Lie Algebras, Algebraic Groups, And Lie Groups - James Milne

D RAMATIS P ERSONÆJACOBI (1804–1851). In his work on partial differential equations, he discovered the Jacobiidentity. Jacobi’s work helped Lie to develop an analytic framework for his geometric ideas.R IEMANN (1826–1866). Defined the spaces that bear his name. The study of these spacesled to the introduction of local Lie groups and Lie algebras.L IE (1842–1899). Founded and developed the subject that bears his name with the originalintention of finding a “Galois theory” for systems of differential equations.K ILLING (1847–1923). He introduced Lie algebras independently of Lie in order to understand the different noneuclidean geometries (manifolds of constant curvature), and heclassified the possible Lie algebras over the complex numbers in terms of root systems. Introduced Cartan subalgebras, Cartan matrices, Weyl groups, and Coxeter transformations.M AURER (1859–1927). His thesis was on linear substitutions (matrix groups). He characterized the Lie algebras of algebraic groups, and essentially proved that group varieties arerational (in characteristic zero).E NGEL (1861–1941). In collaborating with Lie on the three-volume Theorie der Transformationsgruppen and editing Lie’s collected works, he helped put Lie’s ideas into coherentform and make them more accessible.E. C ARTAN (1869–1951). Corrected and completed the work of Killing on the classification of semisimple Lie algebras over C, and extended it to give a classification of theirrepresentations. He also classified the semisimple Lie algebras over R, and he used this toclassify symmetric spaces.W EYL (1885–1955). He was a pioneer in the application of Lie groups to physics. Heproved that the finite-dimensional representations of semisimple Lie algebras and semisimple Lie groups are semisimple (completely reducible).N OETHER (1882–1935).They found a classification of semisimple algebrasH ASSE (1898–1979).over number fields, which leads to a classification ofB RAUER (1901–1977).the classical algebraic groups over the same fields.A LBERT (1905–1972).H OPF (1894–1971). Observed that a multiplication map on a manifold defines a comultiplication map on the cohomology ring, and exploited this to study the ring. This observationled to the notion of a Hopf algebra.N EUMANN (1903–1957). Proved that every closed subgroup of a real Lie group isagain a Lie group.VONW EIL (1906–1998). Foundational work on algebraic groups over arbitrary fields. Classifiedthe classical algebraic groups over arbitrary fields in terms of semisimple algebras withinvolution (thereby winning the all India cocycling championship for 1960). Introducedadéles into the study of arithmetic problems on algebraic groups.C HEVALLEY (1909–1984). He proved the existence of the simple Lie algebras and oftheir representations without using a case-by-case argument. Was the leading pioneer inthe development of the theory algebraic groups over arbitrary fields. Classified the splitsemisimple algebraic groups over any field, and in the process found new classes of finitesimple groups.JACOBSON (1910–1999). Proved that most of the classical results on Lie algebras remaintrue over any field of characteristic zero (at least for split algebras).8

KOLCHIN (1916–1991). Obtained the first significant results on matrix groups over arbitrary fields as preparation for his work on differential algebraic groups.I WASAWA (1917–1998). Found the Iwasawa decomposition, which is fundamental for thestructure of real semisimple Lie groups.H ARISH -C HANDRA (1923–1983). Independently of Chevalley, he showed the existence ofthe simple Lie algebras and of their representations without using a case-by-case argument.With Borel he proved some basic results on arithmetic groups. Was one of the founders ofthe theory of infinite-dimensional representations of Lie groups.B OREL (1923–2003). He applied algebraic geometry to study algebraic groups, therebysimplifying and extending earlier work of Chevalley, who then adopted these methods himself. Borel made many fundamental contributions to the theory of algebraic groups and oftheir arithmetic subgroups.S ATAKE (1927–). He classified reductive algebraic groups over perfect fields (independently of Tits).T ITS (1930–). His theory of buildings gives a geometric approach to the study of algebraicgroups, especially the exceptional simple groups. With Bruhat he used them to study thestructure of algebraic groups over discrete valuation rings.M ARGULIS (1946–). Proved fundamental results on discrete subgroups of Lie groups.9

