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postgraduate thesis: On the Knutson-Woo-Yong maps and some poisson homogeneous spaces
Title | On the Knutson-Woo-Yong maps and some poisson homogeneous spaces |
---|---|
Authors | |
Advisors | Advisor(s):Lu, J |
Issue Date | 2018 |
Publisher | The University of Hong Kong (Pokfulam, Hong Kong) |
Citation | Yu, S. [于世卓]. (2018). On the Knutson-Woo-Yong maps and some poisson homogeneous spaces. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. |
Abstract | Richardson varieties and
Schubert varieties associated to a connected complex semisimple Lie group $G$
are certain closed subvarieties of the flag variety $G/B$ of $G$, and they play very important roles in
Lie theory. In a paper in 2013,
A. Knutson, A. Woo, and A. Yong introduced certain maps that relate the singularities of Richardson varieties with that
of Schubert varieties. In this thesis, the Knutson-Woo-Yong maps were studied from the points of view of Poisson geometry, and
applications of the Knutson-Woo-Yong maps to Poisson
geometry were obtained.
In the first part of the thesis, the Knutson-Woo-Yong maps were generalized to certain homogeneous
space $G/Q$ of $G$ and were shown to be Poisson isomorphisms with respect to a
naturally defined Poisson structure $\pi_{\sG/\sQ}$ on $G/Q$. In the second part of the thesis, the generalized
Knutson-Woo-Yong maps for $G/Q$ were used to define coordinate charts on $G/Q$, and it was shown that in {\it each} of such coordinate
charts, the Poisson structure $\pi_{\sG/\sQ}$ on $G/Q$ gives rise to a
polynomial Poisson algebra that is a {\it symmetric Cauchon-Goodearl-Letzter (CGL) extension (of ${\mathbb{C}}$)}.
Primary examples of homogeneous spaces $G/Q$ of $G$ studied in the thesis were the orbits in the double flag variety $(G/B) \times (G/B_-)$ for the diagonal
action of $G$, which include the flag variety $G/B$ as the closed orbit, and
the symmetric Poisson CGL extensions arising from these examples are certain {\it $T$-mixed products}
of the symmetric Poisson CGL extensions defined by generalized Bruhat cells, where $T$ is a maximal torus of $G$.
Symmetric CGL extensions form a particular class of polynomial Poisson algebras that are iterated Poisson Ore extensions
and have compatible torus actions. They have been recently introduced and studied by K. Goodearl and M. Yakimov in the context of cluster algebras and quantum groups. The results of the thesis show that important Poisson varieties in Lie theory can be covered by symmetric Poisson CGL extensions. |
Degree | Doctor of Philosophy |
Subject | Flag manifolds Poisson algebras |
Dept/Program | Mathematics |
Persistent Identifier | http://hdl.handle.net/10722/261466 |
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | Lu, J | - |
dc.contributor.author | Yu, Shizhuo | - |
dc.contributor.author | 于世卓 | - |
dc.date.accessioned | 2018-09-20T06:43:47Z | - |
dc.date.available | 2018-09-20T06:43:47Z | - |
dc.date.issued | 2018 | - |
dc.identifier.citation | Yu, S. [于世卓]. (2018). On the Knutson-Woo-Yong maps and some poisson homogeneous spaces. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. | - |
dc.identifier.uri | http://hdl.handle.net/10722/261466 | - |
dc.description.abstract | Richardson varieties and Schubert varieties associated to a connected complex semisimple Lie group $G$ are certain closed subvarieties of the flag variety $G/B$ of $G$, and they play very important roles in Lie theory. In a paper in 2013, A. Knutson, A. Woo, and A. Yong introduced certain maps that relate the singularities of Richardson varieties with that of Schubert varieties. In this thesis, the Knutson-Woo-Yong maps were studied from the points of view of Poisson geometry, and applications of the Knutson-Woo-Yong maps to Poisson geometry were obtained. In the first part of the thesis, the Knutson-Woo-Yong maps were generalized to certain homogeneous space $G/Q$ of $G$ and were shown to be Poisson isomorphisms with respect to a naturally defined Poisson structure $\pi_{\sG/\sQ}$ on $G/Q$. In the second part of the thesis, the generalized Knutson-Woo-Yong maps for $G/Q$ were used to define coordinate charts on $G/Q$, and it was shown that in {\it each} of such coordinate charts, the Poisson structure $\pi_{\sG/\sQ}$ on $G/Q$ gives rise to a polynomial Poisson algebra that is a {\it symmetric Cauchon-Goodearl-Letzter (CGL) extension (of ${\mathbb{C}}$)}. Primary examples of homogeneous spaces $G/Q$ of $G$ studied in the thesis were the orbits in the double flag variety $(G/B) \times (G/B_-)$ for the diagonal action of $G$, which include the flag variety $G/B$ as the closed orbit, and the symmetric Poisson CGL extensions arising from these examples are certain {\it $T$-mixed products} of the symmetric Poisson CGL extensions defined by generalized Bruhat cells, where $T$ is a maximal torus of $G$. Symmetric CGL extensions form a particular class of polynomial Poisson algebras that are iterated Poisson Ore extensions and have compatible torus actions. They have been recently introduced and studied by K. Goodearl and M. Yakimov in the context of cluster algebras and quantum groups. The results of the thesis show that important Poisson varieties in Lie theory can be covered by symmetric Poisson CGL extensions. | - |
dc.language | eng | - |
dc.publisher | The University of Hong Kong (Pokfulam, Hong Kong) | - |
dc.relation.ispartof | HKU Theses Online (HKUTO) | - |
dc.rights | The author retains all proprietary rights, (such as patent rights) and the right to use in future works. | - |
dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.subject.lcsh | Flag manifolds | - |
dc.subject.lcsh | Poisson algebras | - |
dc.title | On the Knutson-Woo-Yong maps and some poisson homogeneous spaces | - |
dc.type | PG_Thesis | - |
dc.description.thesisname | Doctor of Philosophy | - |
dc.description.thesislevel | Doctoral | - |
dc.description.thesisdiscipline | Mathematics | - |
dc.description.nature | published_or_final_version | - |
dc.identifier.doi | 10.5353/th_991044040582003414 | - |
dc.date.hkucongregation | 2018 | - |
dc.identifier.mmsid | 991044040582003414 | - |