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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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/11/2043 |
| _version_ | 1797508373594243072 |
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| author | Ihor Mykytyuk |
| author_facet | Ihor Mykytyuk |
| author_sort | Ihor Mykytyuk |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:01:13Z |
| format | Article |
| id | doaj.art-7948e9cc0f6a455aa738ed568c09c013 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T05:01:13Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-7948e9cc0f6a455aa738ed568c09c0132023-11-23T01:43:56ZengMDPI AGSymmetry2073-89942021-10-011311204310.3390/sym13112043Reductions of Invariant bi-Poisson Structures and Locally Free ActionsIhor Mykytyuk0Department of Applied Mathematics, Faculty of Computer Science and Telecommunications, Cracow University of Technology, Warszawska 24, 31155 Cracow, PolandLet <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.https://www.mdpi.com/2073-8994/13/11/2043bi-Poisson structurereductionproper action |
| spellingShingle | Ihor Mykytyuk Reductions of Invariant bi-Poisson Structures and Locally Free Actions Symmetry bi-Poisson structure reduction proper action |
| title | 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_short | Reductions of Invariant bi-Poisson Structures and Locally Free Actions |
| title_sort | reductions of invariant bi poisson structures and locally free actions |
| topic | bi-Poisson structure reduction proper action |
| url | https://www.mdpi.com/2073-8994/13/11/2043 |
| work_keys_str_mv | AT ihormykytyuk reductionsofinvariantbipoissonstructuresandlocallyfreeactions |