Nilpotent Orbits Primitive Ideals And Characteristic Classes

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Author: Walter Borho
Publisher: Springer Science & Business Media
ISBN: 1461245583
Size: 74.22 MB
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Nilpotent Orbits Primitive Ideals And Characteristic Classes by Walter Borho


Original Title: Nilpotent Orbits Primitive Ideals And Characteristic Classes

1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials The "vertices" of this graph are some of the most important objects in representation theory. Each has a theory in its own right, and each has had its own independent historical development. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. (For the special linear group 2 G = SL(n,C), whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. In this case, the nilpotent orbits are classified by partitions of n, given by the sizes of the Jordan blocks.) The closures of the nilpotent orbits are singular in general, and understanding their singularities is an important problem. - The classification of irreducible Weyl group representations is quite old.

Quantum Groups And Their Primitive Ideals

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Author: Anthony Joseph
Publisher: Springer Science & Business Media
ISBN: 3642784003
Size: 68.51 MB
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Quantum Groups And Their Primitive Ideals by Anthony Joseph


Original Title: Quantum Groups And Their Primitive Ideals

by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.

The Book Of Prime Number Records

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Author: Paulo Ribenboim
Publisher: Springer Science & Business Media
ISBN: 1468405071
Size: 49.98 MB
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The Book Of Prime Number Records by Paulo Ribenboim


Original Title: The Book Of Prime Number Records

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquim series established to honor Professors A. J. Coleman and H. W. Ellis and to acknow ledge their long lasting interest in the quality of teaching under graduate students. In another colloquim lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guilllless Book oj Records, remainded me very gently that the most "innumerate" people of the world are of a certain tribe in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes Morris, I'm from Brazil, but my book will contain numbers different from 'one.' " He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name), and consists of about 16 million digits of the number 11. "I assure you Morris, that in spite of the beauty of the apparent randomness of the decimal digits of 11, I'll be sure that my text will include also some words." Acknowledgment. The manuscript of this book was prepared on the word processor by Linda Nuttall. I wish to express my appreciation for the great care, speed, and competence of her work.

The New Book Of Prime Number Records

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Author: Paulo Ribenboim
Publisher: Springer Science & Business Media
ISBN: 1461207592
Size: 76.41 MB
Format: PDF, Mobi
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The New Book Of Prime Number Records by Paulo Ribenboim


Original Title: The New Book Of Prime Number Records

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium senes. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book of Records, reminded me very gently that the most "innumerate" people of the world are of a certain trible in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes, Morris, I'm from Brazil, but my book will contain numbers different from ·one.''' He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name) and consists of about 16 million decimal digits of the number Te. "I assure you, Morris, that in spite of the beauty of the appar ent randomness of the decimal digits of Te, I'll be sure that my text will include also some words." And then I proceeded putting together the magic combina tion of words and numbers, which became The Book of Prime Number Records. If you have seen it, only extreme curiosity could impel you to have this one in your hands. The New Book of Prime Number Records differs little from its predecessor in the general planning. But it contains new sections and updated records.

Dimensions Of Ring Theory

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Author: C. Nastasescu
Publisher: Springer Science & Business Media
ISBN: 9789027724618
Size: 70.21 MB
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Dimensions Of Ring Theory by C. Nastasescu


Original Title: Dimensions Of Ring Theory

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of s9phistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Lebendige Zahlen

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Author: W. Borho
Publisher: Springer-Verlag
ISBN: 3034854072
Size: 53.66 MB
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Lebendige Zahlen by W. Borho


Original Title: Lebendige Zahlen

Die Habilitation ist eine alte Einrichtung der deutschen Universi täten. Nachdem der junge Doktor nach der Promotion einige Jahre wissenschaftlich gearbeitet hat, legt er der Fakultät eine größere Ver öffentlichung, die Habilitationsschrift, mit dem Ziel vor, die Venia Legendi zu erwerben, das Recht, in eigener Verantwortung zu lehren und Vorlesungen zu halten. Im Rahmen einer Fakultätssitzung fin den Habilitationsvortrag und wissenschaftliche Aussprache statt. Einige Wochen später wird das Verfahren durch die öffentliche An trittsvorlesung abgeschlossen, die zugleich den Höhepunkt darstellt. An manchen Universitäten scheint die Bedeutung der Antritts vorlesung zurückgegangen zu sein. In Bonn haben wir uns bemüht, diese Tradition fortzuftihren und die Antrittsvorlesungen zu einem wichtigen Teil des mathematischen Lebens zu machen. In den Vor trägen soll ein anspruchsvolles Thema so dargestellt werden, daß Mathematiker und Naturwissenschaftler, die vielleicht in ganz ande ren Gebieten arbeiten, und auch Studenten, den Ausftihrungen fol gen können. Historische Entwicklungen und die Zusammenhänge der mathematischen Gebiete untereinander und mit Bereichen außer halb der Mathematik sollen erkennbar werden, den Zuhörern soll et was von der Faszination der Mathematik und ihrer Rolle im geisti gen Leben vermittelt werden.

Representations Of Semisimple Lie Algebras In The Bgg Category O

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Author: James E. Humphreys
Publisher: American Mathematical Soc.
ISBN: 0821846787
Size: 42.64 MB
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Representations Of Semisimple Lie Algebras In The Bgg Category O by James E. Humphreys


Original Title: Representations Of Semisimple Lie Algebras In The Bgg Category O

This is the first textbook treatment of work leading to the landmark 1979 Kazhdan-Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. The setting is the module category $\mathscr {O}$ introduced by Bernstein-Gelfand-Gelfand, which includes all highest weight modules for $\mathfrak{g}$ such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of $\mathfrak{g}$. Basic techniques in category $\mathscr {O}$ such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan-Lusztig Conjecture (due to Beilinson-Bernstein and Brylinski-Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: $D$-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category $\mathscr {O}$, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson-Ginzburg-Soergel.

Handbook Of Number Theory Ii

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Author: Jozsef Sandor
Publisher: Springer Science & Business Media
ISBN: 1402025467
Size: 33.20 MB
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Handbook Of Number Theory Ii by Jozsef Sandor


Original Title: Handbook Of Number Theory Ii

This handbook focuses on some important topics from Number Theory and Discrete Mathematics. These include the sum of divisors function with the many old and new issues on Perfect numbers; Euler's totient and its many facets; the Möbius function along with its generalizations, extensions, and applications; the arithmetic functions related to the divisors or the digits of a number; the Stirling, Bell, Bernoulli, Euler and Eulerian numbers, with connections to various fields of pure or applied mathematics. Each chapter is a survey and can be viewed as an encyclopedia of the considered field, underlining the interconnections of Number Theory with Combinatorics, Numerical mathematics, Algebra, or Probability Theory. This reference work will be useful to specialists in number theory and discrete mathematics as well as mathematicians or scientists who need access to some of these results in other fields of research.

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