C HAPTERLie AlgebrasThe Lie algebra of an algebraic group or Lie group is the first linear approximation of thegroup. The study of Lie algebras is much more elementary than that of the groups, andso we begin with it. Beyond the basic results of Engel, Lie, and Cartan on nilpotent andsolvable Lie algebras, the main theorems in this chapter attach a root system to each splitsemisimple Lie algebra and explain how to deduce the structure of the Lie algebra (forexample, its Lie subalgebras) and its representations from the root system.The first nine sections are almost complete except that a few proofs are omitted (references are given). The remaining sections are not yet written. They will extend the theory tononsplit Lie algebras. Specifically, they will cover the following topics. Classification of Lie algebras over R their representations in terms of “enhanced”Dynkin diagrams; Cartan involutions. Classification of forms of a (split) Lie algebra by Galois cohomology groups. Description of all classical Lie algebras in terms of semisimple algebras with involution. Relative root systems, and the classification of Lie algebras and their representationsin terms relative root systems and the anisotropic kernel.In this chapter, we follow Bourbaki’s terminology and exposition quite closely, extracting what we need for the remaining two chapters.Throughout this chapter k is a field.1Definitions and basic propertiesBasic definitionsD EFINITION 1.1 A Lie algebra over a field k is a vector space g over k together with ak-bilinear mapŒ ; W g g ! g(called the bracket) such that(a) Œx; x D 0 for all x 2 g,(b) Œx; Œy; z C Œy; Œz; x C Œz; Œx; y D 0 for all x; y; z 2 g.11I

12CHAPTER I. LIE ALGEBRASA homomorphism of Lie algebras is a k-linear map W g ! g0 such that .Œx; y / D Œ .x/; .y/ for all x; y 2 g:Condition (b) is called the Jacobi identity. Note that (a) applied to Œx C y; x C y showsthat the Lie bracket is skew-symmetric,Œx; y D Œy; x , for all x; y 2 g;(2)and that (2) allows us to rewrite the Jacobi identity asŒx; Œy; z D ŒŒx; y ; z C Œy; Œx; z (3)ŒŒx; y ; z D Œx; Œy; z (4)orŒy; Œx; z A Lie subalgebra of a Lie algebra g is a k-subspace s such that Œx; y 2 s wheneverx; y 2 s (i.e., such that1 Œs; s s). With the bracket, it becomes a Lie algebra.A Lie algebra g is said to be commutative (or abelian) if Œx; y D 0 for all x; y 2 g. Thus,to give a commutative Lie algebra amounts to giving a finite-dimensional vector space.An injective homomorphism is sometimes called an embedding, and a surjective homomorphism is sometimes called a quotient map.We shall be mainly concerned with finite-dimensional Lie algebras. Suppose that g hasa basis fe1 ; : : : ; en g, and writeŒei ; ej DnXlel ;aijlaij2 k;1 i; j n:(5)lD1lThe aij, 1 i; j; l n, are called the structure constants of g relative to the given basis.They determine the bracket on g.D EFINITION 1.2 An ideal in a Lie algebra g is a subspace a such that Œx; a 2 a for all x 2 gand a 2 a (i.e., such that Œg; a a).Notice that, because of the skew-symmetry of the bracketŒg; a a ” Œa; g a ” Œg; a a and Œa; g a— all left (or right) ideals are two-sided ideals.Examples1.3 Up to isomorphism, the only noncommutative Lie algebra of dimension 2 is that withbasis x; y and bracket determined by Œx; y D x (exercise).1.4 Let A be an associative k-algebra. The bracketŒa; b D ab1 Webawrite Œs; t for the k-subspace of g spanned by the brackets Œx; y with x 2 s and y 2 t.(6)

