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Lie Groups Notes Notes for Lie Theory Ting Gong Started Feb. 7, 2020. Last revision March 23, 2020 Contents 1 Matrix Analysis 1 1.1 Matrix groups . .1 1.2 Topological Properties . .4 1.3 Lie groups and homomorphisms . .6 1.4 Matrix Exponential . .8 1.5 Matrix Logarithm . .9 1.6 Polar Decomposition and More . 11 2 Lie Algebra and Representation 14 2.1 Definitions . 14 2.2 Lie algebra of Matrix Group . 16 2.3 Mixed Topics: Homomorphisms and Complexification . 18 i Chapter 1 Matrix Analysis 1.1 Matrix groups Last semester, in our reading group on differential topology, we analyzed some common Lie groups by treating them as an embedded submanifold. And we calculated the dimension by implicit function theorem. Now we discuss an algebraic approach to Lie groups. We assume familiarity in group theory, and we consider Lie groups as topological groups, namely, putting a topology on groups. This is going to be used in matrix analysis. Definition 1.1.1. Let M(n; k) be the set of n × n matrices, where k is a field (such as C; R). Let GL(n; k) ⊂ M(n; k) be the set of n × n invertible matrices with entries in k. Then we call this the general linear group over k. n2 Remark 1.1.2. Now we let k = C, and we can consider M(n; C) as C with the Euclidean topology. Thus we can define basic terms in analysis. Definition 1.1.3. Let fAmg ⊂ M(n; C) be a sequence. Then Am converges to A if each entry converges to the cooresponding entry in A in Euclidean metric space. Definition 1.1.4. A matrix Lie group is a closed subgroup G of GL(n; C). That is, If Am is a convergent sequence in G, then Am converges to some A 2 G or A not invertible. Example 1.1.5. Consider GL(n; C), it is a matrix Lie group since the either the convergent sequence converges to an invertible matrix or a non-invertible one. Example 1.1.6. We define SL(n; k) be the group of n × n invertivle matrices with entries in k with determinant 1. Then if we have a convergent sequence in SL(n; C), converging to A, then det(A) = 1. Thus A 2 SL(n; C). Thus SL(n; C) is a matrix Lie group. Definition 1.1.7. An n × n complex matrix A is unitary if A∗A = I, where A∗ is the adjoint of A, the complex conjugate of the transpose of A. This is equivalent to A∗ = A−1 1 Example 1.1.8. The group of unitary matrices is called the unitary group, and we denote it as U(n). The group of unitary matrices with determinant 1 is called the special unitary group, and we denote it as SU(n). Let fAmg be a convergent sequence in U(n), converging −1 −1 ∗ ∗ ∗ −1 ∗ −1 to A, then Am ! A and Am ! A . And Am = Am , Thus A = A since limit is unique. Thus U(n) is closed subgroup of GL(n; C). Thus a matrix group. And SU(n) = U(n)\SL(n; C). Thus SU(n) is closed and subgroup of GL(n; C). Thus a matrix Lie group. n P Now, we let h·; ·i be the standard inner product on C , by hx; yi = i xiyi. Then we see that the condition on unitary matrix is equivalent to hAx; Ayi = hx; yi. Proposition 1.1.9. If A is unitary, j det(A)j = 1. Proof. det(A∗A) = j det(A)j2 = det(I) = 1 since det(A∗) = det(A) and basic complex multiplication. ut Example 1.1.10. An n × n real matrix A is called orthogonal if AT = A−1. Thus it n preserves inner product on R as unitary matrices do. And by proposition, j det(A)j = ±1. The group of all orthogonal matrices is called the orthogonal group, denoted O(n), and the ones with determinant 1 is called the special orhtogonal group, denoted SO(n). Then O(n) denotes rotations and reflections and SO(n) denotes rotations. These are matrix Lie groups, which can be checked as unitary groups. P Now we define a bilinear form (·; ·) such that (x; y) = i xiyi (called the dot product), n where x; y 2 C . Example 1.1.11. The group of n × n complex valued matrices which preserves the above bilinear form is the complex orthogonal group, O(n; C), which is also a matrix Lie group. Similarly we have another matrix Lie group SO(n; C). n+k Example 1.1.12. Let n; k 2 N, define [·; ·]n;k on R , a symmetric bilinear by [x; y]n;k = x1y1 + ::: + xnyn − xn+1yn+1 − ::: − xn+kyn+k. Then the set of (n + k) × (n + k) real matrices A such that [Ax; Ay]n;k = [x; y] is called the generalized orthogonal group, denoted as O(n; k) is a matrix Lie group. In physics, one is interested in O(3; 1), the Lorentz group. We define SO(n; k) be the subgroup of O(n; k) with determinant 1. 