Generalized Lie Theory in Mathematics, Physics and Beyond

ISBN-13: 9783642099045 / Angielski / Miękka / 2010 / 306 str.

Sergei D. Silvestrov; Eugen Paal; Viktor Abramov

Opis

The aim of this book is to extend the understanding of the fundamental role of generalizations of Lie and related non-commutative and non-associative structures in Mathematics and Physics. This is a thematic volume devoted to the interplay between several rapidly exp- ding research ?elds in contemporary Mathematics and Physics, such as generali- tions of the main structures of Lie theory aimed at quantization and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, n- commutative geometry and applications in Physics and beyond. The speci?c ?elds covered by this volume include: Applications of Lie, non-associative and non-commutative associative structures to generalizations of classical and quantum mechanics and non-linear integrable systems, operadic and group theoretical methods; Generalizations and quasi-deformations of Lie algebras such as color and super Lie algebras, quasi-Lie algebras, Hom Lie algebras, in?nite-dimensional Lie algebras of vector ?elds associated to Riemann surfaces, quasi-Lie algebras of Witt type and their central extensions and deformations important for in- grable systems, for conformal ?eld theory and for string theory; Non-commutative deformation theory, moduli spaces and interplay with n- commutativegeometry, algebraicgeometryandcommutativealgebra, q-deformed differential calculi and extensions of homological methods and structures; Crossed product algebras and actions of groups and semi-groups, graded rings and algebras, quantum algebras, twisted generalizations of coalgebras and Hopf algebra structures such as Hom-coalgebras, Hom-Hopf algebras, and super Hopf algebras and their applications to bosonisation, parastatistics, parabosonic and parafermionic algebras, orthoalgebas and root systems in quantum mechanics;"

Spis treści

Non-Associative and Non-Commutative Structures for Physics.- Moufang Transformations and Noether Currents.- Weakly Nonassociative Algebras, Riccati and KP Hierarchies.- Applications of Transvectants.- Automorphisms of Finite Orthoalgebras, Exceptional Root Systems and Quantum Mechanics.- A Rewriting Approach to Graph Invariants.- Non-Commutative Deformations, Quantization, Homological Methods, and Representations.- Graded q-Differential Algebra Approach to q-Connection.- On Generalized N-Complexes Coming from Twisted Derivations.- Remarks on Quantizations, Words and R-Matrices.- Connections on Modules over Singularities of Finite and Tame CM Representation Type.- Computing Noncommutative Global Deformations Of D-Modules.- Comparing Small Orthogonal Classes.- Groups and Actions.- How to Compose Lagrangian?.- Semidirect Products of Generalized Quaternion Groups by a Cyclic Group.- A Characterization Of A Class Of 2-Groups By Their Endomorphism Semigroups.- Adjoint Representations and Movements.- Applications of Hypocontinuous Bilinear Maps in Infinite-Dimensional Differential Calculus.- Quasi-Lie, Super-Lie, Hom-Hopf and Super-Hopf Structures and Extensions, Deformations and Generalizations of Infinite-Dimensional Lie Algebras.- Hom-Lie Admissible Hom-Coalgebras and Hom-Hopf Algebras.- Bosonisation and Parastatistics.- Deformations of the Witt, Virasoro, and Current Algebra.- Conformal Algebras in the Context of Linear Algebraic Groups.- Lie Color and Hom-Lie Algebras of Witt Type and Their Central Extensions.- A Note on Quasi-Lie and Hom-Lie Structures of ?-Derivations of C=[Z 1 ±1 ,…,Z n ±1 ].- Commutative Subalgebras in Noncommutative Algebras.- Algebraic Dependence of Commuting Elements in Algebras.- Crossed Product-Like and Pre-Crystalline Graded Rings.- Decomposition of the Enveloping Algebra so(5).

Blurb

The goal of this book is to extend the understanding of the fundamental role of generalizations of Lie theory and related non-commutative and non-associative structures in mathematics and physics.
This volume is devoted to the interplay between several rapidly expanding research fields in contemporary mathematics and physics concerned with generalizations of the main structures of Lie theory aimed at quantization and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, non-commutative geometry and applications in physics and beyond.
The book will be a useful source of inspiration for a broad spectrum of researchers and for research students, and includes contributions from several large research communities in modern mathematics and physics.
This volume consists of 5 parts comprising 25 chapters, which were contributed by 32 researchers from 12 different countries. All contributions in the volume have been refereed.