Introduction to Kac-Moody groups and Lie algebras N. Perrin November 9, 2015 2 Introduction This text consists of lecture notes for lectures given in Bonn University. The main inspiration for these notes was the book of V. Kac [Ka90]. I stayed very close from this book at least for the begining of the lecture: definition and first properties of Kac-Moody Lie algebras. In particular I have choosen to keep the definition of g(A) the Kac-Moody Lie algebra associated to a generalised Cartan matrix A as the quotient of the Lie algebra g(A) with the obvious commuting relations but the quotient of the maximal ideal with trivial intersection with the Cartan subalgebra instead taking the definition using Serre relations. This has manye advantages when one wants to identity a Lie algebra as a special Kac-Moody Lie algebra in particular in the explicit constructions of finite dimensional simple Lie algebras (Chapter 13) or of twisted and untwisted affine Lie algebras (Chapter 12 and 14). The second source of inspiration was the book of S. Kumar [Ku02]. In particular we took his point of view to define the Weyl group of a Kac-Moody Lie algebra and also to some extend in the presentation of the invariant bilinear form on a Kac-Moody Lie algebra as well as in the presentation of the representation theory of Kac-Moody Lie algebras and the celebrated character formula. Finally the books [Hu90] and [Bo54] where my main sources for the treatment of Coxeter groups in general even if the link with Weyl group of Kac-Moody Lie algebras is inspired from [Ku02]. As a conclusion, I would like to thank the HCM and the University of Bonn for giving me the opportunity to give these lectures. I also thank the students for their patience when I tried to explain the flavour of some tedious computations that may appear in proofs in the subject. I would finally like to thank Manfred Lehn whose lectures on affine Lie algebras in WS2001-2002 in Cologne was my first and inspirating introduction to the subject. Let me briefly review the content of these lecture notes: in the first part I give a quick introduction to semisimple Lie algebras and semisimple groups. This part should serve as a guide to what we want to obtain for Kac-Moody Lie algebras (and Kac-Moody groups that should be treated in a forthcoming third part). The main references are J.E. Humpreys book [Hu72] for Chapter 1 and T.A. Springer’s book [Sp98] for Chapter 2. In a third chapter, we recall some basic facts and definitions on algebras and Lie algebras like free Lie algebras, enveloping algebras an the Poincar´e-Birkhoff-Witt Theorem. We then enter the subject with the defintion and first properties of Kac-Moody Lie algebras in Chapter 4. In Chapter 5 we define the Weyl group of a Kac-Moody Lie algebra and associated geometric representation. In the next Chapter, we review the theory of Coxeter group and end with a proof of the fact that the Weyl group of a Kac-Moody Lie algebra is a cristallographic Coxeter group. At this point we start to study some particular generalised Cartan matrices especially, in Chapter 7, the so called symmetrisable Cartan matrices A for which the Kac-Moody Lie algebra g(A) has an invariant bilinear form. These are the Kac-Moody Lie algebra for which we prove the character formula. In Chapter 8, we give the classification of generalised Cartan matrices in finite, affine and general type. We also give the possible connected Dynkin diagrams in finite and affine case. In Chapter 9, we give a first description of the root system of Kac-Moody Lie algebras. In particular a 3 4 new phenomenon occurs here: not all roots are in the orbit of simple roots, these are the imaginary roots. We then start with the representation theory of Kac-Moody Lie algebras in Chapter 10. We define Verma modules and intergrable highest weight. In Chapter 11, we define the Casimir operator which exists only if the algebra is symmetrisable. We use this operator to prove the character formula for symmetrisable Kac-Moody Lie algebras. We then start to realise explicitly the Kac-Moody Lie algebras of finte and affine type. We start in Chapter 12 with untwisted affine Lie algebras. Then in Chapter 13 we come back to the finite case and explicitly realise simply laced and non simply laced simple Lie algebras. We use this construction in Chapter 14 to construct twisted affine Lie algebras. The explicit construction in particular induce description of the root system and Weyl groups of the affine Lie algebras. Contents I Quick review of semisimple Lie algebras and semisimple groups 11 1 Semisimple Lie algebras 13 1.1 Semisimple Lie algebras and Killing form . ............ 13 1.2 Cartan subalgebras, roots and Weyl group . ............ 14 1.3 Cartan matrices and Dynkin diagrams . .......... 15 1.3.1 Simpleroots ................................... 