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Exercise 12 Let G be the (parabolic) subgroup of GL4 (R) of matrices whoselower left 2 by 2 block of elements are all zero. Decompose G in an analogousway to the example above.So to classify all connected Lie groups we need to find the simple ones, theunipotent ones, and find how the simple ones can act on the unipotent ones.One might guess that the easiest part of this will be to find the unipotent ones,as these are just built from abelian ones by taking central extensions. Howeverthis turns out to be the hardest part, and there seems to be no good solution.The simple ones can be classified with some effort: we will more or less dothis in the course. Over the complex numbers the complete list is given by thefollowing Dynkin diagrams (where the subscript in the name is the number ofnodes):An ··· Bn ··· CnE8F4 ··· ··· G2 V DnE6E7These Dynkin diagrams are pictures of the Lie groups with the following meaning. Each dot is a copy of SL2 . Two dots are disconnected if the correspondingSL2 s commute, and are joined by a single line if they “overlap by one matrixelement”. Double and triple lines describe more complicated ways they can interact. For each complex simple Lie group there are a finite number of simplereal Lie groups whose complexification is the complex Lie group, and we willlater use this to find the simple Lie groups.For example, sl2 (R) 6' su2 (R), but sl2 (R) C ' su2 (R) C ' sl2 (C).By the way, sl2 (C) is simple as a real Lie algebra, but its complexification issl2 (C) sl2 (C), which is not simple.Dynkin diagrams also classify lots of other things: 3-dimensional rotationgroups, finite crystallographic reflection groups, du Val singularities, Macdonald polynomials, singular fibers of elliptic surfaces or elliptic curves over theintegers,.Exercise 13 Find out what all the things mentioned above are, and find somemore examples of mathematical objects classified by these Dynkin diagrams.(Hint: wikipedia.)5

We also need to know the actions of simple Lie groups on unipotent ones. Wecan at least describe the actions on abelian ones: this is called representationtheory.1.1Infinite dimensional Lie groupsExamples of infinite dimensional Lie groups are diffeomorphisms of manifolds,or gauge groups, or infinite dimensional classical groups. There is not muchgeneral theory of infinite dimensional Lie groups: they are just too complicated.1.2Lie groups and finite groups1. The classification of finite simple groups resembles the classification ofconnected simple Lie groups.For example, P SLn (R) is a simple Lie group, and P SLn (Fq ) is a finitesimple group except when n q 2 or n 2, q 3. Simple finite groupsform about 18 series similar to Lie groups, and 26 or 27 exceptions, calledsporadic groups, which don’t seem to have any analogues for Lie groups.Exercise 14 Show that the projective special linear groups P SL2 (F4 )and P SL2 (F5 ) are isomorphic (or if this is too hard, show they have thesame order). This gives a first hint of some of the complications of finitesimple groups: there are many accidental isomorphisms similar to this.2. Finite groups and Lie groups are both built up from simple and abeliangroups. However, the way that finite groups are built is much more complicated than the way Lie groups are built. Finite groups can containsimple subgroups in very complicated ways; not just as direct factors.For example, there are wreath products. Let G and H be finite simplegroups with an action of H on a set of n points. Then H acts on Gnby permuting the factors. We can form the semi-direct product Gn n H,sometimes denoted GoH. There is no analogue for (finite dimensional) Liegroups. There is an analogue for infinite dimensional Lie groups, which isone reason why the theory becomes hard in infinite dimensions.3. The commutator subgroup of a connected solvable Lie group is nilpotent,but the commutator subgroup of a solvable finite group need not be anilpotent group.Exercise 15 Show that the symmetric group S4 is solvable but its derivedsubgroup is not nilpotent. Show that it cannot be represented as a groupof upper triangular matrices over any field.4 Non-trivial nilpotent finite groups are never subgroups of real upper triangular matrices (with ones on the diagonal).6

