An introduction to the geometry of homogeneous spaces

Size: px
Start display at page:

Download "An introduction to the geometry of homogeneous spaces"

Transcription

1 Proceedings of The Thirteenth International Workshop on Diff. Geom. 13(2009) An introduction to the geometry of homogeneous spaces Takashi Koda Department of Mathematics, Faculty of Science, University of Toyama, Gofuku, Toyama , Japan koda@sci.u-toyama.ac.jp Abstract. This article is a survey of the geometry of homogeneous spaces for students. In particular, we are concerned with classical Lie groups and their root systems. Then we describe curvature tensors of the Riemannian connection and the canonical connection of a homogeneous space G/H in terms of the Lie algebra g. 1 Lie groups 1.1 Definition and examples Definition 1. A set G is called a Lie group if (1) G is an (abstract) group, (2) G is a C manifold, (3) the group operations G G G, G G, (x, y) xy, x x 1 are C maps. Example 1 R n is an (abelian) Lie group under vector addition: (x, y) x + y, x x. Example 2 (Classical Lie groups) Let M(n, R) be the set of all real n n matrices. We may identify it as M(n, R) = R n2. The subset GL(n, R) = {A M(n, R) det A 0} is an open submanifold of M(n, R), and becomes a Lie group under the multiplication of matrices. GL(n, R) is called the general linear group. Definition 2. Let G be a Lie group. A Lie group H is a Lie subgroup of G if (1) H is a submanifold of the manifold G (H need not have the relative topology), (2) H is a subgroup of the (abstract) group G. 121

2 122 Takashi Koda Example 3 SL(n, R) = {A GL(n, R) det A = 1}, dim = n 2 1 O(n) = {A GL(n, R) t AA = A t n(n 1) A = I n }, dim = 2 n(n 1) SO(n) = {A O(n) det A = 1}, dim = 2 U(n) = {A GL(n, C) A A = A A = I n }, dim = n 2 SU(n) = {A U(n) det A = 1}, dim = n 2 1 Sp(n) = {A GL(n, H) AA = A A = I n }, dim = 2n 2 + n where, H = {a + bi + cj + dk a, b, c, d R} denotes the set of all quaternions (i 2 = j 2 = k 2 = 1, ij = ji = k, jk = kj = i, ki = ik = j), and A = t A = t A is the conjugate transpose of a matrix A. Theorem 1.1. (Cartan) Let H be a closed subgroup of a Lie group G. Then H has a structure of Lie subgroup of G. (The topology of H coincides with the relative topology and the structure of Lie subgroup is unique). 1.2 Left translation For any fixed a G, the map L a : G G defined by L a (x) = ax for x G is called the left translation. L a is a C map on G, and furthermore it is a diffeomorphism of G. L 1 a = L a 1. Similarly, the right translation R a by a G is a diffeomorphism defined by R a (x) = xa for x G. The inner automorphism i a : G G i a := L a R a 1, i a (x) = axa 1 (x G) is C map on G. Definition 3. Let G be a Lie group. A vector field X on G is left invariant if (L a ) X = X, where ((L a ) X)f := X(f L a ) (f C (G)).

3 An introduction to the geometry of homogeneous spaces 123 More precisely, (L a ) x X x = X ax for x G ((L a ) x X x )f = X x (f L a ) for f C (G) We denote by g the set of all left invariant vector fields on G. Proposition 1.2. (1) For X, Y g and λ, µ R, we have λx + µy g. (2) For X, Y g, we have [X, Y ] g. Remark. For vector fields X, Y on a manifold, bracket [X, Y ] is defined by [X, Y ]f := X(Y f) Y (Xf) (f : C function). 1.3 Lie algebras Definition 4. An n-dimensional vector space g over a field K = R or C with a bracket [, ] : g g g is called a Lie algebra if it satisfies (1) [, ] is bilinear, (2) [X, X] = 0 for X g and (3) (Jacobi identity) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 for X, Y, Z g. Example M(n, R), M(n, C) and M(n, H) with a commutator [, ] defined by [X, Y ] = XY Y X are Lie algebras over R. Definition 5. A vector subspace h of a Lie algebra g is called a Lie subalgebra if [h, h] h. Example sl(n, R) = {X M(n, R) trace X = 0} M(n, R) o(n) = {X M(n, R) X + t X = 0} M(n, R) so(n) = {X o(n) trace X = 0} o(n) u(n) = {X M(n, C) X + X = O} M(n, C) su(n) = {X u(n) trace X = 0} u(n)

