## Principal Structures and Methods of Representation TheoryThe main topic of this book can be described as the theory of algebraic and topological structures admitting natural representations by operators in vector spaces. These structures include topological algebras, Lie algebras, topological groups, and Lie groups. The book is divided into three parts. Part I surveys general facts for beginners, including linear algebra and functional analysis. Part II considers associative algebras, Lie algebras, topological groups, and Lie groups,along with some aspects of ring theory and the theory of algebraic groups. The author provides a detailed account of classical results in related branches of mathematics, such as invariant integration and Lie's theory of connections between Lie groups and Lie algebras. Part III discusses semisimple Liealgebras and Lie groups, Banach algebras, and quantum groups. This is a useful text for a wide range of specialists, including graduate students and researchers working in mathematical physics and specialists interested in modern representation theory. It is suitable for independent study or supplementary reading. Also available from the AMS by this acclaimed author is Compact Lie Groups and Their Representations. |

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### Contents

3 | |

6 | |

Vector spaces 3 Elements of linear algebra | 14 |

Functional calculus 5 Unitary spaces | 26 |

Tensor products | 38 |

Smodules | 44 |

omments to Chapter 1 | 47 |

General Theory | 49 |

Local Lie groups | 203 |

Connected Lie groups | 209 |

Representations of Lie groups | 214 |

Examples and exercises | 219 |

Comments to Chapter 5 | 224 |

Special Topics | 225 |

Semisimple Lie Algebras | 227 |

Classification | 233 |

Associative Algebras | 51 |

Semisimple modules | 58 |

Group algebras | 64 |

Systems of generators | 70 |

Tensor algebras | 75 |

Formal series | 80 |

Weyl algebras | 86 |

Elements of ring theory | 93 |

Comments to Chapter 2 | 98 |

Lie Algebras | 99 |

Solvable Lie algebras | 105 |

Bilinear forms | 109 |

The algebra Ug | 115 |

Semisimple Lie algebras | 120 |

Free Lie algebras | 125 |

Examples of Lie algebras | 130 |

Comments to Chapter 3 | 137 |

Topological Groups | 139 |

Topological vector spaces | 145 |

Topological modules | 152 |

Invariant measures | 157 |

Group algebras | 164 |

Compact groups | 170 |

Solvable groups | 175 |

Algebraic groups | 181 |

Comments to Chapter 4 | 185 |

Lie Groups | 187 |

Lie groups | 192 |

Formal groups | 198 |

Verma modules | 238 |

Finitedimensional gmodules | 244 |

The algebra Zg | 250 |

The algebra Fºxtg | 256 |

Comments to Chapter 6 | 262 |

Semisimple Lie Groups | 263 |

Compact Lie groups | 268 |

Maximal tori | 272 |

Semisimple Lie groups | 277 |

The algebra AG | 283 |

The classical groups | 289 |

Reduction problems | 294 |

Comments to Chapter 7 | 300 |

Banach Algebras | 301 |

The commutative case | 307 |

Spectral theory | 312 |

Calgebras | 317 |

Representations of Calgebras | 323 |

Von Neumann algebras | 329 |

The algebra CG | 335 |

Abelian groups | 340 |

Comments to Chapter 8 | 346 |

Quantum Groups | 347 |

Appendix A Root Systems | 391 |

Appendix B Banach Spaces | 403 |

The Algebra BH | 413 |

421 | |

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Principal Structures and Methods of Representation Theory Dmitriĭ Petrovich Zhelobenko No preview available - 2006 |

### Common terms and phrases

Accordingly algebra g Applying associative algebra automorphism Banach basis bilinear form Borel C*-algebra called Cartan matrix Cartan subalgebra coincides commutative components condition contains continuous convergence Corollary decomposition defined definition direct sum dual easy to check elements a e equivalent everywhere dense example Exercise field F finite follows functions f g e G gives rise GL(n GL(V group G homomorphism Hopf algebras ideal identity implies induction isomorphism Lemma Let G Lie algebra Lie groups linear hull linearly independent module monomials multiplication neighborhood nilpotent nondegenerate norm normal notation Note operator a e orthogonal particular polynomial PROOF quotient reduces relations representation respectively seminorm semisimple Similarly simple simply connected solvable subgroup of G submodule subset subspace symmetric Theorem theory topological group topology unitary vector space Verma module Weyl

### Popular passages

Page 4 - ... Jordanform representations of square matrices, and to compute functions of a matrix, in particular, exponential functions of a matrix (see Section 2-9, Concluding Remarks1). In Section 2-2 we introduce the concepts of field and linear space over a field. The fields we shall encounter in this book are the field of real numbers, the field of complex numbers, and the field of rational functions. In order to have a representation of a vector in a linear space, we introduce, in Section 2-3, the concept...