1 ExamplesA typical example of a Lie group is the group GL2(R) of invertible 2 by 2matrices, and a Lie group is defined to be something that resembles this. Itskey properties are that it is a smooth manifold and a group and these structuresare compatible. So we define a Lie group to be a smooth manifold that is also agroup, such that the product and inverse are smooth maps. All manifolds willbe smooth and metrizable unless otherwise stated.We start by trying to list all Lie groups.Example 1 Any discrete group is a 0-dimensional Lie group.This already shows that listing all Lie groups is hopeless, as there are toomany discrete groups. However we can split a Lie group into two: the componentof the identity is a connected normal subgroup, and the quotient is discrete.Although a complete description of the discrete part is hopeless, we can goquite far towards classifying the connected Lie groups.Example 2 The real numbers under addition are a 1-dimensional commutativeLie group. Similarly so is any finite dimensional real vector space under addition.Example 3 The circle group S1of all complex numbers of absolute value 1 isa Lie group, also abelian.We have essentially found all the connected abelian Lie groups: they areproducts of copies of the circle and the real numbers. Foe example, the non-zero complex numbers form a Lie group, which (via the exponential map andpolar decomposition) is isomorphic to the product of a circle and the reals.Example 4 The general linear group GLn(R) is the archetypal example of anon-commutative Lie group. This has 2 components as the determinant can bepositive or negative. Similarly we can take the complex general linear group.The classical groups are roughly the subgroups of general linear groups thatpreserve bilinear or hermitian forms. The compact orthogonal groups On(R)preserve a positive definite symmetric bilinear form on a real vector space. Wedo not have to restrict to positiver definite forms: in special relativity we getthe Lorentz group O1,3(R) preserving an indefinite form. The symplectic groupSp2n(R) preserves a symplectic form and is not compact. The unitary groupUnpreserves a hermitian form on Cnand is compact as it is a closed subgroupof the orthogonal group on R2n. Again we do not have to restict to positivedefinite Hermitean forms, and there are non-compact groups Um,npreserving|z1|2+ ··· + |zm|2− z2m+1− ···.There are many variations of these groups obtained by tweaking abeliangroups at the top and bottom. We can kill off the abelian group at the topof many of them by taking matrices of determinant 1: this gives special linear,special orthogonal r groups and so on. (“Special” usually means determinat1). Alternatively we can make the abelian group at the top bigger: the generalsymplectic group GSp is the group of matrices that multiply a symplectic formby a non-zero constant. We can also kill off the abelian group at the bottom(often the center) by quotienting out by it: this gives projective general linear5groups and so on. (The word “projective” usually means quotient out by thecenter, and comes from the fact that the projective general linear group actson projective space.) Finally we can make the center bigger by taking a centralextension. For example, the spin groups are double covers of the special orthog-onal groups. The spin group double cover of SO3(R) can be constructed usingquaternions.Exercise 5 If z = a + bi + cj + dk is a quaternion show that zz is real, wherez = a − bi − cj − dk. Show that z 7→ |z| =√zz is a homomorphism of groupsfrom non-zero quaternions to positive reals. Show that the quaternions forma division ring; in other words check that every non-zero quaternion has aninverse.Exercise 6 Identify R3with the set of imaginary quaternions bj + cj + dk.Show that the group of unit quaternions S3acts on this by conjugation, andgives a homomorphism S37→ SO3(R) whose kernel has order 2.A typical example of a solvable Lie group is the group of upper triangularmatrices with nonzero determinant. (Recall that solvable means the group canbe split into abelian groups.) It has a subgroup consisting of matrices with 1son the diagonal: this is a typical example of a nilpotent Lie group. (Nilpotentmeans that if you keep killing the center you eventually kill the whole group.We will see later that a connected Lie group is nilpotent if all elements of its liealgebra are nilptent matrices: this is where the name “nilpotent” comes from.)Exercise 7 Check that these groups are indeed solvable and nilpotent.Exercise 8 Show that any finite group of prime-p ower order pnis nilpotent,and find a non-abelian example of order p3for any prime p. (Hint: show thatany conjugacy class not in the center has order divisible by p, and deduce thatthe center has order divisible by p unless the group is trivial.)Exercise 9 The Moebius group consists of all ismorphisms from the complexunit disk to itself: z 7→ (az + b)/(cz + d) with ad − bc = 1, a =d, b = c.Show that this is the group P SU1,1. Similarly show that the group of conformaltransformations of the upper half plane is P SL2(R). Since the upper half planeis isomorphic to the unit disc, we see that the groups P SU1,1and P SL2(R) areisomorphic. This illustrates one of the confusing things about Lie groups: thereare a bewildering number of unexpected isomorphisms between them in smalldimensions.Exercise 10 Show that there is a (nontrivial!) homomorphism from SL2(R)to the group O2,1(R), and find the image and kernel. (Consider the action ofthe groups SL2(R) on the 3-dimensional symmetric square S2(R2) and showthat this action preserves a quadratic form of signature (2, 1).)Klein c laimed at one point that geometry should be identified with grouptheory: a geometry is determined by its group of symmetries. (This fails for Rie-mannian geometry.) For example, affine geometry consists of the properties ofspace invariant under the group of affine transformsations, projective geometryis properties of projective space invariant under projective transformations, and6so on. The group of affine transformations in n dimensicanons is a semidirectproduct Rn.GLn(R). This can b e identified with the subgroup of GLn+1fixing avector (sometimes called the mirabolic subgroup). For example, in 1-dimensionwe get a non-abelian 2-dimensional Lie group of transformations x 7→ ax + bwith a 6=