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Torsors, Reductive Group Schemes and Extended Affine Lie Algebras

2013-10-23 Mathematics
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Release : 2013-10-23
Genre : Mathematics
Kind : eBook
Book Rating : 742/5 ( reviews)

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Book Synopsis Torsors, Reductive Group Schemes and Extended Affine Lie Algebras by : Philippe Gille

Download or read book Torsors, Reductive Group Schemes and Extended Affine Lie Algebras written by Philippe Gille. This book was released on 2013-10-23. Available in PDF, EPUB and Kindle. Book excerpt: The authors give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that the authors take draws heavily from the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows the authors to find a bridge between multiloop algebras and the work of F. Bruhat and J. Tits on reductive groups over complete local fields.

Algebraic Groups

2017-09-21 Mathematics
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Release : 2017-09-21
Genre : Mathematics
Kind : eBook
Book Rating : 155/5 ( reviews)

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Book Synopsis Algebraic Groups by : J. S. Milne

Download or read book Algebraic Groups written by J. S. Milne. This book was released on 2017-09-21. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Introduction to Affine Group Schemes

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 178/5 ( reviews)

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Book Synopsis Introduction to Affine Group Schemes by : W.C. Waterhouse

Download or read book Introduction to Affine Group Schemes written by W.C. Waterhouse. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: Ah Love! Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre' KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme.

On the Spectra of Quantum Groups

2014-04-07 Mathematics
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Release : 2014-04-07
Genre : Mathematics
Kind : eBook
Book Rating : 74X/5 ( reviews)

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Book Synopsis On the Spectra of Quantum Groups by : Milen Yakimov

Download or read book On the Spectra of Quantum Groups written by Milen Yakimov. This book was released on 2014-04-07. Available in PDF, EPUB and Kindle. Book excerpt: Joseph and Hodges-Levasseur (in the A case) described the spectra of all quantum function algebras on simple algebraic groups in terms of the centers of certain localizations of quotients of by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. The author determines the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it he deduces a more explicit description of all prime ideals of than the previously known ones and an explicit parametrization of .

Cohomology for Quantum Groups via the Geometry of the Nullcone

2014-04-07 Mathematics
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Release : 2014-04-07
Genre : Mathematics
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Book Rating : 758/5 ( reviews)

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Book Synopsis Cohomology for Quantum Groups via the Geometry of the Nullcone by : Christopher P. Bendel

Download or read book Cohomology for Quantum Groups via the Geometry of the Nullcone written by Christopher P. Bendel. This book was released on 2014-04-07. Available in PDF, EPUB and Kindle. Book excerpt: In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G -modules stipulates that p=h. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H (u ? ,C) of the small quantum group.

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