1. Definitions and basic properties13is k-bilinear, and it makes A into a Lie algebra ŒA because Œa; a is obviously 0 and theJacobi identity can be proved by a direct calculation. In fact, on expanding out the left sideof the Jacobi identity for a; b; c one obtains a sum of 12 terms, 6 with plus signs and 6 withminus signs; by symmetry, each permutation of a; b; c must occur exactly once with a plussign and exactly once with a minus sign.1.5 In the special case of (1.4) in which A D Mn .k/, we obtain the Lie algebra gln . Thusgln consists of the n n matrices A with entries in k endowed with the bracketŒA; B D ABBA:Let Eij be the matrix with 1 in the ij th position and 0 elsewhere. These matrices form abasis for gln , and8 Eij 0 if j D i 0Ei 0 j if i D j 0ŒEij ; Ei 0 j 0 D(7):0otherwise.More generally, let V be a k-vector space. From A D Endk-linear .V / we obtain the Liealgebra glV of endomorphisms of V withŒ ; ˇ D ı ˇˇ ı :1.6 Let A be an associative k-algebra such that k D k1 is contained the centre of A. Aninvolution of A is a k-linear map a 7! a W A ! A such that.a C b/ D a C b ;.ab/ D b a ;a D afor all a; b 2 A. When is an involution of A,defŒA; D fa 2 A j a C a D 0gis a Lie k-subalgebra of ŒA , because it is a k-subspace andŒa; b D .abba/ D b a a b D baab D Œb; a :1.7 Let V be a finite-dimensional vector space over k, and letˇW V V ! kbe a nondegenerate k-bilinear form. Define W End.V / ! End.V / byˇ.av; v 0 / D ˇ.v; a v 0 /;a 2 End.V /, v; v 0 2 V:Then .a C b/ D a C b and .ab/ D b a . If ˇ is symmetric or skew-symmetric, then is an involution, and ŒEnd.V /; is the Lie algebra g D x 2 glV j ˇ.xv; v 0 / C ˇ.v; xv 0 / D 0 all v; v 0 2 V :

14CHAPTER I. LIE ALGEBRAS1.8 The following are all Lie subalgebras of gln :sln D fA 2 Mn .k/ j trace.A/ D 0g;on D fA 2 Mn .k/ j A is skew symmetric, i.e., A C At D 0g;ˇ spn D A 2 Mn .k/ ˇ 0 I A C At 0 I D 0 ;I 0I 0bn D f.cij / j cij D 0 if i j g (upper triangular matrices);nn D f.cij / j cij D 0 if i j g (strictly upper triangular matrices);dn D f.cij / j cij D 0 if i j g (diagonal matrices).To see that sln is a Lie subalgebra of gln , note that, for n n matrices A D .aij / andB D .bij /,Xtrace.AB/ Daij bj i D trace.BA/.(8)1 i;j nTherefore ŒA; B D AB BA has trace zero. Similarly, the endomorphisms with trace 0 ofa finite-dimensional vector space V form a Lie subalgebra slV of glV . Both on and spn arespecial cases of (1.7).N OTATION 1.9 We write ha; b; : : :i for Span.a; b; : : :/, and we write ha; b; : : : jRi for theLie algebra with basis a; b; : : : and the bracket given by the rules R. For example, the Liealgebra in (1.3) can be written hx; y j Œx; y D xi.N OTES Although Lie algebras have been studied since the 1880s, the term “Lie algebra” was introduced by Weyl only in 1934. Previously people had spoken of “infinitesimal groups” or used evenless precise terms. See Bourbaki LIE, Historical Note to Chapters 1–3, IV.Derivations; the adjoint mapD EFINITION 1.10 Let A be a k-algebra (not necessarily associative). A derivation of A isa k-linear map DW A ! A such thatD.ab/ D D.a/ b C a D.b/for all a; b 2 A:(9)The composite of two derivations need not be a derivation, but their bracketŒD; E D D ı EE ıDis, and so the set of k-derivations A ! A is a Lie subalgebra Derk .A/ of glA . For example,if the product on A is trivial, then the condition (9) is vacuous, and so Derk .A/ D glA .D EFINITION 1.11 Let g be a Lie algebra. For a fixed x in g, the linear mapy 7! Œx; y W g ! gis called the adjoint (linear) map of x, and is denoted adg .x/ or ad.x/ (we sometimes omitthe parentheses) .For each x, the map adg .x/ is a k-derivation of g because (3) can be rewritten asad.x/Œy; z D Œad.x/y; z C Œy; ad.x/z :

