetries from the solution manifold

2015 ◽  
Vol 12 (08) ◽  
pp. 1560016 ◽  
Author(s):  
Víctor Aldaya ◽  
Julio Guerrero ◽  
Francisco F. Lopez-Ruiz ◽  
Francisco Cossío
Keyword(s):  
Symplectic Structure ◽  
Vector Fields ◽  
Sigma Model ◽  
Canonical Quantization ◽  
General Concept ◽  
Noether Theorem ◽  
Preliminary Step ◽  
Hamiltonian Vector Fields ◽  
Solution Manifold

We face a revision of the role of symmetries of a physical system aiming at characterizing the corresponding Solution Manifold (SM) by means of Noether invariants as a preliminary step towards a proper, non-canonical, quantization. To this end, "point symmetries" of the Lagrangian are generally not enough, and we must resort to the more general concept of contact symmetries. They are defined in terms of the Poincaré–Cartan form, which allows us, in turn, to find the symplectic structure on the SM, through some sort of Hamilton–Jacobi (HJ) transformation. These basic symmetries are realized as Hamiltonian vector fields, associated with (coordinate) functions on the SM, lifted back to the Evolution Manifold through the inverse of this HJ mapping, that constitutes an inverse of the Noether Theorem. The specific examples of a particle moving on S3, at the mechanical level, and nonlinear SU(2)-sigma model in field theory are sketched.

scholarly journals Hamiltonian Vector Fields on Weil Bundles

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Norbert Mahoungou Moukala ◽  
Basile Guy Richard Bossoto
Keyword(s):  
Vector Fields ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Fields

A weight-homogenous condition to the real Jacobian conjecture in

2021 ◽  
pp. 1-9
Author(s):  
Francisco Braun ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Hamiltonian Vector ◽  
Jacobian Conjecture ◽  
Local Diffeomorphism ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Real Linear

Abstract It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

scholarly journals On Lie algebras of vector fields with invariant submanifolds

1974 ◽  
Vol 55 ◽  
pp. 91-110 ◽  
Author(s):  
Akira Koriyama
Keyword(s):  
Lie Algebras ◽  
Compact Support ◽  
Vector Fields ◽  
Finite Dimensional ◽  
Invariant Submanifolds ◽  
Infinitesimal Automorphisms

It is known (Pursell and Shanks [9]) that an isomorphism between Lie algebras of infinitesimal automorphisms of C∞ structures with compact support on manifolds M and M′ yields an isomorphism between C∞ structures of M and M’.Omori [5] proved that this is still true for some other structures on manifolds. More precisely, let M and M′ be Hausdorff and finite dimensional manifolds without boundary. Let α be one of the following structures:

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