## Układy całkowalne na przestrzeniach Banacha Lie-Poissona związanych z grassmannianem Sato

### Tomasz Goliński

#### Abstract

In this thesis we have studied Banach Lie–Poisson spaces C × L 1 res and iR × u 1 res predual to the central extensions of Banach Lie algebras Lres and ures related to restricted Grassmannian Grres, which is an infinite dimensional strong symplectic manifold. These extensions are defined by Schwinger term. Moreover space iR×u 1 res contains Grres as a strong symplectic leaf. On these Banach Lie–Poisson spaces we have constructed non-trivial hierarchy of non-linear Hamilton differential equations. These equations possess infinite family of integrals of motion in involution, which was obtained by the Magri method. They also admit Lax representation. The hierarchy obtained in this thesis includes e.g. operator Riccati-type equations. Some examples of solutions for particular cases of the equations included in investigated hierarchy have been presented at the end of the thesis. We have also proved that these equations are linear in homogenous coordinates on restricted Grassmannian Grres.Diploma type | Doctor of Philosophy | ||||

Author |
Tomasz Goliński (FMIS)
Tomasz Goliński
| ||||

Title in Polish | Układy całkowalne na przestrzeniach Banacha Lie-Poissona związanych z grassmannianem Sato | ||||

Language | pl polski | ||||

Certifying Unit | Faculty of Mathematics and Information Science (FMIS) | ||||

Discipline | mathematics / (mathematics domain) / (physical sciences) | ||||

Defense Date | 10-06-2010 | ||||

End date | 24-06-2010 | ||||

Supervisor |
Anatol Odzijewicz (FMIS)
Anatol Odzijewicz
| ||||

Internal reviewers |
Zbigniew Pasternak-Winiarski (FMIS / DDG)
Zbigniew Pasternak-Winiarski
| ||||

External reviewers |
Andriy Panasyuk
Andriy Panasyuk
| ||||

Pages | 94 | ||||

Keywords in English | xxx | ||||

Abstract in English | In this thesis we have studied Banach Lie–Poisson spaces C × L 1 res and iR × u 1 res predual to the central extensions of Banach Lie algebras Lres and ures related to restricted Grassmannian Grres, which is an infinite dimensional strong symplectic manifold. These extensions are defined by Schwinger term. Moreover space iR×u 1 res contains Grres as a strong symplectic leaf. On these Banach Lie–Poisson spaces we have constructed non-trivial hierarchy of non-linear Hamilton differential equations. These equations possess infinite family of integrals of motion in involution, which was obtained by the Magri method. They also admit Lax representation. The hierarchy obtained in this thesis includes e.g. operator Riccati-type equations. Some examples of solutions for particular cases of the equations included in investigated hierarchy have been presented at the end of the thesis. We have also proved that these equations are linear in homogenous coordinates on restricted Grassmannian Grres. | ||||

Thesis file |
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Citation count* | 5 (2020-09-18) |

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