Lie Algebras of Bounded Operators

ISBN-13: 9783034895200 / Angielski / Miękka / 2012 / 219 str.

Daniel Beltita; Mihai Sabac

Opis

In several proofs from the theory of finite-dimensional Lie algebras, an essential contribution comes from the Jordan canonical structure of linear maps acting on finite-dimensional vector spaces. On the other hand, there exist classical results concerning Lie algebras which advise us to use infinite-dimensional vector spaces as well. For example, the classical Lie Theorem asserts that all finite-dimensional irreducible representations of solvable Lie algebras are one-dimensional. Hence, from this point of view, the solvable Lie algebras cannot be distinguished from one another, that is, they cannot be classified. Even this example alone urges the infinite-dimensional vector spaces to appear on the stage. But the structure of linear maps on such a space is too little understood; for these linear maps one cannot speak about something like the Jordan canonical structure of matrices. Fortunately there exists a large class of linear maps on vector spaces of arbi- trary dimension, having some common features with the matrices. We mean the bounded linear operators on a complex Banach space. Certain types of bounded operators (such as the Dunford spectral, Foia decomposable, scalar generalized or Colojoara spectral generalized operators) actually even enjoy a kind of Jordan decomposition theorem. One of the aims of the present book is to expound the most important results obtained until now by using bounded operators in the study of Lie algebras.

Spis treści

I Preliminaries.- A Lie Algebras.- § 1 Basic facts.- § 2 Ideals, solvability, nilpotence, radical and semisimplicity.- § 3 SP classes of solvable Lie algebras.- § 4 Radical splitting theorems.- § 5 Cartan subalgebras.- § 6 The Killing form and compact Lie algebras.- § 7 Lie *-algebras.- B Complexes.- § 8 Generalities.- § 9 Banach space complexes.- § 10 Koszul complexes.- § 11 Operations with the Koszul complexes: duality and tensor products.- C Spectral Theory in Complex Banach Space.- § 12 General spectral theory and decomposable operators.- § 13 The transformation of the spectral maximal subspaces by bounded operators.- § 14 Special classes of decomposable operators.- Notes.- II The Commutators and Nilpotence Criteria.- § 15 An asymptotic formula for the commutators.- § 16 Nilpotence criteria in an associative algebra.- § 17 Quasinilpotence and nilpotence criteria in complex Banach algebras.- § 18 Nilpotent elements in LM-decomposable Lie subalgebras of an associative algebra.- § 19 Nilpotent elements in LM-decomposable Lie algebras of bounded linear operators.- Notes.- III Infinite-dimensional Variants of the Lie and Engel Theorems.- § 20 Weights for operator algebras.- § 21 Invariant subspaces for LM-decomposable Lie algebras of bounded operators.- § 22 The irreducible representations of an LM-decomposable Lie algebra. Infinite-dimensional variant of Lie's Theorem on a complex Banach space.- § 23 The associative envelope of a Lie algebra of quasinilpotent operators.- § 24 Commutativity modulo the Jacobson radical.- Notes.- IV Spectral Theory for Solvable Lie Algebras of Operators.- § 25 Spectral theory for representations of Lie algebras.- § 26 Spectral theory for systems of operators generating nilpotent Lie algebras.- § 27 The Cartan-Taylor spectrum of a locally solvable Lie algebra of operators.- § 28 Lie ideals of generalized spectral operators.- Notes.- V Semisimple Lie Algebras of Operators.- § 29 Lie subalgebra with involution consisting of bounded operators on a complex Banach space. Normal elements given by a space of self-adjoint operators.- § 30 Individual spectral properties in ideally finite semisimple Lie algebras of operators.- § 31 Semisimple Lie algebras of compact quasinilpotent operators.- Notes.- List of Symbols.