scholarly journals Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 88-102
Author(s):  
Yat Sun Poon ◽  
John Simanyi
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Complex Structure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Poisson Cohomology ◽  
Abelian Complex Structure ◽  
Gerstenhaber Algebras ◽  
Holomorphic Poisson Cohomology ◽  
Necessary And Sufficient

AbstractIt is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure.We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.

MOMENTUM MAPPINGS AND POISSON COHOMOLOGY

1996 ◽  
Vol 07 (03) ◽  
pp. 329-358 ◽  
Author(s):  
VIKTOR L. GINZBURG
Keyword(s):  
Poisson Structure ◽  
Poisson Manifold ◽  
Existence Problem ◽  
Necessary And Sufficient Condition ◽  
Momentum Mapping ◽  
Lie Algebra Cohomology ◽  
Poisson Cohomology ◽  
Poisson Actions ◽  
Poisson Lie Groups ◽  
Necessary And Sufficient

We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentum mapping to be unique is given. The existence problem is solved under some extra hypotheses, for example, when the action preserves the Poisson structure. In this case, the problem is closely related to the triviality of the induced group action on the Poisson cohomology. This action is shown to be trivial whenever the group is compact or semisimple. Conceptually, these results rely upon a version of “Poisson calculus” developed here to make one-forms on a Poisson manifold induce a “flow” preserving the Poisson structure. In the general case, obstructions to the existence of an infinitesimal version of an equivariant momentum mapping are found. Using Lie algebra cohomology with coefficients in Fréchet modules, we show that the obstructions vanish, and the infinitesimal mapping exists, when the group is compact semisimple. We also prove the rigidity of compact group actions preserving the Poisson structure on a compact manifold and calculate the Poisson cohomology of the Poisson homogeneous space [Formula: see text].

scholarly journals A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

2017 ◽  
Vol 4 (1) ◽  
pp. 137-154 ◽  
Author(s):  
Yat Sun Poon ◽  
John Simanyi
Keyword(s):  
Spectral Sequence ◽  
Complex Structure ◽  
Adjoint Action ◽  
Poisson Structures ◽  
Dolbeault Cohomology ◽  
Poisson Cohomology ◽  
Abelian Complex Structure ◽  
Nijenhuis Bracket ◽  
Holomorphic Poisson Cohomology ◽  
Type Decomposition

Abstract A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

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