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Compact Lie Groups and Their Representations - download pdf or read online

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By D. P. Zelobenko

ISBN-10: 0821815903

ISBN-13: 9780821815908

The contents of this quantity are a little diversified from the normal connotations of the identify. First, the writer, taking into consideration the desires of the physicist, has attempted to make the exposition as easy as attainable. the necessity for an trouble-free exposition has motivated the distribution of the cloth; the ebook is split into 3 principally autonomous elements, prepared so as of accelerating trouble. along with compact Lie teams, teams with different topological constitution (``similar'' to compact teams in a few experience) are thought of. admired between those are reductive complicated Lie teams (including semisimple groups), bought from compact Lie teams via analytic continuation, and in addition their actual types (reductive actual Lie groups). the idea of finite-dimensional illustration for those periods of teams is built, striving every time attainable to stress the ``compact origin'' of those representations, i.e. their analytic dating to representations of compact Lie teams. additionally studied are infinite-dimensional representations of semisimple advanced Lie algebras. a few features of the idea of infinite-dimensional representations of Lie teams are offered in a quick survey.

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Extra resources for Compact Lie Groups and Their Representations

Sample text

Il = vy und -;. Z = uz. Ist zum Beispiel E ein elektrischer Vektor, so gilt E Eo mit und analog fiir Ey und Ez. Fiir das Skalarprodukt zweier physikalischer Vektoren -; E Uo [ a 1 und U; E Uo [ b1 errechnet man wegen der Bilinearităt und wegen x· x = il· il = z· = 1 und x· il = il· = x = O sofort z -; . 6 fiir das Skalarprodukt eines Ortsvektors -+ r WAS SIND VEKTOREN? E 49 Ao mit einem elektrischen Vektor E E Go. Dies ist die erste der nutzlichen Vektoridentităten auf S. 44 im Berkeley Physics Course (6), die anderen haben mit dem Vektorprodukt zu tun, das wir erst im nâchsten Paragraphen betrachten werden.

Aber, so argwohnen Sie vielleicht, miissen wir dafiir nicht einen hohen Preis bezahlen? Ist nicht die lineare Algebra des abstrakten Vektorraums viel schwieriger als die lineare Algebra des R n ? - Keineswegs, antworte ich Ihnen, fast gar nicht: In mancher Hinsicht sogar einfacher und iibersichtlicher. Aber ganz umsonst bekommen wir den groBen Vorteil doch nicht, und besonders am Anfang haben wir im abstrakten Vektorraum einige Aussagen zu iiberpriifen, die sich in einem n-tupeloder Funktionenraum von selbst verstiinden, und es mag befremdlich und ein klein wenig beunruhigend wirken, daB solche Sachen nun beweisbediirftig sein sollen.

Der Nachteil dieser Definition ist nur, daf3 man dazu eine Vorrede braucht, deshalb habe ich sie im Haupttext nicht benutzt. h. daf3 fiir alle x, y, z, fiir die die Verkniipfung erklart ist, gilt: (1) (x+y)+z=x+(y+z) und(2) x+y=y+x. Wennes nun nochein "neutrales Element" O gibt und zu jedem x ein Negativ, dann nennt man die betreffende Menge zusammen mit der Verkniipfllng + eine abelsche Gruppe (nach dem norwegischen Mathematiker Niels Henrik Abel (1802-1829)): DEFINITION: Eine abelsche Gruppe ist ein Paar (A, +) bestehend aus einer Menge A und einer Verkniipfung + : A x A -> A, so daf3 gilt: (1) Fiiralle a,b,c E A ist (a+b)+c=a+(b+c).

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Compact Lie Groups and Their Representations by D. P. Zelobenko


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