1. Definitions and basic properties15Moreover, adg is a homomorphism of Lie algebras g ! Der.g/ because (4) can be rewrittenasad.Œx; y /z D ad.x/.ad.y/z/ ad.y/.ad.x/z/:The kernel of adg W g ! Derk .g/ is the centre of g,z.g/ D fx 2 g j Œx; g D 0g:The derivations of g of the form ad x are said to be inner (by analogy with the inner automorphisms of a group).An ideal in g is a subspace stable under all inner derivations of g. A subspace stableunder all derivations is called a characteristic ideal. For example, the centre z.g/ of gis a characteristic ideal of g. An ideal a in g is, in particular, a subalgebra of g; if a ischaracteristic, then every ideal in a is also an ideal in g.The isomorphism theoremsWhen a is an ideal in a Lie algebra g, the quotient vector space g a becomes a Lie algebrawith the bracketŒx C a; y C a D Œx; y C a.The following statements are straightforward consequences of the similar statements forvector spaces.1.12 (Existence of quotients). The kernel of a homomorphism g ! g0 of Lie algebras isan ideal, and every ideal a is the kernel of a quotient map g ! q.1.13 (Homomorphism theorem). The image of a homomorphism W g ! g0 of Lie algebrasis a Lie subalgebra g of g0 , and defines an isomorphism of g Ker. / onto g; in particular, every homomorphism of Lie algebras is the composite of a surjective homomorphismwith an injective homomorphism.1.14 (Isomorphism theorem). Let h and a be Lie subalgebras of g. If Œh; a a, then h C ais a Lie subalgebra of g, h \ a is an ideal in h, and the mapx C h \ a 7! x C aW h h \ a ! .h C a/ ais an isomorphism.1.15 (Correspondence theorem). Let a be an ideal in a Lie algebra g. The map h 7! h ais a bijection from the set of Lie subalgebras of g containing a to the set of Lie subalgebrasof g a. A Lie subalgebra h containing a is an ideal if and only if h a is an ideal in g a, inwhich case the mapg h ! .g a/ .h a/is an isomorphism.

16CHAPTER I. LIE ALGEBRASNormalizers and centralizersFor a subalgebra h of g, the normalizer and centralizer of h in g areng .h/ D fx 2 g j Œx; h hgcg .h/ D fx 2 g j Œx; h D 0g:These are both subalgebras of g, and ng .h/ is the largest subalgebra containing h as an ideal.When h is commutative, cg .h/ is the largest subalgebra of g containing h in its centre.Extensions; semidirect productsAn exact sequence of Lie algebras0!a!g!b!0is called an extension of b by a. The extension is said to be central if a is contained in thecentre of g, i.e., if Œg; a D 0.Let a be an ideal in a Lie algebra g. Each element g of g defines a derivation a 7! Œg; a of a, and this defines a homomorphismW g ! Der.a/;g 7! ad.g/ja.If there exists a Lie subalgebra q of g such that g ! g a maps q isomorphically onto g a,then I claim that we can reconstruct g from a, q, and jq. Indeed, each element g of g canbe written uniquely in the formg D a C q;a 2 a;q 2 qI— here q must be the unique element of Q mapping to g C a in g a and a must be gThus we have a one-to-one correspondence of setsgq.1-1! a q;which is, in fact, an isomorphism of k-vector spaces. If g D a C q and g 0 D a0 C q 0 , thenŒg; g 0 D Œa C q; a0 C q 0 D Œa; a0 C Œa; q 0 C Œq; a0 C Œq; q 0 D Œa; a0 C q a0 q 0 a C Œq; q 0 ;which proves the claim.D EFINITION 1.16 A Lie algebra g is a semidirect product of subalgebras a and q, denotedg D a o q, if a is an ideal in g and the quotient map g ! g a induces an isomorphismq ! g a.We have seen that, from a semidirect product g D a o q, we obtain a triple.a; q; W q ! Derk .a//;and that the triple determines g. We now show that every triple .a; q; / consisting of twoLie algebras a and q and a homomorphism W q ! Derk .a/ arises from a semidirect product.As a k-vector space, we let g D a q, and we define Œ.a; q/; .a0 ; q 0 / D Œa; a0 C q a0 q 0 a; Œq; q 0 .(10)

