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postgraduate thesis: Computing the standard Poisson structure on BottSamelson varieties incoordinates
Title  Computing the standard Poisson structure on BottSamelson varieties incoordinates 

Authors  
Issue Date  2012 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Elek, B.. (2012). Computing the standard Poisson structure on BottSamelson varieties in coordinates. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833005 
Abstract  BottSamelson varieties associated to reductive algebraic groups are much studied in representation theory and algebraic geometry. They not only provide resolutions of singularities for Schubert varieties but also have interesting geometric properties of their own. A distinguished feature of BottSamelson varieties is that they admit natural affine coordinate charts, which allow explicit computations of geometric quantities in coordinates.
Poisson geometry dates back to 19th century mechanics, and the more recent theory of quantum groups provides a large class of Poisson structures associated to reductive algebraic groups. A holomorphic Poisson structure Π on BottSamelson varieties associated to complex semisimple Lie groups, referred to as the standard Poisson structure on BottSamelson varieties in this thesis, was introduced and studied by J. H. Lu. In particular, it was shown by Lu that the Poisson structure Π was algebraic and gave rise to an iterated Poisson polynomial algebra associated to each affine chart of the BottSamelson variety. The formula by Lu, however, was in terms of certain holomorphic vector fields on the BottSamelson variety, and it is much desirable to have explicit formulas for these vector fields in coordinates.
In this thesis, the holomorphic vector fields in Lu’s formula for the Poisson structure Π were computed explicitly in coordinates in every affine chart of the BottSamelson variety, resulting in an explicit formula for the Poisson structure Π in coordinates. The formula revealed the explicit relations between the Poisson structure and the root system and the structure constants of the underlying Lie algebra in any basis. Using a Chevalley basis, it was shown that the Poisson structure restricted to every affine chart of the BottSamelson variety was defined over the integers. Consequently, one obtained a large class of iterated Poisson polynomial algebras over any field, and in particular, over fields of positive characteristic. Concrete examples were given at the end of the thesis. 
Degree  Master of Philosophy 
Subject  Poisson manifolds. Lie groups Schubert varieties Root systems (Algebra) Coordinates. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/173875 
HKU Library Item ID  b4833005 
DC Field  Value  Language 

dc.contributor.author  Elek, Balázes.   
dc.date.issued  2012   
dc.identifier.citation  Elek, B.. (2012). Computing the standard Poisson structure on BottSamelson varieties in coordinates. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833005   
dc.identifier.uri  http://hdl.handle.net/10722/173875   
dc.description.abstract  BottSamelson varieties associated to reductive algebraic groups are much studied in representation theory and algebraic geometry. They not only provide resolutions of singularities for Schubert varieties but also have interesting geometric properties of their own. A distinguished feature of BottSamelson varieties is that they admit natural affine coordinate charts, which allow explicit computations of geometric quantities in coordinates. Poisson geometry dates back to 19th century mechanics, and the more recent theory of quantum groups provides a large class of Poisson structures associated to reductive algebraic groups. A holomorphic Poisson structure Π on BottSamelson varieties associated to complex semisimple Lie groups, referred to as the standard Poisson structure on BottSamelson varieties in this thesis, was introduced and studied by J. H. Lu. In particular, it was shown by Lu that the Poisson structure Π was algebraic and gave rise to an iterated Poisson polynomial algebra associated to each affine chart of the BottSamelson variety. The formula by Lu, however, was in terms of certain holomorphic vector fields on the BottSamelson variety, and it is much desirable to have explicit formulas for these vector fields in coordinates. In this thesis, the holomorphic vector fields in Lu’s formula for the Poisson structure Π were computed explicitly in coordinates in every affine chart of the BottSamelson variety, resulting in an explicit formula for the Poisson structure Π in coordinates. The formula revealed the explicit relations between the Poisson structure and the root system and the structure constants of the underlying Lie algebra in any basis. Using a Chevalley basis, it was shown that the Poisson structure restricted to every affine chart of the BottSamelson variety was defined over the integers. Consequently, one obtained a large class of iterated Poisson polynomial algebras over any field, and in particular, over fields of positive characteristic. Concrete examples were given at the end of the thesis.   
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  Creative Commons: AttributionNonCommerical 3.0 Hong Kong License   
dc.source.uri  http://hub.hku.hk/bib/B4833005X   
dc.subject.lcsh  Poisson manifolds.   
dc.subject.lcsh  Lie groups   
dc.subject.lcsh  Schubert varieties   
dc.subject.lcsh  Root systems (Algebra)   
dc.subject.lcsh  Coordinates.   
dc.title  Computing the standard Poisson structure on BottSamelson varieties incoordinates   
dc.type  PG_Thesis   
dc.identifier.hkul  b4833005   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4833005   
dc.date.hkucongregation  2012   