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-69121 Heidelberg, Germany Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Santa Clara Singapore Paris Tokyo
Contents Introduction 1 1. Lie Algebras 5 1.1 Basic Definitions 6 1.2 Universal Enveloping Algebra 13 1.3 Poincare - Birkhoff - Witt Theorem 18 1.4 Free Lie Algebras 19 1.5 Classification of Semisimple Finite-Dimensional Complex Lie Algebras 21 2. Eie Superalgebras 43 2.1 Basic Definitions 44 2.2 Universal Enveloping Superalgebra 49 2.3 Free Lie Superalgebras 54 2.4 Classification of Classical Lie Superalgebras 55 3. Coalgebras and Z2-Graded Hopf Algebras 73 3.1 Coalgebras over Commutative Rings 74; 3.2 Z 2 -Graded Bialgebras and Z 2 -Graded Hopf Algebras 84; 3.3 Comodules V 89 3.4 Duality of Finite-Dimensional Z2-Graded Hopf Algebras... 91 3.5 Examples of Z2-Graded Hopf Algebras 93 3.6 Smash Product of Z 2 -Graded Hopf Algebras 96 3.7 Graded Star Operations on Z 2 -Graded Hopf Algebras 100 3.8 Z 2 -Graded Lie Coalgebras and Lie Bialgebras 101 3.9 Quasitriangular Z 2 -Graded Lie Bialgebras 104 3.10 Duals of Quasitriangular Z 2 -Graded Lie Bialgebras 106 3.11 Deformed Tensor Product of Semiclassical MCR-Type Algebras 109 4. Formal Power Series with Homogeneous Relations 113 4.1 Polynomials in Finitely Many Commuting Indeterminates.. 113 4.2 Power Series in Finitely Many Commuting Indeterminates. 116
4.3 Power Series in Finitely Many Indeterminates with Homogeneous Relations 125 4.4 Tensor Product of Formal Power Series with Homogeneous Relations 129 5. Z 2 -Graded Lie - Cartan Pairs 133 5.1 Tensor Products over Graded-Commutative Algebras 134 5.2 Projective-Finite Modules 143 5.3 Lie - Cartan Pairs 147 5.4 Real DifferentiaLForms 155 5.5 Cohomologies of Lie - Cartan Pairs 162 5.6 Z 2 -Graded Lie - Cartan Pairs 167 5.7 Real Z 2 -Graded Differential Forms 183 5.8 Cohomologies of Z 2 -Graded Lie - Cartan Pairs 189 6. Real Lie - Hopf Superalgebras 191 6.1 Linear Forms on Graded-Commutative Associative Superalgebras 192 6.2 Real Lie - Hopf Superalgebras., 195 6.3 Z 2 -Graded Distributions with Finite Support 198 6.4 Lie Supergroups as Real Lie - Hopf Superalgebras 202 6.5 Linear Supergroups 206 7. Universal Differential Envelope 213 7.1 Non-Unital Universal Differential Envelope 216 7.2 Explicit Construction of the Unital Universal Differential Envelope 219 7.3.R-Endomorphisms of Q{A) 225 7.4 Hochschild and Cyclic Cohomology 232 7.5 Graded Tensor Product of Bigraded Differential Algebras.. 237 7.6 Universal Differential Envelope of the Graded Tensor Product of Associative Superalgebras 238 7.7 Universal Differential Envelope of a Commutative Ring... 241 7.8 Universal Differential Envelopes of Finite-Dimensional Clifford Algebras 244 7.9 Differential Envelopes of Real Scalar Fields 248 7.10 Universal Differential Envelope as Factor Algebra 252 7.11 Extension of Graded Star Operations to the Universal Differential Envelope 255 7.12 Cycles over Associative Superalgebras 259 7.13 Fredholm Modules 262 7.14 Connes Modules 271
8. Quantum Groups 279 8.1 Duality of Z 2 -Graded Hopf Algebras 284 8.2 Quasitriangular Z 2 -Graded Hopf Algebras 285 8.3 Matrices with Non-Commuting Components 288 8.4 Transformations of the Quantum Plane 293 8.5 Transformations of the Quantum Superplane 300 8.6 Transformations of Quantum Superspace 307 8.7 Topological Z 2 -Graded Hopf Algebras 313 8.8 g-deformation of si (2, C) 320 8.9 g-deformation of Simple Finite-Dimensional Complex Lie Algebras 327 8.10 Finite-Dimensional Quantum Double 331 8.11 Universal fl-matrix of U q {A m ) 332 8.12 Universal.R-Matrix of g-deformed Simple Lie Algebras... 341 8.13 Quantum Weyl Group 347 8.14 g-deformation of Oscillator Algebras 350 8.15 Oscillator and Spinor Representations of g-deformed Universal Enveloping Algebras 356 8.16 Main Commutation Relations and Matrix Quantum Semigroups 359 8.17 Fundamental Representation of U q (A2) 361 8.18 Fundamental Representation of C/ q (B 2 ) 363 8.19 q-deformation of Basic Classical Lie Superalgebras 366 8.20 Universal.R-Matrix of U q (B(0,1)) / 373 8.21 Duals of Quasitriangular Z 2 -Graded Hopf Algebras 374 8.22 Deformed Tensor Product of Matrix Quantum Super-Semigroups 380 8.23 Deformed Tensor Product of, Quantum Super-Vector Spaces - 382 8.24 Covariant Differential Calculus on Quantum Superspaces.. 385 8.25 Fundamental Representations of g-deformed Universal Enveloping Algebras 389 8.26 Generic Representations of g-deformed Universal Enveloping Algebras 393 8.27 Cyclic Representations of g-deformed Universal Enveloping Algebras 397 9. Categorial Viewpoint 405 9.1 Categories and Functors 406 9.2 Abelian Categories 414 9.3 Quasitensor Categories 419 9.4 Rigid Quasitensor Categories 425 XI
XII Bibliography 433 Monographs 433 Contributions to Journals 436 Contributions to Proceedings 449 Notation 453 Index..., 461