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Hans Samelson, Notes on Lie Algebras Hans Samelson Notes on Lie Algebras Third Corrected Edition To Nancy Preface to the New Edition This is a revised edition of my “Notes on Lie Algebras" of 1969. Since that time I have gone over the material in lectures at Stanford University and at the University of Crete (whose Department of Mathematics I thank for its hospitality in 1988). The purpose, as before, is to present a simple straightforward introduc- tion, for the general mathematical reader, to the theory of Lie algebras, specifically to the structure and the (finite dimensional) representations of the semisimple Lie algebras. I hope the book will also enable the reader to enter into the more advanced phases of the theory. I have tried to make all arguments as simple and direct as I could, with- out entering into too many possible ramifications. In particular I use only the reals and the complex numbers as base fields. The material, most of it discovered by W. Killing, E. Cartan and H. Weyl, is quite classical by now. The approach to it has changed over the years, mainly by becoming more algebraic. (In particular, the existence and the complete reducibility of representations was originally proved by Analysis; after a while algebraic proofs were found.) — The background needed for these notes is mostly linear algebra (of the geometric kind; vector spaces and linear transformations in preference to column vectors and matrices, although the latter are used too). Relevant facts and the no- tation are collected in the Appendix. Some familiarity with the usual gen- eral facts about groups, rings, and homomorphisms, and the standard basic facts from analysis is also assumed. The first chapter contains the necessary general facts about Lie algebras. Semisimplicity is defined and Cartan’s criterion for it in terms of a certain quadratic form, the Killing form, is developed. The chapter also brings the representations of sl(2, C), the Lie algebra consisting of the 2 × 2 complex matrices with trace 0 (or, equivalently, the representations of the Lie group SU(2), the 2 × 2 special-unitary matrices M, i.e. with M · M ∗ = id and detM = 1). This Lie algebra is a quite fundamental object, that crops up at many places, and thus its representations are interesting in themselves; in addition these results are used quite heavily within the theory of semisim- ple Lie algebras. The second chapter brings the structure of the semisimple Lie algebras (Cartan sub Lie algebra, roots, Weyl group, Dynkin diagram,...) and the classification, as found by Killing and Cartan (the list of all semisimple Lie viii algebras consists of (1) the special- linear ones, i.e. all matrices (of any fixed dimension) with trace 0, (2) the orthogonal ones, i.e. all skewsym- metric matrices (of any fixed dimension), (3) the symplectic ones, i.e. all matrices M (of any fixed even dimension) that satisfy MJ = −JMT with a certain non-degenerate skewsymmetric matrix J, and (4) five special Lie algebras G2, F4, E6, E7, E8, of dimensions 14, 52, 78, 133, 248, the “excep- tional Lie algebras", that just somehow appear in the process). There is also a discussion of the compact form and other real forms of a (com- plex) semisimple Lie algebra, and a section on automorphisms. The third chapter brings the theory of the finite dimensional representations of a semisimple Lie algebra, with the highest or extreme weight as central notion. The proof for the existence of representations is an ad hoc ver- sion of the present standard proof, but avoids explicit use of the Poincaré- Birkhoff-Witt theorem. Complete reducibility is proved, as usual, with J.H.C. Whitehead’s proof (the first proof, by H. Weyl, was analytical-topological and used the exis- tence of a compact form of the group in question). Then come H. Weyl’s formula for the character of an irreducible representation, and its conse- quences (the formula for the dimension of the representation, Kostant’s formula for the multiplicities of the weights and algorithms for finding the weights, Steinberg’s formula for the multiplicities in the splitting of a tensor product and algorithms for finding them). The last topic is the determination of which representations can be brought into orthogonal or symplectic form. This is due to I.A. Malcev; we bring the much simpler approach by Bose-Patera. Some of the text has been rewritten and, I hope, made clearer. Errors have been eliminated; I hope no new ones have crept in. Some new ma- terial has been added, mainly the section on automorphisms, the formulas of Freudenthal and Klimyk for the multiplicities of weights, R. Brauer’s algorithm for the splitting of tensor products, and the Bose-Patera proof mentioned above. The References