4 124 Takashi Koda 1.4 Lie algebra of a Lie group Let X, Y be left invariant vector fields on a Lie group G, then hence we get (L a ) X = X, (L a ) Y = Y for a G, (L a ) [X, Y ] = [X, Y ]. In general, if ϕ : M N is a C map and vector fields X, Y on M and X, Y on N satisfy ϕ X = X, ϕ Y = Y, then we have ϕ [X, Y ] = [X, Y ]. Proposition 1.3. The set g of all left invariant vector fields on a Lie group G is a Lie algebra with respect to the Lie bracket [X, Y ] of vector fields X, Y on G. g is called the Lie algebra of a Lie group G. Proposition 1.4. The Lie algebra g of a Lie group G may be identified with the tangent space T e G at the identity e G. and conversely g X X e T e G (isomorphism between vector spaces) T e G X e X g defined by X x := (L x ) e X e (x G). Example An element x in GL(n, R) M(n, R) = R n2 may be expressed as x = (x i j), then x i j are global coordinates of GL(n, R). A left invariant vector field X gl(n, R) on GL(n, R) may be written as X = n Xj i x i i,j=1 j Then X e T e GL(n, R) may be identified with the n n matrix (Xj i (e)) M(n, R).. n gl(n, R) X = Xj i x i i,j=1 j (X i j(e)) M(n, R) Then we may see that the Lie algebra gl(n, R) of GL(n, R) is isomorphic to M(n, R) as Lie algebras: (gl(n, R), [, ]) = (M(n, R), [, ])

5 An introduction to the geometry of homogeneous spaces 125 Theorem 1.5. Let G be a Lie group and g its Lie algebra. (1) If H is a Lie subgroup of G, h is a Lie subalgebra of g (2) If h is a Lie subalgebra, there exists a unique Lie subgroup H of G such that the Lie algebra of H is isomorphic to h. 1.5 Exponential maps Definition 6. A homomorphism ϕ : R G is called a 1-parameter subgroup of a Lie group G. ϕ(t) is a curve in G such that ϕ(0) = e, ϕ(s + t) = ϕ(s)ϕ(t). For any 1-parameter subgroup ϕ : R G, we consider its initial tangent vector ( ) d ϕ (0) = ϕ 0 T e G dt This is a one-to-one correspondense. For each left invariant vector field X g on G, we denote by ϕ X (t) the 1- parameter subgroup with the initial tangent vector ϕ X (0) = X e T e G. Definition 7. The map exp : g G defined by 0 exp X := ϕ X (1) is called the exponential map of a Lie group G. Example For each X M(n, R), the series of matrix e X = I + X + X2 2! converges in M(n, R) = R n2. The map exp : M(n, R) M(n, R), + + Xn n! is called the exponential map of matrices. If [X, Y ] = 0, exp(x + Y ) = exp X exp Y and we may see that Hence + X exp X = e X exp O = I, exp( X) = (exp X) 1. exp : M(n, R) GL(n, R). This coincides with the exponential map of the Lie group GL(n, R).

