Preliminary Exam Notes

ly one can define a Lie group as a smooth manifold which is also a group such that inversion and multiplication are smooth maps. A special case we focus on is any (topologically) closed subgroup of any SL(`,R). The major examples of “classical” Lie groups are SL(n,R), SO(m,n) (with g Im,ng = Im,n), SU(m,n) (same as SO but with conjugate transpose and complex elements), and Sp(2m,R) (similar to SO but with J2m with zero blocks on the diagonal and ±I on the off diagonal blocks). Some other Lie groups are S and S (circle group and modulus 1 quaternions) but not S (by the hairy ball theorem, there is no Lie algebra). A discrete subgroup Γ of a Lie group G is called a lattice if the Haar volume of G/Γ is finite. Examples: any discrete subgroup of a compact Lie group; also SL(n,Z) ⊂ SL(n,R). The rank of a Lie group is the dimension of any Cartan subgroup of it. A Cartan subgroup is a maximal nilpotent subgroup C ≤ G such that each finite-index normal subgroup D of C has finite index in NG(D), the normalizer of D in G. For (connected) matrix groups, this is equivalent to the centralizer of a maximal torus of G, or as a connected closed nilpotent subgroup C which coincides with the identity component of NG(C). Rank is a useful invariant which helps classify Lie groups, and certain phenomena only happen for higher-rank Lie groups. The geometry of Lie groups is very well-understood. Since semisimple Lie groups decompose to simple ones, it is useful to tell if it semisimple. Not only is the theory elegant in its own right, drawing together many areas of mathematics, but it has profound applications across mathematics. For our purposes, connections to symmetric spaces and moduli spaces will be the most useful. 2.2.2 The Lie algebra of a Lie group In the abstract, a Lie algebra is a vector space together with a Lie bracket (binary operation) with a few axioms. For our purposes, it will suffice to define it as a vector subspace of the set of n × n (real) matrices closed under commutators. The Lie algebra of a Lie group is the collection of left-invariant (under pushforward by the derivative of leftmultiplication) vector fields. It is isomorphic to the tangent space at the identity, so it is of course a vector space. The Lie bracket is the same as commutators of matrices, which act as differential operators. CLaim: the Lie bracket gives back a left-invariant vector field. Example: The Lie algebra sl(`,R) of SL(`,R) is the collection of trace-0 matrices. To see why, notice that 1 = det(I + tA) = 1 + t(Tr(A) + · · · ), so d/dt(det(I + tA))|t=0 = Tr(A) = 0, since the determinant must be