Lecture Notes in Physics

Transcription

1 Lecture Notes in Physics New Series m: Monographs Editorial Board' H. Araki, Kyoto, Japan E. Brdzin, Paris, France J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Zurich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Kippenhahn, Gottingen, Germany H. A. Weidenmuller, Heidelberg, Germany J. Wess, Miinchen, Germany J. Zittartz, Koln, Germany Managing Editor W. Beiglbock Assisted^by Mrs. Sabine Landgraf c/o Springer-Verlag, Physics Editorial Department II Tiergartenstrasse 17, D Heidelberg, Germany Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Santa Clara Singapore Paris Tokyo

2 Contents Introduction 1 1. Lie Algebras Basic Definitions Universal Enveloping Algebra Poincare - Birkhoff - Witt Theorem Free Lie Algebras Classification of Semisimple Finite-Dimensional Complex Lie Algebras Eie Superalgebras Basic Definitions Universal Enveloping Superalgebra Free Lie Superalgebras Classification of Classical Lie Superalgebras Coalgebras and Z2-Graded Hopf Algebras Coalgebras over Commutative Rings 74; 3.2 Z 2 -Graded Bialgebras and Z 2 -Graded Hopf Algebras 84; 3.3 Comodules V Duality of Finite-Dimensional Z2-Graded Hopf Algebras Examples of Z2-Graded Hopf Algebras Smash Product of Z 2 -Graded Hopf Algebras Graded Star Operations on Z 2 -Graded Hopf Algebras Z 2 -Graded Lie Coalgebras and Lie Bialgebras Quasitriangular Z 2 -Graded Lie Bialgebras Duals of Quasitriangular Z 2 -Graded Lie Bialgebras Deformed Tensor Product of Semiclassical MCR-Type Algebras Formal Power Series with Homogeneous Relations Polynomials in Finitely Many Commuting Indeterminates Power Series in Finitely Many Commuting Indeterminates. 116

3 4.3 Power Series in Finitely Many Indeterminates with Homogeneous Relations Tensor Product of Formal Power Series with Homogeneous Relations Z 2 -Graded Lie - Cartan Pairs Tensor Products over Graded-Commutative Algebras Projective-Finite Modules Lie - Cartan Pairs Real DifferentiaLForms Cohomologies of Lie - Cartan Pairs Z 2 -Graded Lie - Cartan Pairs Real Z 2 -Graded Differential Forms Cohomologies of Z 2 -Graded Lie - Cartan Pairs Real Lie - Hopf Superalgebras Linear Forms on Graded-Commutative Associative Superalgebras Real Lie - Hopf Superalgebras., Z 2 -Graded Distributions with Finite Support Lie Supergroups as Real Lie - Hopf Superalgebras Linear Supergroups Universal Differential Envelope Non-Unital Universal Differential Envelope Explicit Construction of the Unital Universal Differential Envelope R-Endomorphisms of Q{A) Hochschild and Cyclic Cohomology Graded Tensor Product of Bigraded Differential Algebras Universal Differential Envelope of the Graded Tensor Product of Associative Superalgebras Universal Differential Envelope of a Commutative Ring Universal Differential Envelopes of Finite-Dimensional Clifford Algebras Differential Envelopes of Real Scalar Fields Universal Differential Envelope as Factor Algebra Extension of Graded Star Operations to the Universal Differential Envelope Cycles over Associative Superalgebras Fredholm Modules Connes Modules 271

4 8. Quantum Groups Duality of Z 2 -Graded Hopf Algebras Quasitriangular Z 2 -Graded Hopf Algebras Matrices with Non-Commuting Components Transformations of the Quantum Plane Transformations of the Quantum Superplane Transformations of Quantum Superspace Topological Z 2 -Graded Hopf Algebras g-deformation of si (2, C) g-deformation of Simple Finite-Dimensional Complex Lie Algebras Finite-Dimensional Quantum Double Universal fl-matrix of U q {A m ) Universal.R-Matrix of g-deformed Simple Lie Algebras Quantum Weyl Group g-deformation of Oscillator Algebras Oscillator and Spinor Representations of g-deformed Universal Enveloping Algebras Main Commutation Relations and Matrix Quantum Semigroups Fundamental Representation of U q (A2) Fundamental Representation of C/ q (B 2 ) q-deformation of Basic Classical Lie Superalgebras Universal.R-Matrix of U q (B(0,1)) / Duals of Quasitriangular Z 2 -Graded Hopf Algebras Deformed Tensor Product of Matrix Quantum Super-Semigroups Deformed Tensor Product of, Quantum Super-Vector Spaces Covariant Differential Calculus on Quantum Superspaces Fundamental Representations of g-deformed Universal Enveloping Algebras Generic Representations of g-deformed Universal Enveloping Algebras Cyclic Representations of g-deformed Universal Enveloping Algebras Categorial Viewpoint Categories and Functors Abelian Categories Quasitensor Categories Rigid Quasitensor Categories 425 XI

5 XII Bibliography 433 Monographs 433 Contributions to Journals 436 Contributions to Proceedings 449 Notation 453 Index..., 461