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postgraduate thesis: Computing the standard Poisson structure on Bott-Samelson varieties incoordinates

TitleComputing the standard Poisson structure on Bott-Samelson varieties incoordinates
Authors
Issue Date2012
PublisherThe University of Hong Kong (Pokfulam, Hong Kong)
Citation
Elek, B.. (2012). Computing the standard Poisson structure on Bott-Samelson varieties in coordinates. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833005
AbstractBott-Samelson 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 Bott-Samelson 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 Bott-Samelson varieties associated to complex semisimple Lie groups, referred to as the standard Poisson structure on Bott-Samelson 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 Bott-Samelson variety. The formula by Lu, however, was in terms of certain holomorphic vector fields on the Bott-Samelson 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 Bott-Samelson 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 Bott-Samelson 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.
DegreeMaster of Philosophy
SubjectPoisson manifolds.
Lie groups
Schubert varieties
Root systems (Algebra)
Coordinates.
Dept/ProgramMathematics

 

DC FieldValueLanguage
dc.contributor.authorElek, Balázes.-
dc.date.issued2012-
dc.identifier.citationElek, B.. (2012). Computing the standard Poisson structure on Bott-Samelson varieties in coordinates. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833005-
dc.description.abstractBott-Samelson 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 Bott-Samelson 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 Bott-Samelson varieties associated to complex semisimple Lie groups, referred to as the standard Poisson structure on Bott-Samelson 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 Bott-Samelson variety. The formula by Lu, however, was in terms of certain holomorphic vector fields on the Bott-Samelson 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 Bott-Samelson 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 Bott-Samelson 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.languageeng-
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)-
dc.relation.ispartofHKU Theses Online (HKUTO)-
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.source.urihttp://hub.hku.hk/bib/B4833005X-
dc.subject.lcshPoisson manifolds.-
dc.subject.lcshLie groups-
dc.subject.lcshSchubert varieties-
dc.subject.lcshRoot systems (Algebra)-
dc.subject.lcshCoordinates.-
dc.titleComputing the standard Poisson structure on Bott-Samelson varieties incoordinates-
dc.typePG_Thesis-
dc.identifier.hkulb4833005-
dc.description.thesisnameMaster of Philosophy-
dc.description.thesislevelMaster-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.5353/th_b4833005-
dc.date.hkucongregation2012-

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