1.3Lie groups and algebraic groupsBy algebraic group, we mean an algebraic variety which is also a group, suchas GLn (R). Any real algebraic group is a Lie group. Most of the connected Liegroups we have seen so far are real algebraic groups. Since they are so similar,we’ll list some differences.1. Unipotent and semisimple abelian algebraic groups are totally different,1 but for Lie groups they are anearly the same. For example R ' {( 0 1 )}0 is unipotent and R 'is semisimple. As Lie groups, they0 a 1are closely related (nearly the same), but the Lie group homomorphismexp : R R is not algebraic (polynomial), so they look quite differentas algebraic groups.2. Abelian varieties are different from affine algebraic groups. For example,consider the (projective) elliptic curve y 2 x3 x withgroup its usuala b22witha b 1.operation and the group of matrices of the form baBoth are isomorphic to S 1 as Lie groups, but they are completely differentas algebraic groups; one is projective and the other is affine.3. Some Lie groups do not correspond to ANY algebraic group. We give twoexamples here.The Heisenberg group is the subgroup of symmetries of L2 (R) generated bytranslations (f (t) 7 f (t x)), multiplication by e2πity (f (t) 7 e2πity f (t)),and multiplication by e2πiz (f (t) 7 e2πiz f (t)). The general element is ofthe form f (t) 7 e2πi(yt z) f (t x). This can also be modeled as , 1 x z 1 0 n 0 1 y 0 1 0 n Z 0 0 10 0 1It has the property that in any finite dimensional representation, the center(elements with x y 0) acts trivially, so it cannot be isomorphic to anyalgebraic group.The metaplectic group. Let’s try to find all connected groups with Liealgebra sl2 (R) { ac db a d 0}. There are two obvious ones: SL2 (R)and P SL2 (R). There aren’t any other ones that can be represented asgroups of finite dimensional matrices. However, if you look at SL2 (R),you’ll find that it is not simply connected. To see this, we will use Iwasawadecomposition (which we will only prove in some special cases).Theorem 16 (Iwasawa decomposition) If G is a connected semisimple Lie group, then there are closed subgroups K, A, and N , with Kcompact, A abelian, and N unipotent, such that the multiplication mapK A N G is a surjective diffeomorphism. Moreover, A and N aresimply connected.In the case of SLn , this is the statement that any basis can be obtaineduniquely by taking an orthonormal basis (K SOn ), scaling by positivereals (A is the group of diagonal matrices with positive real entries), and7

shearing (N is the group 1. . .0 1 ). This is exactly the result of theGram-Schmidt process.The upshot is that G ' K A N (topologically), and A and N do notcontribute to the fundamental group, so the fundamental group of G isthe same as that of K. In our case, K SO2 (R) is isomorphic to a circle,so the fundamental group of SL2 (R) is Z. So the universal cover SL2 (R) has center Z. Any finite dimensional rep resentation of SL2 (R) factors through SL2 (R), so none of the covers ofSL2 (R) can be written as a group of finite dimensional matrices. Representing such groups is a pain.(This seems to be repeated later.) Most of the examples of Lie groupsso far are algebraic: this means roughly that they are algebraic varietiesover the reals. Much of the theory of algebraic groups is similar to that ofLie groups, but there are some differences. In particular some Lie groupsare not algebraic. One example is the group of upper triangular 3 by 3unipotent matrices modulo Z. This is the Heisenberg or Weyl group: it hasa nice infinite dimensional representation generated by translations andmultiplication by eiyx acting on L2 functions on the reals. it is somewhateasier to study the representations of its Lie algebra, which has a basis of3 elements X, Y , Z with [X, Y ] Z. We can represent this in infinitedimensions by putting X x, Y d/dx, Z 1 acting on polynomials(this is Leibniz’s rule!) In quantum mechanics this algebra turns up a lotwhere X is the position operator and Y the momentum operator. But ithas not finite dimensional representations with Z acting as 1 because thetrace of Z must be 0.Also abelian algebraic groups are much more subtle than abelian Liegroups. For example, over the complex numbers, the Lie groups C, C ,and an elliptic curve are all quite similar: they differ by quotienting outcopies of Z so look the same locally. However as algebraic groups they aretotally different and have nothing to do with each other: the first is unipotent, the second is semisimple, and the third is a (complete) variety. Thepoint is that the maps between the corresponding Lie groups that makethem similar are transcendental functions (exponentials or elliptic functions) that do not exist in the algebraic setting. Rather oddly, non-abelianalgebraic groups are easier to handle than abelian ones. The reason is thatnon-abelian ones can be studied using the adjoint representation on theirLie algebras, while for abelian ones this representation is trivial and givesno useful information: abelian varieties are abelian algebraic groups, andare VERY hard to understand over the rationals.The most important case is the metaplectic group M p2 (R), which is theconnected double cover of SL2 (R). It turns up in the theory of modularforms of half-integral weight and has a representation called the metaplectic representation.p-adic Lie groups are defined in a similar way to Lie groups using p-adicmanifolds rather than smooth manifolds. They turn up a lot in number theoryand algebraic geometry. For example, Galois groups act on varieties defined over8