15 1.3.2 Cartanmatrices ................................ 16 1.3.3 Serre’spresentation . ...... 16 1.3.4 Dynkindiagrams ................................ 17 1.4 Representation theory of semisimple Lie algebras . ................ 17 1.4.1 Weights of a representation . ....... 18 1.4.2 Weightlattice................................. 18 1.4.3 Dominantweights ............................... 19 1.4.4 Vermamodules.................................. 19 1.4.5 Characterformula .............................. 20 2 Semisimple groups 21 2.1 Algebraicgroups ................................. ...... 21 2.1.1 Firstproperties............................... ..... 21 2.1.2 Semisimplegroups .............................. 22 2.2 Some subgroups of G ..................................... 22 2.2.1 Maximaltorusandrootsystems . ...... 22 2.2.2 Borelsubgroups ................................ 23 2.3 Charactersandlinebundles . ......... 23 II Kac-Moody Lie algebras 25 3 Some facts on associative algebras 27 3.1 Freealgebras .................................... ..... 27 3.2 Envelopingalgebras .............................. ....... 27 4 Kac-Moody Lie algebras 29 4.1 Lie algebras associated to a complex square matrix . ............... 29 4.1.1 Realization of a matrix . ..... 29 4.1.2 The Lie algebra g(A)................................. 30 4.1.3 The Lie algebra g(A)................................. 32 4.2 Kac-MoodyLiealgebrase . .. .. .. .. .. .. .. .. ....... 34 5 6 CONTENTS 4.2.1 Generalized Cartan matrices . ....... 34 4.2.2 Isomorphisms of Kac-Moody Lie algebras . ......... 35 4.2.3 Serrerelations ................................ 35 4.2.4 Ideals in the Kac-Moody Lie algebras . ........ 37 5 Weyl group 39 5.1 locally finite and nilpotent elements . ............ 39 5.2 Integrable representations and Weyl group . .............. 42 5.2.1 Integrable representations . ......... 42 5.2.2 Definition of the Weyl group and action on integrable representations . 42 5.3 Using integrable representations to construct groups . .................. 45 6 Coxeter groups 47 6.1 Definition ....................................... 47 6.1.1 Coxetersystems ................................ 47 6.1.2 Lengthfunction................................ 47 6.2 Geometric representation . ......... 48 6.2.1 Rootsystem .................................... 49 6.2.2 Geometric interpretation of length function . ............. 50 6.3 Exchangeconditions .............................. ....... 52 6.3.1 Reflections .................................... 52 6.3.2 Strongexchangecondition. ....... 52 6.4 Weyl groups of Kac-Moody Lie algebras . .......... 53 6.4.1 Equivalentdefinitions . ...... 53 6.5 DominantchambersandTitscone . ........ 55 7 Invariant bilinear form on g(A) 57 7.1 SymmetrisableCartanmatrices . .......... 57 7.2 Invariantbilinearforms . ......... 58 8 Classification of Cartan matrices 63 8.1 Finite,affineandindefinitecase. .......... 63 8.2 Finiteandaffinecases .............................. ...... 66 8.2.1 Firstresults.................................. 66 8.2.2 Examples of finite and affine type matrices . ........ 68 8.2.3 Classification of finite and affine type matrices . ........... 76 9 Real and imaginary roots 79 9.1 Definitionsandfirstproperties . .......... 79 9.1.1 realroots ..................................... 79 9.1.2 Imaginaryroots................................ 80 9.1.3 Isotropicroots ................................ 83 10 The category O 85 10.1 Definition of the category O ................................. 85 10.2 Highestweightmodules . ........ 85 10.3Vermamodules ................................... ..... 86 10.4 Lowestweightmodules. ........ 87 10.5 Integrable highest weight modules . ............ 89 CONTENTS 7 10.6Filtration ..................................... ...... 89 10.7 CharacterformulaforVermamodules . ........... 90 11 Casimir operator and character formula 93 11.1 Casimiroperator ................................ ....... 93 11.1.1 Someformulas ................................. 93 11.1.2 Casimiroperator . .. .. .. .. .. .. .. .. .. ..... 94 11.2 Characterformula ............................... ....... 97 12 Untwisted affine Lie algebras 101 12.1 Someresultsonfiniterootsystems . ........... 101 12.2 UntwistedaffineLiealgebras . ......... 104 12.2.1 Construction of affine Lie algebras . ......... 104 12.2.2 The Lie algebra g isanaffineKac-Moodyalgebra. 105 12.2.3 AffineWeylgroup ................................ 107 12.3 Application: Jacobi tripleb product formula . ................ 109 13 Explicit construction of finite dimensional Lie algebras 111 13.1 Simplylacedcase................................ ....... 111 13.2 Nonsimplylacedcase ............................. ....... 114 13.3 The case A2n ......................................... 120 14 Twisted affine Lie algebras 121