6 126 Takashi Koda 1.6 Adjoint representations Let G be a Lie group and i a denotes the inner automorphism by a G. We denote its differential by i a = L a R a 1 : G G, x axa 1 Ad(a) = Ad G (a) := (i a ). For any left invariant vector field X g, Ad(a)X = (i a ) X is also left invariant. satisfies Thus we have a homomorphism Ad(a) : g g (linear) [Ad(a)X, Ad(a)Y ] = Ad(a)[X, Y ]. Ad : G GL(g), which called the adjoint represenation of G. Proposition 1.6. For any a G and X g, satisfies a Ad(a) exp(ad(a)x) = i a (exp X). Let g be a Lie algebra with a bracket [, ]. For any X, the linear map (Jacobi identity). The homomorphism ad(x) : g g, Y ad(x)y := [X, Y ] ad(x)[y, Z] = [ad(x)y, Z] + [Y, ad(x)z] ad : g gl(g), is called the adjoint representation of g. Proposition 1.7. For any X, Y g, Ad(exp X)Y = e ad(x) Y = X ad(x) k=0 Definition 8. Let g be a Lie algebra. The bilinear form 1 k! ad(x)k Y. B(X, Y ) := tr(ad(x) ad(y )) (X, Y g) is called the Killing form of g, where the right-hand-side means the trace of the linear map ad(x) ad(y ) : g g.

7 An introduction to the geometry of homogeneous spaces 127 Example sl(n, R) B(X, Y ) = 2n tr(xy ) o(n) B(X, Y ) = (n 2) tr(xy ) su(n) B(X, Y ) = 2n tr(xy ) sp(n) B(X, Y ) = (2n + 2) tr(xy ) 1.7 Toral subgroups and maximal tori Definition 9. Let G be a connected Lie group. A subgroup S of G which is isomorphic to S 1 S }{{} 1 is called a toral subgroup of G, or simply a torus. k A toral subgroup S is an arcwise-connected compact abelian Lie group. Definition 10. A torus T is a maximal torus of G if T is any other torus with T T G = T = T Theorem 1.8. Any torus S of G is contained in a maximal torus T. Theorem 1.9. (Cartan Weil Hopf) Let G be a compact connected Lie group. Then (1) G has a maximal torus T. (2) G = xt x 1. x G (3) Any other maximal torus T of G is conjugate to T, i.e., a G s.t. T = at a 1 (4) Any maximal torus of G is a maximal abelian subgroup of G. The dimension l = dim T of T is called the rank of G. Example G = U(n). T := e iθ e iθn is a toral subgroup of U(n). θ 1,, θ n R = S 1 S 1 }{{} n

8 128 Takashi Koda Proposition Any unitary matrix A U(n) can be diagonalized by a unitary matrix P U(n), P 1 AP = e iθ e iθ n =: diag[eiθ 1,, e iθ n ] Example G = SO(n). We denote by R θ the rotation matrix ( cos θ sin θ R θ = sin θ cos θ ) and if n = 2m put T = R θ R θm θ 1,, θ m R = S 1 S 1 }{{} m and if n = 2m + 1 we put T = R θ Rθm 0 1 θ 1,, θ m R = S 1 S 1 }{{} m then T is a maximal torus of SO(n). Proposition Any orthogonal matrix A O(n) may be put into the following

9 An introduction to the geometry of homogeneous spaces 129 form by an orthogonal matrix Q O(n), t QAQ = 1 cos θ 1 sin θ 1 sin θ 1 cos θ 1... cos θ 0 r sin θ r sin θ r cos θ r Remark. R 0 = ( ) = I 2, R π = ( ) = I 2, 1.8 Root systems Let G be a compact connected Lie group. Let (ρ, V ) be any representation of G. ρ : G GL(V ) (homomorphism) (V is a vector space over R or C). We may choose a ρ(g)-invariant inner product on V. For any ρ(g)-invariant subspace W V, we have V = W W Therefore, (ρ, V ) is completely reducible. (ρ(g)-invariant decomposition) (ρ, V ) = (ρ 1, V 1 ) (ρ k, V k ) where (ρ i, V i ) is irreducible. In particular, the adjoint representation (Ad, g) of G is completely reducible, and we have a decomposition where g = g 0 g 1 g m g 0 = {X g ad(y )X = 0 for Y g} (center of g) and g i is an ideal of g (i.e., [g, g i ] g i ) and g i is irreducible.