number fields, and therefore act on their (etale) cohomology groups, which arevector spaces over p-adic fields, So we get representations of Galois groups intop-adic Lie groups. These see much more of the Galois group than representationsinto real Lie groups, which tend to have finite images.1.4Lie groups and Lie algebrasLie groups can have a rather complicated global structure. For example, whatdoes GLn (R) look like as a topological space, and what are its homology groups?Lie algebras are a way to linearize Lie groups. The Lie algebra is just the tangentspace to the identity, with a Lie bracket [, ] which is a sort of ghost of thecommutator in the Lie group. The Lie algebra is almost enough to determinethe connected component of the Lie group. (Obviously it cannot see any of thecomponents other than the identity.)Exercise 17 The Lie algebra of GLn (R) is Mn (R), with Lie bracket [A, B] AB BA, corresponding to the fact that if A and B are small then the commutator (1 A)(1 B)(1 A) 1 (1 B) 1 1 [A, B] higher order terms.Show that [A, B] [B, A] and prove the Jacobi identity[[A, B], C] [[B, C], A] [[C, A], B] 0An abstract Lie algebra over a field is a vector space with a bracket satisfyingthese two identities.We have an equivalence of categories between simply connected Lie groupsand Lie algebras. The correspondence cannot detect Non-trivial components of G. For example, SOn and On have the sameLie algebra. Discrete normal (therefore central) subgroups of G. If Z G is anydiscrete normal subgroup, then G and G/Z have the same Lie algebra.For example, SU (2) has the same Lie algebra as P SU (2) ' SO3 (R).If G̃ is a connected and simply connected Lie group with Lie algebra g, then anyother connected group G with Lie algebra g must be isomorphic to G̃/Z, whereZ is some discrete subgroup of the center. Thus, if you know all the discretesubgroups of the center of G̃, we can read off all the connected Lie groups withthe given Lie algebra.Let’s find all the groups with the algebra so4 (R). First let’s find a simplyconnected group with this Lie algebra. You might guess SO4 (R), but that isn’tsimply connected. The simply connected one is S 3 S 3 as we saw earlier (itis a product of two simply connected groups, so it is simply connected). Thecenter of S 3 is generated by 1, so the center of S 3 S 3 is (Z/2Z)2 , the Kleinfour group. There are three subgroups of order 2Therefore, there are 5 groups with Lie algebra so4 .Formal groups are intermediate between Lie algebras and Lie groups: Weget maps (Lie groups) to (Formal groups) to (Lie algebras). In characteristic 0there is little difference between formal groups and Lie algebras, but over moregeneral rings there is a big difference. Roughly speaking, Lie algebras seem to9

be the “wrong” objects in this case as they do not see enough of the group, andformal groups seem to be a good replacement.A 1-dimensional formal group is a power series F (x, y) x y · · · thatis associative in the obvious sense F (x, F (y, z)) F (F (x, y), z). For exampleF (x, y) x y is a formal group called the additive formal group.Exercise 18 Show that F (x, y) x y xy is a formal group, and checkthat over the rationals it is isomorphic to the additive formal group (in otherwords there is a power series with rational coefficients such that F (f (x), f (y)) f (x y)). These formal groups are not isomorphic over the integers.Higher dimensional formal groups are defined similarly, except they are givenby several power series in several variables.1.5Important Lie groupsDimension 1: There are just R and S 1 R/Z.Dimension 2: The abelian groups are quotients of R2 by some discrete subgroup; there are three cases: R2 , R2 /Z R S 1 , and R2 /Z2 S 1 S 1 .There is also a non-abelian group, the group of all matrices of the forma b 1 , where a 0. The Lie algebra is the subalgebra of 2 2 matrices of the0 a x, which is generated by two elements H and X, with [H, X] 2X.form h0 hDimension 3: There are some abelian and solvable groups, such as R2 nR1 , orthe direct sum of R1 with one of the two dimensional groups. As the dimensionincreases, the number of solvable groups gets huge, so we ignore them from hereon.You get the group SL2 (R), which is the most important Lie group of all. Wesaw earlier that SL2 (R) has fundamental group Z. The double cover M p2 (R)is important. The quotient P SL2 (R) is simple, and acts on the open upper halfplane by linear fractional transformationsClosely related to SL2 (R) is the compact group SU2 . We know that SU2 'S 3 , and it covers SO3 (R), with kernel 1. After we learn about Spin groups, wewill see that SU2 Spin3 (R). The Lie algebra su2 is generated by three elementsX, Y , and Z with relations [X, Y ] 2Z, [Y, Z] 2X, and [Z, X] 2Y .1The Lie algebras sl2 (R) and su2 are non-isomorphic, but when we complexify,they both become isomorphic to sl2 (C).There is another interesting 3 dimensional algebra. The Heisenberg algebrais the Lie algebra of the Heisenberg group. It is generated by X, Y, Z, with[X, Y ] Z and Z central. You can think of this as strictly upper triangularmatrices.Dimension 6: (nothing interesting happens in dimensions 4,5) We get thegroup SL2 (C). Later, we will see that it is also called Spin1,3 (R).Dimension 8: We have SU3 (R) and SL3 (R). This is the first time we get a2-dimensional root system.Dimension 14: G2 , which we will discuss a little.Dimension 248: E8 , which we will discuss in detail.This class is mostly about finite dimensional algebras, but let’s mention someinfinite dimensional Lie groups or Lie algebras. 0 1explicit representation is given by X 10 , Y 3cross product on R gives it the structure of this Lie algebra.1 An100 ii 0 , and Z i 00 i . The

