Background
Jantzen, Jens Carsten was born on October 18, 1948 in Stoertewerker-Koog, Schleswig-Holstein, Federal Republic of Germany. Came to the United States, 1988. Son of Ewald and Annelene (Steensen) Jantzen.
(Back in print from the AMS, the first part of this book i...)
Back in print from the AMS, the first part of this book is an introduction to the general theory of representations of algebraic group schemes. Here, Janzten describes important basic notions: induction functors, cohomology, quotients, Frobenius kernels, and reduction mod $p$, among others. The second part of the book is devoted to the representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, the Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and line bundles on them. This is a significantly revised edition of a modern classic. The author has added nearly 150 pages of new material describing later developments and has made major revisions to parts of the old text. It continues to be the ultimate source of information on representations of algebraic groups in finite characteristics. The book is suitable for graduate students and research mathematicians interested in algebraic groups and their representations. Algebra, as a subdiscipline of mathematics, arguably has a history going back some 4000 years to ancient Mesopotamia.The history, however, of what is recognized today as high school algebra is much shorter, extending back to the sixteenth century, while the history of what practicing mathematicians call "modern algebra" is even shorter still. The present volume provides a glimpse into the complicated and often convoluted history of this latter conception of algebra by juxtaposing twelve episodes in the evolution of modern algebra from the early nineteenth-century work of Charles Babbage on functional equations to Alexandre Grothendieck's mid-twentieth-century metaphor of a "rising sea" in his categorical approach to algebraic geometry. In addition to considering the technical development of various aspects of algebraic thought, the historians of modern algebra whose work is united in this volume explore such themes as the changing aims and organization of the subject as well as the often complex lines of mathematical communication within and across national boundaries. Among the specific algebraic ideas considered are the concept of divisibility and the introduction of non-commutative algebras into the study of number theory and the emergence of algebraic geometry in the twentieth century.The resulting volume is essential reading for anyone interested in the history of modern mathematics in general and modern algebra in particular. It will be of particular interest to mathematicians and historians of mathematics. Information for our distributors: Co-published with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.
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(Es sei 9 eine endlich dimensionale Lie-Algebra uber dem K...)
Es sei 9 eine endlich dimensionale Lie-Algebra uber dem Korper der komple xen Zahlen. In der Darstellungstheorie von gist eine der am einfachsten zu stellenden Fragen die nach einer Beschreibung aller irreduziblen Darstellungen von 9 oder (iiquivalent dazu) aller einfacher Moduln uber der universellen ein hullenden Algebra U (g) von g. Eine einfache Antwort auf diese Frage hat man nur, wenn 9 kommutativ ist. Hier ist auch U(g) kommutativ, also entsprechen die Isomorphieklassen einfa cher U (g)-Moduln eindeutig den maximal en Idealen in U (g). Da hier U (g) zur Algebra der polynomialen Funktionen auf dem Dualraum g* von 9 isomorph ist, werden diese maximalen Ideale nach dem schwachen Nullstellensatz durch die Punkte von g* klassifiziert. Jede irreduzible Darstellung von gist demnach eindimensional, jede Linearform auf 9 legt soleh eine Darstellung fest. Fur andere Lie-Algebren sind die Verhiiltnisse viel komplizierter. 1st 9 zum Beispiel einfach, so ist bisher nur fUr g=Glz eine Klassifikation der irreduzib len Darstellungen bekannt (vorgelegt von R Block), die jedoch weit davon ent femt ist, iihnlich explizit wie die im kommutativen Fall zu sein. Fur noch gro Bere Lie-Algebren scheint selbst eine solehe Klassifikation nicht erreichbar zu sein. Es scheint daher sinnvoll, zuniichst ein einfacheres Problem zu losen, das im kommutativen Fall mit dem alten zusammenfiillt. Dies ist die Untersuchung der primitiven Ideale von U(g), das heiBt der Annullatoren in U(g) der einfa chen U(g)-Moduln. Man mag hoffen, daraus auch Informationen uber die moglichen einfachen Moduln zu erhalten.
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(The book contains several well-written, accessible survey...)
The book contains several well-written, accessible survey papers in many interrelated areas of current research. These areas cover various aspects of the representation theory of Lie algebras, finite groups of Lie types, Hecke algebras, and Lie superalgebras. Geometric methods have been instrumental in representation theory, and these proceedings include surveys on geometric as well as combinatorial constructions of the crystal basis for representations of quantum groups. Humphreys' paper outlines intricate connections among irreducible representations of certain blocks of reduced enveloping algebras of semi-simple Lie algebras in positive characteristic, left cells in two sided cells of affine Weyl groups, and the geometry of the nilpotent orbits. All these papers provide the reader with a broad picture of the interaction of many different research areas and should be helpful to those who want to have a glimpse of current research involving representation theory.
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Jantzen, Jens Carsten was born on October 18, 1948 in Stoertewerker-Koog, Schleswig-Holstein, Federal Republic of Germany. Came to the United States, 1988. Son of Ewald and Annelene (Steensen) Jantzen.
Doctor of Philosophy, University Bonn, Federal Republic of Germany, 1973.
Assistant professor University Bonn, 1973-1978, associate professor, 1978-1985. Professor University Hamburg, Federal Republic of Germany, 1985-1988. Professor mathematics University Oregon, Eugene, 1988-1995, University Aarhus, Denmark, since 1995.
Lecturer in field.
(Back in print from the AMS, the first part of this book i...)
(The book contains several well-written, accessible survey...)
(Es sei 9 eine endlich dimensionale Lie-Algebra uber dem K...)
Member American Mat. Society, German Mathematics Society, London Mathematics Society.