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Lie Group, Lie Algebra and Their Representations

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Lie Group, Lie Algebra and Their Representations Lie Group, Lie Algebra and their Representations Prof. Totaro ([email protected]) Typeset by Aaron Chan ([email protected]) Last update: June 8, 2010 Recommended Books: A. Kirillov - An introduction to Lie groups and Lie algebras J-P. Serre - Complex semisimple Lie algebra W. Fulton, J. Harris - Representation theory Kirillov is the closest to what we will cover, Fulton-Harris is longer but with lots of example, which provides a good way to understand representation theory. This course fit in especially well with Differential Geometry and Algebraic Topology. Definition 1 A Lie group is a group which is also a smooth manifold Example: (R,+) is a Lie group of dimension 1 1 S = fz 2 C : jzj = 1g under multiplication Definition 2 n n+1 2 2 The n-sphere S = f(x0; : : : ; xn)g 2 R jx0 + ··· + xn = 1g is an n-manifold Many interesting Lie groups act on S2 Example: 3 SO(3)=group of rotation in R (this is non-abelian) 2 P GL(2; C) acts on S = C [f1g as Mobius transformation a 0 Here SO(3) ⊆ P GL(2; ) = GL(2; )= a 2 × C C 0 a C 2 z 7! 2z, say, is in P GL(2; C) acting on S = C [f1g Examples of Lie groups n • (R ; +) any n 2 N (or any finite dimensional real vector space) × • R = fx 2 R jx 6= 0g under multiplication × • C = fx 2 C jx 6= 0g under multiplication • GL(n; R) = fA 2 Mn(R)j det A 6= 0g under mulitiplication • GL(V ) - General Linear group, where V is a finite dimensional vector space n n • SL(n; R) = fA 2 Mn(R)j det A = 1g = ff : R ! R linear and preserves volumeg (Special Linear group) • O(n) (Orthogonal group) • Sp(2n; R) (Sympletic group) • U(n) (Unitary group) • SU(n) (Special Unitary group) Remark: S0;S1;S3 are the only spheres that are also Lie groups 1 Orthogonal group > O(n) = fA 2 Mn(R)jAA = 1g n n = ff : R ! R jf linear and preserves distances g n n n = ff : R ! R jhf(x); f(y)i = hx; yi 8x; y 2 R g n where the standard inner product on R is h(x1; : : : ; xn); (y1; : : : ; yn)i = x1y1 + x2y2 + ··· + xnyn 2 R Elements of O(n) includes rotations and reflections × Note that det is a homomorphism det : GL(n; R) ! R and this restricts to det : O(n) ! {±1g since, A 2 O(n) ) 1 = det(1) = det(AA>) = det(A) det(A>) = det(A)2 Definition 3 Special orthogonal group > SO(n) = fA 2 Mn(R)jAA = 1; det A = 1g n Elements include rotations but not reflections (on R ) SO(n) is a subgroup of index 2 in O(n). In fact, O(n) has 2 connected component, the one containing 1 is SO(n) Also note that SO(2) =∼ S1 Symplectic group 2n 2n 2n Sp(2n; R) = ff : R ! R jw(x; y) = w(f(x); f(y))8x; y 2 R g 2n where w is a non-degenerate alternating bilinear form on R : n 0 0 0 0 X 0 0 w (q1; : : : ; qn; p1; : : : ; pn); (q1; : : : ; qn; p1; : : : ; pn) = qipi − piqi i=1 for some choice of basis. n Remark: Any non-degenerate alternating bilinear form w on R must have n even, and after a change of basis, such a form is given by above formula Example: Sp(2n; R) ⊂ SL(2n; R) Sp(2; R) = SL(2; R) = the group of area preserving linear maps Unitary group n n U(n) = ff : C ! C linear and preserves distanceg (= GL(n; C) \ O(2n)) Definition 4 n The standard inner product on C is the nondegenerate positive definite Hermitian form n X h(z1; : : : ; zn); (w1; : : : ; wn)i = z1w1 i=1 n 2n p Note that the length of a vector z 2 C (= R ) is kzk = hz; zi 2 2 2 (as z = xiy; zz = jzj = x + y ) 2 So, we have n n n U(n) = ff : C ! C linear jhf(x); f(y)i = hx; yi 8x; y 2 C g ∗ ∗ = fA 2 GL(n; C)jAA = 1g (A = A>) Special Unitary group 1 × The det of a unitary matrix gives a homomorphism det : U(n) ! S ⊂ C SU(n) = ker det jU(n) Example 1 U(1) = fz 2 M1(C)jzz = 1g = S a b 2 2 SU(2) = a; b 2 C; jaj + jbj = 1 −b a 3 2 2 2 2 Wee see that SU(2) is diffeomorphic to S = f(x0; x1; x2; x3)jx0 + x1 + x2 + x3 = 1g Remark: S0;S1;S3 are the only sphere that are also Lie groups Some Basics of Smooth Manifold Definition 5 n n A subset M ⊆ R is called k-dimensional manifold (in R ) if for every point x 2 M, the following condition is satisfied: n (M) There is an oen set U 3 x, and open set V ⊂ R , and a diffeomorphism h : U ! V s.t. k h(U \ M) = V \ (R ×{0g) = fx 2 V jxk+1 = ··· xn = 0g Definition 6 n 1 Let U ⊆ R ; n > 0 be an open set, a smooth function f : U ! R (or C ) if all partial derivatives @r f (r ≥ 0) @ ··· @ xi1 xir are defined and continuous on U m n For U ⊆ R open, a smooth mapping f : U ! R is a function s.t. f = (f1; : : : ; fm), with fi : U ! R smooth function n n For U; V ⊆ R , a diffeomorphism f : U ! V of degree n is a smooth map (on R ) with a smooth inverse m n m n The derivative dfjx of a smooth map f : U(⊆ R ) ! R at a point x 2 U is a linear map R ! R given by matrix ( @fi ) @xj 0 @f1 ··· @f1 1 0 u 1 0v 1 @x1 @xm 1 1 B . .. C B . C B . C @ . A @ . A = @ . A @f @f n ··· 1 um vn @x1 @x1 composite of the smooth maps is smooth, and d(g ◦ f)jx = dgjf(x) ◦ dfjx Theorem 7 (Inverse Function Theorem) n n Let U ⊂ R open, f : U ! R be a smooth map. Suppose dfjx is an isomorphism, for some x 2 U Then 9V ⊂ U, V 3 x, s.t. f(V ) is open and f is a diffeomorphism from V to f(V ) Theorem 8 (Implicit Function Theorem) m n Let U ⊆ R open, f : U ! R smooth map. Suppose dfjx is surjective at a point x 2 U (n ≤ m) n Then 9V ⊆ U; V 3 x and diffeomorphism φ : W ! V (W ⊆ R open), s.t. f(φ(x1; : : : ; xn)) = (x1; : : : ; xn) 3 Definition 9 A submersion is a smooth map which has derivatives being surjective everywhere N N A smooth submanifold X ⊆ R of dimension n is a subset s.t. 8x 2 X; 9 nbhd U 3 x, U ⊆ R and a N−n −1 submersion F : U ! R s.t. X \ U = F (0) ⊆ U Example: n n+1 Claim: The sphere S ⊆ R is a smooth n-dimensional submanifold Proof n −1 S = F (0), where n+1 F : R ! R 2 2 (x0; : : : ; xn) 7! x0 + ··· + xn − 1 We have to check that F is a submersion at points x 2 Sn: dF = (2x0; 2x1;:::; 2xn) (row matrix) n This is surjective whenever (x0; : : : ; xn) 6= (0;:::; 0) ) surjective everywhere on S Example: 2 X = f(x; y) 2 R jxy = ag is a smooth 1-dimensional submanifold for nonzero a 2 R, but not for a = 0, here we have: 2 F : R ! R (x; y) 7! xy − a Example: 2 2 X = fx 2 R jx = 0g is an 0-dimensional submanifold of R, but x : R ! R is not a submersion at 0. To prove that X is a 0-dimensional submanifold, you have to notice that X = fx 2 R jx = 0g Definition 10 N The tangent space to a smooth n-dimensional submanifold X ⊆ R at a point x 2 X (if we describe −1 N−n X as X = F (V ) for some submersion F : V ! R ) is defined as: N N−n TxX = ker(dF jx : R ! R ) N This is an n-dimensional linear subspace of R N Let X ⊂ R be a smooth n-dimensional submanifold. A function f : X ! R is smooth , near each point x 2 X, f is the restriction of a smooth function N N on an open nbhd of X in R (0-dimensional submanifold of R = discrete subset) M N For submanifold X ⊆ R , Y ⊆ R (dim=m; n resp.) a smooth map f : X ! Y has derivative dfjx : TxX ! Tf(x)Y which is a linear map A diffeomorphism between 2 submanifolds is a smooth map with smooth inverse. N Fact: (Hausdorff countable basis) Every smooth manifold is diffeomorphic to a submanifold of R M N M N M+N For submanifold X ⊆ R ;Y ⊆ R (dim=m; n resp.), the product X × Y ⊆ R × R = R is a smooth submanifold. It has the product topology Lie Group Definition 11 A Lie group G is a smooth manifold which is also a group s.t. multiplication : G × G ! G (g; h) 7! gh inverse : G ! G g 7! g−1 are smooth maps. We have a point 1 2 G (the identity) 4 N Note that a Lie group need not be connected. (0-dimensional submanifold of R =discrete subset) In particular, we can view any group (say countable) as a 0-dimensional Lie group.
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