Differential Geometry Lie Groups And Symmetric Spaces Over General Base Fields And Rings

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Differential Geometry Lie Groups And Symmetric Spaces Over General Base Fields And Rings

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Author : Wolfgang Bertram
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Differential Geometry Lie Groups And Symmetric Spaces Over General Base Fields And Rings written by Wolfgang Bertram and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

Invariant Differential Operators For Quantum Symmetric Spaces

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Author : Gail Letzter
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Invariant Differential Operators For Quantum Symmetric Spaces written by Gail Letzter and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.

The Generalized Triangle Inequalities In Symmetric Spaces And Buildings With Applications To Algebra

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Author : Michael Kapovich
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

The Generalized Triangle Inequalities In Symmetric Spaces And Buildings With Applications To Algebra written by Michael Kapovich and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over $\mathbb{Q}$ and its complex Langlands' dual. The authors give a new proof of the Saturation Conjecture for $GL(\ell)$ as a consequence of their solution of the corresponding saturation problem for the Hecke structure constants for all split reductive algebraic groups over $\mathbb{Q}$.

Sum Formula For Sl 2 Over A Totally Real Number Field

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Author : Roelof W. Bruggeman
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-01-21

Sum Formula For Sl 2 Over A Totally Real Number Field written by Roelof W. Bruggeman and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-01-21 with Mathematics categories.

The authors prove a general form of the sum formula $\mathrm{SL}_2$ over a totally real number field. This formula relates sums of Kloosterman sums to products of Fourier coefficients of automorphic representations. The authors give two versions: the spectral sum formula (in short: sum formula) and the Kloosterman sum formula. They have the independent test function in the spectral term, in the sum of Kloosterman sums, respectively.

The Recognition Theorem For Graded Lie Algebras In Prime Characteristic

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Author : Georgia Benkart
language : en
Publisher: American Mathematical Soc.
Release Date : 2009

The Recognition Theorem For Graded Lie Algebras In Prime Characteristic written by Georgia Benkart and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.

"Volume 197, number 920 (second of 5 numbers)."

Galois Extensions Of Structured Ring Spectra Stably Dualizable Groups

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Author : John Rognes
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Galois Extensions Of Structured Ring Spectra Stably Dualizable Groups written by John Rognes and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

Developments And Trends In Infinite Dimensional Lie Theory

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Author : Karl-Hermann Neeb
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-10-17

Developments And Trends In Infinite Dimensional Lie Theory written by Karl-Hermann Neeb and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-17 with Mathematics categories.

This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups. Contributors: B. Allison, D. Beltiţă, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

The History Of Continua

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Author : Stewart Shapiro
language : en
Publisher: Oxford University Press (UK)
Release Date : 2021

The History Of Continua written by Stewart Shapiro and has been published by Oxford University Press (UK) this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with Mathematics categories.

Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.

Toroidal Dehn Fillings On Hyperbolic 3 Manifolds

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Author : Cameron Gordon
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Toroidal Dehn Fillings On Hyperbolic 3 Manifolds written by Cameron Gordon and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

The authors determine all hyperbolic $3$-manifolds $M$ admitting two toroidal Dehn fillings at distance $4$ or $5$. They show that if $M$ is a hyperbolic $3$-manifold with a torus boundary component $T 0$, and $r,s$ are two slopes on $T 0$ with $\Delta(r,s) = 4$ or $5$ such that $M(r)$ and $M(s)$ both contain an essential torus, then $M$ is either one of $14$ specific manifolds $M i$, or obtained from $M 1, M 2, M 3$ or $M {14}$ by attaching a solid torus to $\partial M i - T 0$.All the manifolds $M i$ are hyperbolic, and the authors show that only the first three can be embedded into $S3$. As a consequence, this leads to a complete classification of all hyperbolic knots in $S3$ admitting two toroidal surgeries with distance at least $4$.

Rank One Higgs Bundles And Representations Of Fundamental Groups Of Riemann Surfaces

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Author : William Mark Goldman
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Rank One Higgs Bundles And Representations Of Fundamental Groups Of Riemann Surfaces written by William Mark Goldman and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of $X$. The authors describe the moduli spaces and their geometry in terms of the Riemann period matrix of $X$.