1. Automorphisms of a Hilbert space form a Lie group.2. Diffeomorphisms of a manifold form a Lie group. There is some physicsstuff related to this.3. Gauge groups are (continuous, smooth, analytic, or whatever) maps froma manifold M to a group G.4. The Virasoro algebra is generated by Ln for n Z and c, with relations3 n[Ln , Lm ] (n m)Ln m δn m,0 n 12c, where c is central (called thecentral charge). If we set c 0, we get (complexified) vector fields on S 1 , . Thus, the Virasoro algebra is a centralwhere we think of Ln as ieinθ θextension0 cC Virasoro Vect(S 1 ) 0.5. Affine Kac-Moody algebras, which are more or less central extensions ofcertain gauge groups over the circle.2Lie algebrasLie groups such as GLn (R) are quite complicated nonlinear objects. A Liealgebra is a way of linearizing a Lie group, which is often easier to handle.Roughly speaking, the addition and Lie bracket of the Lie algebra are given bythe lowest order terms in the product and commutator of the Lie group. Bya minor miracle (the Campbell-Baker-Hausdorff formula) we do not need anyhigher order terms: the Lie algebra is enough to determine the group productlocally. We first recall some background about vector fields and differentialoperators on a manifold. We will then define the Lie algebra of a Lie group tobe the left invariant vector fields on the group.For any algebra over a ring we define the Lie bracket [a, b] to be ab ba. Itsatisfies the identities [a, b] is bilinear [a, b] [b, a] [[a, b], c] [[b, c], a] [[c, a], b] 0 (Jacobi identity)Definition 19 A Lie algebra over a ring is a module with a bracket satisfyingthe conditions above, in other words it is bilinear, skew symmetric, and satisfiesthe Jacobi identity.These conditions make sense in any additive tensor category, so for examplewe can define Lie algebras of sheaves, or graded Lie algebras. An interestingvariation is Lie superalgebras, where we use the tensor category of supermodulesover a ring or field. Some authors add the non-linear condition that [a, a] 0.Example 20 The basic example of a Lie algebra is given by taking V to be anassociative algebra and defining [a, b] to be ab ba.11

The Lie algebra of a Lie group can be defined as its tangent space at theidentity, with the Lie bracket given by the lowest order part of the commutator.The lowest-order terms of the group law are just given by addition on the Liealgebra, as can be seen in GLn (R): the product of 1 A and 1 B is 1 (A B)to first order. However defining the Lie bracket in terms of the commutator isa little messy, and it is technically more convenient to define the Lie algebra asthe left invariant vector fields on the manifold.There are several different ways to think of vector fields: Informally, a vector field is a little tangent vector at each point. A vector field is informally an infinitesimal diffeomorphism, where we getan infinitesimal diffeomorphism from a vector field by pushing each pointslightly in the direction of the vector field. More formally, a vector field is a section of the tangent bundle or sheaf. A vector field is a normalized differential operator of order at most 1 A vector fie

a Lie group, also abelian. We have essentially found all the connected abelian Lie groups: they are products of copies of the circle and the real numbers. For example, the non-zero complex numbers form a Lie group, which (via the exponential map and polar decomposition) is isomorphic to the product of a circle and the reals.