The Geometry of Jordan and Lie Structures

by Wolfgang Bertram

Overview

0. In this work of we the Lie- and Jordan on an study interplay theory and ona level.Weintendtocontinue ittoa algebraic geometric systematicstudy ofthe role Jordan inharmonic In the of theoryplays analysis. fact, applications the of Jordan to theharmonic on cones theory algebras analysis symmetric (cf. of the wereatthe theauthor'sworkinthisarea. Then monograph[FK94]) origin Jordan in of turned the causal algebras up study many symmetric (see spaces Section and clearthat all soon itbecame XI.3), "generically" symmetric spaces have Since a relation toJordan Jordan does not significant theory. theory (yet) to the standard tools inharmonic the is text belong analysis, present designed to self-contained introduction to Jordan for readers a provide theory having basic Lie and Our ofview some on knowledge groups symmetric spaces. point is introduce first the relevant structures geometric: throughout we geometric anddeducefromtheir identities fortheassociated propertiesalgebraic algebraic structures. Thus our differs from related ones presentation (cf. e.g. [FK94], the fact thatwe do not take an axiomatic definition ofsome [Lo77], [Sa80]) by Jordan structureasour Let us nowanoverviewof algebraic startingpoint. give the See alsothe introductions the contents. at ofeach given beginning chapter. 0.1. Lie and Jordan Ifwe the associative algebras algebras. decompose of the matrix in its and product algebra M(n, R) symmetric skew-symmetric parts, - XY YX XY YX + XY= + (0.1) 2 2 then second the term leads to the Lie with algebra gf(n, R) product [X, Y] XY- and first the termleadstotheJordan M with YX, algebra (n, R) product - X Y= + (XY YX).

Product Details

ISBN-13:	9783540414261
Publisher:	Springer Berlin Heidelberg
Publication date:	02/08/2001
Series:	Lecture Notes in Mathematics , #1754
Edition description:	2000
Pages:	274
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

The Jordan-Lie functor.- I: Symmetric spaces and the Lie-functor.- II: Prehomogeneous symmetric spaces and Jordan algebras.- III: The Jordan-Lie functor.- IV: The classical spaces.- V: Non-degenerate spaces.- Conformal group and global theory.- VI: Integration of Jordan structures.- VII: The conformal Lie algebra.- VIII: Conformal group and conformal completion.- IX: Liouville theorem and fundamental theorem.- X: Algebraic structures of symmetric spaces with twist.- XI: Spaces of the first and of the second kind.- XII: Tables.- XIII: Further topics.