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Bott-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. / published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 51-52).

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-13T06:54:55Z (GMT). No. of bitstreams: 1 Silva_VitorMorettoFernandesda_M.pdf: 773065 bytes, checksum: 87c9c9b184da7470d32ab571c3b760d1 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho definimos quivers e discutimos como a categoria de módulos sobre uma álgebra conexa qualquer pode ser associada à categoria de representa equações de um quiver. Estudamos também o sistema de raízes da forma quadrática de Cartan associada a um quiver, que tem bijeção com os vetores dimensão de representação indecomponíveis. Estudamos a demonstração do Teorema de Gabriel, que caracteriza todos os quivers que têm quantidade finita de representações indecomponíveis a partir da forma de Cartan. O Teorema de Gabriel também define quando um quiver tem tipo manso. Apresentamos também a demonstração do Teorema de Ovsienko, que sob certas condições, caracteriza os quivers com relações que têm quantidade finita de representações indecomponíveis a partir da forma de Brenner / Abstract: In this work, we define quivers and their representations, and discuss how the category of modules over an arbitrary connected associative algebra can be associated to the category of representations of a quiver. We also study the root system of the Cartan quadratic form associated to a quiver, which is bijective with the set of dimension vectors for which an indecomposable representation exists. The proof of Gabriel's Theorem, which characterizes all quivers with finitely many indecomposable representations in terms of its Cartan form, is presented. Gabriel's Theorem also defines when a quiver is of tame time. Finally, we also describe a theorem due to Ovsienko which, under certain conditions, characterize the quivers with relations that admit only finitely many indecomposable representations in terms of its Brenner form / Mestrado / Mestre em Matemática