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Lie Algebras, Algebraic Groups, and Lie Groups J.S. Milne Version 2.00 May 5, 2013 These notes are an introduction to Lie algebras, algebraic groups, and Lie groups in characteristic zero, emphasizing the relationships between these objects visible in their cat- egories of representations. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. BibTeX information: @misc{milneLAG, author={Milne, James S.}, title={Lie Algebras, Algebraic Groups, and Lie Groups}, year={2013}, note={Available at www.jmilne.org/math/} } v1.00 March 11, 2012; 142 pages. V2.00 May 1, 2013; 186 pages. Please send comments and corrections to me at the address on my website http://www.jmilne.org/math/. The photo is of a grotto on The Peak That Flew Here, Hangzhou, Zhejiang, China. Copyright c 2012, 2013 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Table of Contents Table of Contents3 Preface........................................ 5 I Lie Algebras 11 1 Definitions and basic properties........................ 11 2 Nilpotent Lie algebras: Engel’s theorem................... 26 3 Solvable Lie algebras: Lie’s theorem..................... 33 4 Semisimple Lie algebras ........................... 40 5 Representations of Lie algebras: Weyl’s theorem .............. 46 6 Reductive Lie algebras; Levi subalgebras; Ado’s theorem.......... 56 7 Root systems and their classification..................... 66 8 Split semisimple Lie algebras......................... 77 9 Representations of split semisimple Lie algebras . 100 10 Real Lie algebras...............................102 11 Classical Lie algebras.............................103 II Algebraic Groups 105 1 Algebraic groups ...............................106 2 Representations of algebraic groups; tensor categories . 110 3 The Lie algebra of an algebraic group ....................118 4 Semisimple algebraic groups.........................134 5 Reductive groups...............................146 6 Algebraic groups with unipotent centre....................151 7 Real algebraic groups.............................155 8 Classical algebraic groups ..........................155 III Lie groups 157 1 Lie groups...................................157 2 Lie groups and algebraic groups .......................158 3 Compact topological groups .........................161 A Arithmetic Subgroups 163 1 Commensurable groups............................164 2 Definitions and examples...........................164 3 Questions...................................165 4 Independence of and L............................165 5 Behaviour with respect to homomorphisms . 166 6 Adelic` description of congruence subgroups . 167 3 7 Applications to manifolds...........................168 8 Torsion-free arithmetic groups ........................169 9 A fundamental domain for SL2 . 169 10 Application to quadratic forms........................170 11 “Large” discrete subgroups..........................171 12 Reduction theory ...............................172 13 Presentations .................................175 14 The congruence subgroup problem......................175 15 The theorem of Margulis...........................176 16 Shimura varieties...............................178 Bibliography 181 Index 185 4 Preface [Lie] did not follow the accepted paths. I would compare him rather to a pathfinder in a primal forest who always knows how to find the way, whereas others thrash around in the thicket. moreover, his pathway always leads past the best vistas, over unknown mountains and valleys. Friedrich Engel. Lie algebras are an essential tool in studying both algebraic groups and Lie groups. Chapter I develops the basic theory of Lie algebras, including the fundamental theorems of Engel, Lie, Cartan, Weyl, Ado, and Poincare-Birkhoff-Witt.´ The classification of semisim- ple Lie algebras in terms of the Dynkin diagrams is explained, and the structure of semisim- ple Lie algebras and their representations described. In Chapter II we apply the theory of Lie algebras to the study of algebraic groups in characteristic zero. As Cartier (1956) noted, the relation between Lie algebras and algebraic groups in characteristic zero is best understood through their categories of representations. For example, when g is a semisimple Lie algebra, the representations of g form a tan- nakian category Rep.g/ whose associated affine group G is the simply connected semisim- ple algebraic group G with Lie algebra g. In other words, Rep.G/ Rep.g/ (1) D with G a simply connected semisimple algebraic group having Lie algebra g. It is possible to compute the centre of G from Rep.g/, and to identify the subcategory of Rep.g/ corre- sponding to each quotient of G by a finite subgroup. This makes it possible to read off the entire theory of semisimple algebraic groups and their representations from the (apparently simpler) theory of semisimple Lie