2n Pn Example 1.1.13. We define a skew-symmetric bilinear form on R by !(x; y) = j=1(xjyn+j− xn+jyj). The set of all 2n × 2n real matrices A such that !(Ax; Ay) = !(x; y) is the real symplectic group denoted as Sp(n; R), and this is a matrix Lie group. 0 I Let Ω = be a 2n × 2n matrix, then !(x; y) = hx; Ωyi by direct computation. −I 0 Therefore !(Ax; Ay) = hAx; ΩAyi = hx; AT ΩAyi = hx; Ωyi = !(x; y) iff AT ΩA = Ω. Thus −ΩAT Ω = A−1. Thus, one can show that det(A)2 = 1. We can define similarly the Complex Symplectic Group Sp(n; C), where all the identity above holds. Finally, the compact symplectic group Sp(n) = Sp(n; C) \ U(2n). Then this preserves both the bilinear form ! and the inner product we defined above. 2 n Example 1.1.14. The Euclidean group E(n) is the group of all transformation of R that can be expressed by translations and orthogonal linear transformation. Therefore, n we can write elements in E(n) as pairs fx; Rg with x 2 R and R 2 O(n). An we let n fx; Rg act on R by fx; Rgy = Ry + x. Then the operation of this group is given by fx1;R1gfx2;R2gy = R1(R2y + x2) + x1 = R1R2y + x1 + R1x2 = fx1 + Rx2;R1R2gy. And thus, the inverse is given by fx; Rg−1 = {−R−1x; R−1g. Thus, we observe that an element R x of E(n) is isomorphic to the matrix , R 2 O(n). Therefore, E(n) is isomorphic to 0 1 the closed subgroup of GL(n + 1; R). We define P(n; 1) the Poincare group (inhomogeneous Lorentz group) be the group of A x all transformations of n+1, where an element is in the form where A 2 O(n; 1); x 2 R 0 1 n+1 R . Example 1.1.15. In this example, we introduce some examples that are more common. 21 a b3 The set of all 3 × 3 real matrices of the form 40 1 c5. This set forms a group called the 0 0 1 Heisenberg group. (This is useful in physics as a model for Heisenberg-Weyl commutation relation) One can associate R−f0g with multiplication to be GL(1; R), C−f0g with multiplication 1 to be GL(1; C). And S the sphere, to be U(1). And if we consider R with addition, then we can associate it with the set of matrices with positive determinant by the exponential x n map: x 7! e . Similarly we can do this with R . We end this section with a careful examination of the Compact Symplectic groups by showing that is it the "unitary group over the quaternions". Firstly, define conjugate-linear 2n 2n n 2n map J : C ! C by J(α; β) = (−β;¯ α¯) where α; β 2 C and (α; β) 2 C . Then we 2 2n observe J = −I, and that if z; w 2 C , then !(z; w) = hJz; wi = −hz; Jwi = −hJw; zi. Proposition 1.1.16. If U 2 U(2n), then U 2 Sp(n) if and only if U commutes with J. 2n ∗ Proof. Let U 2 U(2n). Let z; w 2 C , then !(Uz; Uw) = hJUz; Uwi = hU JU; wi = hU −1JUz; wi, and we also have !(z; w) = hJz; wi. Then U preserves ! if and only if U −1JU = J. Thus UJ = JU. ut One should know the quaternion group H in group theory, with eight elements such that i2 = j2 = k2 = −1. In fact the quarternion group can be written in matrix form i 0 0 1 0 i i = ; j = ; k = . Then the quarternion algebra is the space of real 0 −i −1 0 i 0 linear combinations of I; i; j; k. 2n Since J is conjugate linear, then J(iz) = −iJ(z). Then for z 2 C , iJ = −Ji. Then if we define K = iJ, then one can check that iI; J; K satisfy the relations as i, j, k. 3 Therefore, we observe that U 2 Sp(n) commutes with this defined i,j,k. Thus, we conclude that U 2 Sp(n) iff U is quaternion linear and preserves the norm. That is why we describe Sp(n) as the unitary group over the quaternions.
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