at the end of the text contain a some- what expanded list of books and original contributions. In the text I use “iff" for “if and only if", “wr to" for “with respect to" and “resp." for “respectively". A reference such as “Theorem A" indicates Theorem A in the same section; a reference §m.n indicates section n in chapter m; and Ch.m refers to chapter√ m. The symbol [n] indicates item n in the References. The symbol “ " indicates the end of a proof, argument or discussion. I thank Elizabeth Harvey for typing and TEXing and for support in my effort to learn TEX, and I thank Jim Milgram for help with PicTeXing the diagrams. Hans Samelson, Stanford, September 1989 Preface to the Old Edition These notes are a slightly expanded version of lectures given at the Uni- versity of Michigan and Stanford University. Their subject, the basic facts about structure and representations of semisimple Lie algebras, due mainly to S. Lie, W. Killing, E. Cartan, and H. Weyl, is quite classical. My aim has been to follow as direct a path to these topics as I could, avoiding de- tours and side trips, and to keep all arguments as simple as possible. As an example, by refining a construction of Jacobson’s, I get along without the enveloping algebra of a Lie algebra. (This is not to say that the enveloping algebra is not an interesting concept; in fact, for a more advanced devel- opment one certainly needs it.) The necessary background that one should have to read these notes con- sists of a reasonable firm hold on linear algebra (Jordan form, spectral theorem, duality, bilinear forms, tensor products, exterior algebra,. ) and the basic notions of algebra (group, ring, homomorphism,. , the Noether isomorphism theorems, the Jordan-Hoelder theorem,. ), plus some no- tions of calculus. The principal notions of linear algebra used are collected, not very systematically, in an appendix; it might be well for the reader to glance at the appendix to begin with, if only to get acquainted with some of the notation. I restrict myself to the standard fields: R = reals, C = complex numbers (a¯ denotes the complex-conjugate of a); Z denotes the integers; Zn is the cyclic group of order n. “iff” means “if and only if”; “w.r.to” means “with respect to”. In the preparation of these notes, I sub- stituted my own version of the Halmos-symbol√ that indicates the end of a proof or an argument; I use “ ”. The bibliography is kept to a mini- mum; Jacobson’s book contains a fairly extensive list of references and some historical comments. Besides the standard sources I have made use of mimeographed notes that I have come across (Albert, van Est, Freuden- thal, Mostow, J. Shoenfield). Stanford, 1969 x Contents Preface to the New Edition vii Preface to the Old Edition ix 1 Generalities 1 1.1 Basic definitions, examples . 1 1.2 Structure constants . 5 1.3 Relations with Lie groups . 5 1.4 Elementary algebraic concepts . 7 1.5 Representations; the Killing form . 12 1.6 Solvable and nilpotent . 17 1.7 Engel’s theorem . 19 1.8 Lie’s theorem . 20 1.9 Cartan’s first criterion . 22 1.10 Cartan’s second criterion . 23 1.11 Representations of A1 ..................... 25 1.12 Complete reduction for A1 . 29 2 Structure Theory 33 2.1 Cartan subalgebra . 33 2.2 Roots . 35 2.3 Roots for semisimple g .................... 36 2.4 Strings . 37 2.5 Cartan integers . 38 2.6 Root systems, Weyl group . 40 2.7 Root systems of rank two . 43 2.8 Weyl-Chevalley normal form, first stage . 46 2.9 Weyl-Chevalley normal form . 48 2.10 Compact form . 51 2.11 Properties of root systems . 59 2.12 Fundamental systems . 66 2.13 Classification of fundamental systems . 68 2.14 The simple Lie algebras . 73 2.15 Automorphisms . 84 xii 3 Representations 89 3.1 The Cartan-Stiefel diagram . 89 3.2 Weights and weight vectors . 94 3.3 Uniqueness and existence . 98 3.4 Complete reduction . 102 3.5 Cartan semigroup; representation ring . 105 3.6 The simple Lie algebras . 107 3.7 The Weyl character formula . 116 3.8 Some consequences of the character formula . 122 3.9 Examples . 128 3.10 The character ring . 134 3.11 Orthogonal and symplectic representations . 137 Appendix 147 References 153 Index 155 Symbol Index 160 1 Generalities 1.1 Basic definitions, examples A multiplication or product on a vector space V is a bilinear map from V × V to V . Now comes the definition of the central notion of this book: A Lie algebra consists of a (finite dimensional) vector space, over a field F, and a multiplication on the vector space (denoted by [], pronounced “bracket”, the image of a pair (X, Y ) of vectors denoted by [XY ] or [X, Y ]), with the properties (a) [XX] = 0, (b) [X[YZ]] + [Y [ZX]] + [Z[XY ]] = 0 for all elements X, resp X, Y, Z, of our vector space.
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