algebras. For a general Lie algebra g, we consider the category Repnil.g/ of representations of g such that the elements in the largest nilpotent ideal of g act as nilpotent endomorphisms. Ado’s theorem assures us that g has a faithful such representation, and from this we are able to deduce a correspondence between algebraic Lie algebras and algebraic groups with unipotent centre. Let G be a reductive algebraic group with a split maximal torus T . The action of T on the Lie algebra g of G induces a decomposition M g h g˛; h Lie.T /, D ˚ ˛ R D 2 of g into eigenspaces g˛ indexed by certain characters ˛ of T , called the roots. A root ˛ determines a copy s˛ of sl2 in g. From the composite of the exact tensor functors (1) Rep.G/ Rep.g/ Rep.s˛/ Rep.S˛/, ! ! D we obtain a homomorphism from a copy S˛ of SL2 into G. Regard ˛ as a root of S˛; then its coroot ˛_ can be regarded as an element of X .T /. The system .X .T /;R;˛ ˛_/ 7! is a root datum. From this, and the Borel fixed point theorem, the entire theory of split reductive groups over fields of characteristic zero follows easily. 5 Although there are many books on algebraic groups, and even more on Lie groups, there are few that treat both. In fact it is not easy to discover in the expository literature what the precise relation between the two is. In Chapter III we show that all connected complex semisimple Lie groups are algebraic groups, and that all connected real semisimple Lie groups arise as covering groups of algebraic groups. Thus readers who understand the theory of algebraic groups and their representations will find that they also understand much of the theory of Lie groups. Again, the key tool is tannakian duality. Realizing a Lie group as an algebraic group is the first step towards understanding the discrete subgroups of the Lie group. We discuss the discrete groups that arise in this way in an appendix. At present, only the split case is covered in Chapter I, only the semisimple case is covered in detail in Chapter II, and only a partial summary of Chapter III is available. Notations; terminology We use the standard (Bourbaki) notations: N 0;1;2;::: ; Z ring of integers; Q D f g D D field of rational numbers; R field of real numbers; C field of complex numbers; Fp D D D Z=pZ field with p elements, p a prime number. For integers m and n, m n means that D j m divides n, i.e., n mZ. Throughout the notes, p is a prime number, i.e., p 2;3;5;:::. 2 D Throughout k is the ground field, usually of characteristic zero, and R always denotes a commutative k-algebra. A k-algebra A is a k-module equipped with a k-bilinear (mul- tiplication) map A A k. Associative k-algebras are required to have an element 1, ! and c1 c k is contained in the centre of the algebra. Unadorned tensor products are f j 2 g over k. Notations from commutative algebra are as in my primer. When k is a field, ksep denotes a separable algebraic closure of k and kal an algebraic closure of k. The dual Homk-linear.V;k/ of a k-module V is denoted by V _. The transpose of a matrix M is de- noted by M t . We define the eigenvalues of an endomorphism of a vector space to be the roots of its characteristic polynomial. We use the terms “morphism of functors” and “natural transformation of functors” in- terchangeably. When F and F 0 are functors from a category, we say that “a homomorphism F .a/ F .a/ is natural in a” when we have a family of such maps, indexed by the objects ! 0 a of the category, forming a natural transformation F F . For a natural transformation ! 0 ˛ F F , we often write ˛R for the morphism ˛.R/ F .R/ F .R/. When its action on W ! 0 W ! 0 morphisms is obvious, we usually describe a functor F by giving its action R F .R/ on objects. Categories are required to be locally small (i.e., the morphisms between any two objects form a set), except for the category A of functors A Set. A diagram A B C _ ! ! ⇒ is said to be exact if the first arrow is the equalizer of the pair of arrows; in particular, this means that A B is a monomorphism. ! The symbol denotes a surjective map, and , an injective map. ! We use the following conventions: X YX is a subset of Y (not necessarily proper); X def YX is defined to be Y , or equals Y by definition; D X YX is isomorphic to Y ; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism); ' Passages designed to prevent the reader from falling into a possibly fatal error are sig- nalled by putting the symbol A in the margin. ASIDES may be skipped; NOTES are often reminders to the author. 6 Prerequisites The only prerequisite for Chapter I (Lie algebras) is the algebra normally taught in first- year graduate courses and in some advanced undergraduate courses. Chapter II (algebraic groups) makes use of some algebraic geometry from the first 11 chapters of my notes AG, and Chapter III (Lie groups) assumes some familiarity with manifolds.
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