Unimodular graded Poisson Hopf algebras

Bifractional transformations which lead to quantities that interpolate between other known quantities, are considered. They do not form a group, and groupoids are used to described their mathematical structure. Bifractional coherent states and bifractional Wigner functions are also defined.The properties of the bifractional coherent states are studied. The bifractional Wigner functions are used in generalizations of the Moyal star formalism. A generalized Berezin formalism in this context, is also studied.

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“…• In section II we review briefly for later use, the mathematical structure of groupoids [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

The groupoid of bifractional transformations

Agyo

Lei

Vourdas

2017

Journal of Mathematical Physics

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Reflection Groups and Rigidity of Quadratic Poisson Algebras

Gaddis

Veerapen²,

Wang³

2021

Algebr Represent Theor

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Reflection Groups and Rigidity of Quadratic Poisson Algebras

Gaddis¹,

Veerapen²,

Wang³

2020

Preprint

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In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of the Shephard-Todd-Chevalley theorem is proved stating that the fixed Poisson subring A G is skew-symmetric if and only if G is generated by reflections. For many other wellknown families of quadratic Poisson algebras, we show that G contains limited or even no reflections. This kind of Poisson rigidity result ensures that the corresponding fixed Poisson subring A G is not isomorphic to A as Poisson algebras unless G is trivial. Group actions are ubiquitous in mathematics and theoretical physics. To study the symmetry of an algebraic object it is often useful to understand what groups act on it. Invariants of the action of a finite group on a commutative polynomial ring have played a major role in the development of commutative algebra. The notion of a Poisson bracket was introduced by French mathematician Siméon Denis Poisson in the search for integrals of motion in Hamiltonian mechanics. Recently, Poisson algebras have become deeply entangled with noncommutative geometry, integrable systems, topological field theories, and representation theory of noncommutative algebras. In order to study the symmetry involved in Poisson brackets, we propose in this paper the study of group actions on Poisson algebras. Let k be an algebraically closed field of characteristic zero. There is an intricate connection between quadratic Poisson algebras and quantum polynomial rings through deformation theory and semiclassical limits. For example, it is conjectured that the primitive ideal space of the quantized coordinate ring O q (G) of a semisimple Lie group G and the Poisson primitive ideal space of the classical coordinate ring O(G), with their respective Zariski topologies, are homeomorphic. Motivated by noncommutative invariant theory, the work in this paper is a study of certain invariant theory questions related to quadratic Poisson algebras. Specifically, we investigate the Shephard-Todd-Chevalley (STC) theorem in this context. The classical STC theorem gives necessary and sufficient conditions for the fixed subring k[x 1 , . . . , x n ] G under a finite subgroup G of GL n (k) to be a polynomial ring again. In some sense, the problem in the Poisson setting might seem trivial. For any polynomial Poisson algebra A and a finite subgroup G of all Poisson graded automorphisms of A, the fixed Poisson subring A G is again a Poisson polynomial ring when G is generated by (classical) reflections. That is, the problem reduces to a direct application of the STC theorem. However, inspired by the work of [12,

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Unimodular graded Poisson Hopf algebras

Cited by 3 publications

References 31 publications

The groupoid of bifractional transformations

The groupoid of bifractional transformations

Reflection Groups and Rigidity of Quadratic Poisson Algebras

Reflection Groups and Rigidity of Quadratic Poisson Algebras

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