The Geometry of Jordan and Lie Structures: 1754 (Lecture Notes in Mathematics, 1754) - Softcover

Bertram, Wolfgang

Softcover

ISBN 10: 3540414266 ISBN 13: 9783540414261

Publisher: Springer, 2000

Synopsis

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).

About this title

Synopsis

The geometry of Jordan and Lie structures tries to answer the following question: what is the integrated, or geometric, version of real Jordan algebras, - triple systems and - pairs? Lie theory shows the way one has to go: Lie groups and symmetric spaces are the geometric version of Lie algebras and Lie triple systems. It turns out that both geometries are closely related via a functor between them, called the Jordan-Lie functor, which is constructed in this book. The reader is not assumed to have any knowledge of Jordan theory; the text can serve as a self-contained introduction to (real finite-dimensional) Jordan theory.

Edition details

• Publisher: Springer
• Publication date: 2000
• Language: English
• ISBN 10: 3540414266
• ISBN 13: 9783540414261
• Binding: Paperback
• Number of pages: 292