1. Definitions and basic properties17P ROPOSITION 1.17 The bracket (10) makes g into a Lie algebra.P ROOF. Routine verification.2We denote g by a o q. The extension0 ! a ! a o q ! q ! 0is central if and only if a is commutative and is the zero map.Examples1.18 Let D be a derivation of a Lie algebra a. Let q be the one-dimensional Lie algebrak, and letg D a o q,where is the map c 7! cDW q ! Derk .a/. For the element x D .0; 1/ of g, adg .x/ja D D,and so the derivation D of a has become an inner derivation in g.1.19 Let V be a finite-dimensional k-vector space. When we regard V as a commutativeLie algebra, Derk .V / D glV . Let be the identity map glV ! Derk .V /. Then V o glV isa Lie algebra, denoted af.V /.2 An element of af.V / is a pair .v; x/ with v 2 V and u 2 glV ,and the bracket isŒ.v; u/; .v 0 ; u0 / D .u.v 0 / u0 .v/; Œu; u0 /:Let h be a Lie algebra, and let W h ! af.V / be a k-linear map. We can write D . ; /with W h ! V and W h ! glV linear maps, and is a homomorphism of Lie algebras if andonly if is a homomorphism of Lie algebras and .Œx; y / D .x/ .y/ .y/ .x/(11)for all x; y 2 h (we have written a v for a.v/, a 2 glV , v 2 V ).Let V 0 D V k, and leth D fw 2 glV 0 j w.V 0 / V g.Then h is a Lie subalgebra of glV 0 . Define W h ! glV ; W h ! V; .w/ D wjV; .w/ D w.0; 1/:Then is a homomorphism of Lie algebras, and . ; / satisfies (11), and so W h ! af.V /;w 7! . .w/; .w//is a homomorphism of Lie algebras. The map is bijective, and its inverse sends .v; u/ 2af.V / to the element.v 0 ; c/ 7! .u.v 0 / C cv; 0/of h. See Bourbaki LIE, I, 1, 8, Ex. 2.2 Itis the Lie algebra of the group of affine transformations of V — see Chapter II.

18CHAPTER I. LIE ALGEBRASThe universal enveloping algebraRecall (1.4) that an associative k-algebra A becomes a Lie algebra ŒA with the bracketŒa; b D ab ba. Let g be a Lie algebra. Among the pairs consisting of an associativek-algebra A and a Lie algebra homomorphism g ! ŒA , there is one, W g ! ŒU.g/ , that isuniversal:gLieU.g/ LieassociativeHom.g; ŒA / ' Hom.U.g/; A/: ı AIn other words, every Lie algebra homomorphism g ! ŒA extends uniquely to a homomorphism of associative algebras A ! U.g/. The pair .U.g/; / is called the universalenveloping algebra of g.The algebra U.g/ can be constructed as follows. Recall that the tensor algebra T .V / ofa k-vector space V isT .V / D k V .V V / .V V V / with the k-algebra structure.x1 xr / .y1 ys / D x1 xr y1 ys :It has the property that every k-linear map V ! A with A an associative k-algebra extendsuniquely to a k-algebra homomorphism T .V / ! A. We define U.g/ to be the quotient ofT .g/ by the two-sided ideal generated by the tensorsx yy xŒx; y ;x; y 2 g:(12)Every k-linear map W g ! A with A an associative k-algebra extends uniquely to k-algebrahomomorphism T .g/ ! A, which factors through U.g/ if and only if is a Lie algebrahomomorphism g ! ŒA .If g is commutative, then (12) is just the relati

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. References