Reductions of Invariant bi-Poisson Structures and Locally Free Actions

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>,</mo><msub><mi>ω</mi><mn>...

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Autor principal:	Ihor Mykytyuk
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	bi-Poisson structure reduction proper action Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/7948e9cc0f6a455aa738ed568c09c013
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id	oai:doaj.org-article:7948e9cc0f6a455aa738ed568c09c013
record_format	dspace
institution	DOAJ
collection	DOAJ
language	EN
topic	bi-Poisson structure reduction proper action Mathematics QA1-939
spellingShingle	bi-Poisson structure reduction proper action Mathematics QA1-939 Ihor Mykytyuk Reductions of Invariant bi-Poisson Structures and Locally Free Actions
description	Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>,</mo><msub><mi>ω</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi><mn>2</mn></msub><mo>,</mo><mrow><mo>{</mo><msup><mi>η</mi><mi>t</mi></msup><mo>}</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> be a manifold with a bi-Poisson structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msup><mi>η</mi><mi>t</mi></msup><mo>}</mo></mrow></semantics></math></inline-formula> generated by a pair of <i>G</i>-invariant symplectic structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mn>2</mn></msub></semantics></math></inline-formula>, where a Lie group <i>G</i> acts properly on <i>X</i>. We prove that there exists two canonically defined manifolds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub><mo>,</mo><msup><mi>G</mi><mi>i</mi></msup><mo>,</mo><msubsup><mi>ω</mi><mn>1</mn><mi>i</mi></msubsup><mo>,</mo><msubsup><mi>ω</mi><mn>2</mn><mi>i</mi></msubsup><mo>,</mo><mrow><mo>{</mo><msubsup><mi>η</mi><mi>i</mi><mi>t</mi></msubsup><mo>}</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula> such that (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></semantics></math></inline-formula> is a submanifold of an open dense subset <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub><mo>⊂</mo><mi>X</mi></mrow></semantics></math></inline-formula>; (2) symplectic structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>ω</mi><mn>1</mn><mi>i</mi></msubsup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>ω</mi><mn>2</mn><mi>i</mi></msubsup></semantics></math></inline-formula>, generating a bi-Poisson structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>η</mi><mi>i</mi><mi>t</mi></msubsup><mo>}</mo></mrow></semantics></math></inline-formula>, are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>i</mi></msup></semantics></math></inline-formula>- invariant and coincide with restrictions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>1</mn></msub><msub><mrow><mo>\|</mo></mrow><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>2</mn></msub><msub><mrow><mo>\|</mo></mrow><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></msub></mrow></semantics></math></inline-formula>; (3) the canonically defined group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>i</mi></msup></semantics></math></inline-formula> acts properly and <i>locally freely</i> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></semantics></math></inline-formula>; (4) orbit spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub><mo>/</mo><mi>G</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub><mo>/</mo><msup><mi>G</mi><mi>i</mi></msup></mrow></semantics></math></inline-formula> are canonically diffeomorphic smooth manifolds; (5) spaces of <i>G</i>-invariant functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>i</mi></msup></semantics></math></inline-formula>-invariant functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></semantics></math></inline-formula> are isomorphic as Poisson algebras with the bi-Poisson structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msup><mi>η</mi><mi>t</mi></msup><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>η</mi><mi>i</mi><mi>t</mi></msubsup><mo>}</mo></mrow></semantics></math></inline-formula> respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a <i>locally free</i> action of a symmetry group.
format	article
author	Ihor Mykytyuk
author_facet	Ihor Mykytyuk
author_sort	Ihor Mykytyuk
title	Reductions of Invariant bi-Poisson Structures and Locally Free Actions
title_short	Reductions of Invariant bi-Poisson Structures and Locally Free Actions
title_full	Reductions of Invariant bi-Poisson Structures and Locally Free Actions
title_fullStr	Reductions of Invariant bi-Poisson Structures and Locally Free Actions
title_full_unstemmed	Reductions of Invariant bi-Poisson Structures and Locally Free Actions
title_sort	reductions of invariant bi-poisson structures and locally free actions
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/7948e9cc0f6a455aa738ed568c09c013
work_keys_str_mv	AT ihormykytyuk reductionsofinvariantbipoissonstructuresandlocallyfreeactions
_version_	1718410285834829824
spelling	oai:doaj.org-article:7948e9cc0f6a455aa738ed568c09c0132021-11-25T19:06:16ZReductions of Invariant bi-Poisson Structures and Locally Free Actions10.3390/sym131120432073-8994https://doaj.org/article/7948e9cc0f6a455aa738ed568c09c0132021-10-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2043https://doaj.org/toc/2073-8994Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>,</mo><msub><mi>ω</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi><mn>2</mn></msub><mo>,</mo><mrow><mo>{</mo><msup><mi>η</mi><mi>t</mi></msup><mo>}</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> be a manifold with a bi-Poisson structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msup><mi>η</mi><mi>t</mi></msup><mo>}</mo></mrow></semantics></math></inline-formula> generated by a pair of <i>G</i>-invariant symplectic structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mn>2</mn></msub></semantics></math></inline-formula>, where a Lie group <i>G</i> acts properly on <i>X</i>. We prove that there exists two canonically defined manifolds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub><mo>,</mo><msup><mi>G</mi><mi>i</mi></msup><mo>,</mo><msubsup><mi>ω</mi><mn>1</mn><mi>i</mi></msubsup><mo>,</mo><msubsup><mi>ω</mi><mn>2</mn><mi>i</mi></msubsup><mo>,</mo><mrow><mo>{</mo><msubsup><mi>η</mi><mi>i</mi><mi>t</mi></msubsup><mo>}</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula> such that (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></semantics></math></inline-formula> is a submanifold of an open dense subset <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub><mo>⊂</mo><mi>X</mi></mrow></semantics></math></inline-formula>; (2) symplectic structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>ω</mi><mn>1</mn><mi>i</mi></msubsup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>ω</mi><mn>2</mn><mi>i</mi></msubsup></semantics></math></inline-formula>, generating a bi-Poisson structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>η</mi><mi>i</mi><mi>t</mi></msubsup><mo>}</mo></mrow></semantics></math></inline-formula>, are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>i</mi></msup></semantics></math></inline-formula>- invariant and coincide with restrictions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>1</mn></msub><msub><mrow><mo>\|</mo></mrow><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>2</mn></msub><msub><mrow><mo>\|</mo></mrow><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></msub></mrow></semantics></math></inline-formula>; (3) the canonically defined group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>i</mi></msup></semantics></math></inline-formula> acts properly and <i>locally freely</i> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></semantics></math></inline-formula>; (4) orbit spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub><mo>/</mo><mi>G</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub><mo>/</mo><msup><mi>G</mi><mi>i</mi></msup></mrow></semantics></math></inline-formula> are canonically diffeomorphic smooth manifolds; (5) spaces of <i>G</i>-invariant functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>i</mi></msup></semantics></math></inline-formula>-invariant functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msup><mi>L</mi><mi>i</mi></msup></msub></semantics></math></inline-formula> are isomorphic as Poisson algebras with the bi-Poisson structures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msup><mi>η</mi><mi>t</mi></msup><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msubsup><mi>η</mi><mi>i</mi><mi>t</mi></msubsup><mo>}</mo></mrow></semantics></math></inline-formula> respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a <i>locally free</i> action of a symmetry group.Ihor MykytyukMDPI AGarticlebi-Poisson structurereductionproper actionMathematicsQA1-939ENSymmetry, Vol 13, Iss 2043, p 2043 (2021)

Reductions of Invariant bi-Poisson Structures and Locally Free Actions

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