10 130 Takashi Koda Proposition Let A, B be two diagonalizable matrices. If [A, B] = 0, then A and B are simultaneously diagonalizable, i.e., x 0 s.t. Ax = λx, Bx = µx (λ, µ : eigenvalues) Let G be a compact connected Lie group, and S a torus of G. We choose an Ad(S)-invariant inner product on g. Then the representation (Ad, g) of S is completely reducible. If we consider the complexification g C = g R C and the representation (Ad, g C ) of S, the eigenvalues of Ad(s) are complex numbers with absolute value 1, and corresponding eigenspaces in g C are of dim C = 1. Then g may be decomposed into irreducible subspaces. g = V 0 V 1 V m where Ad(s) V0 = id on V 0 and for some linear form θ k : s R ( ) cos θk (v) sin θ Ad(exp v) Vk = k (v) sin θ k (v) cos θ k (v) on V k for v s w.r.t. an orthonormal basis on V k. These θ k are unique up to sign and order, and do not depent on the inner product and an orthonormal basis. Definition 11. Linear forms ±θ k (k = 1,, m) are called roots of G relative to S. If S is a maximal torus T of G, they are called simply the roots of G. The set of all roots of G is called the root system of G. Remark. Sometimes we define roots as ± 1 2π θ k so as to take Z-values on the unit lattice exp 1 (e) t. Example G = SU(n) and T is the maximal torus defined as before. n T = {diag[e ix 1,, e ix n ] x k R, x k = 0} k=1 The Lie algebra t of T is n t = {diag[ix 1,, ix n ] x k R, x k = 0} k=1

11 An introduction to the geometry of homogeneous spaces 131 x 1,, x n t are deined in natural way. linear form diag[ix 1,, ix n ] x k is denoted by x k We take the inner product (, ) on g defined by (X, Y ) := 1 tr (XY ) 2 and for 1 j < k n, 1 l n, put 1 U jk = 1, U jk = i i, 2El = 2 i which form an orthonormal basis of g. Then we may see that for any v = diag[ix 1,, ix n ] t ( ) cos (xj x Ad(exp v) = k ) sin (x j x k ) sin (x j x k ) cos (x j x k ) on span R {U jk, U jk }. Thus the root system of SU(n) is = {±(x j x k ) 1 j < k n} Definition 12. Let {λ 1,, λ l } be a basis of t. The lexicographic order λ 1 > λ 2 > > λ l > 0 is defined by l v = v i v 1 = = v k 1 = 0 λ i > 0 v k > 0 for some k k=1 A root α is positive (resp. negative) if α > 0 (resp. α < 0) relative to the lexicographic order w.r.t. the basis. = + A positive root α is called a simple root if it can not be expressed as a sum of two positive roots. The set {α 1,, α l } of all simple roots of G is called the simple root system of G (where l = rank G). Example G = SU(n) and T a maximal torus of all diagonal matrices. {x 1,, x n 1 } is a basis of t. Consider the lexicographic order x 1 > x 2 > > x n 1 > 0. Then

12 132 Takashi Koda + = {x j x k 1 j < k n} = { (x j x k ) 1 j < k n} and the simple root system is positive root system negative root system {α j = x j x j+1 1 j n 1} 1.9 Root systems again Here, we shall calculate the root system in terms of Lie algebra. Let G be a compact connected semisimple Lie group, T a maximal torus of G, g the Lie algebra of G, t the Lie algebra of T. g C := g R C, t C := t R C We extend [, ] complex-linearly on g C, then g C becomes a complex Lie algebra. t C becomes a maximal abelian Lie subalgebra of g C. The Killing form B C of g C is the natural extension of the Killing form B of g. If we take an Ad(G)-invariant inner product (, ) on g, we can make an Ad(G)- invariant Hermitian inner product (, ) on g C in natural way. (ad(v)x, Y ) + (X, ad(v)y ) = 0 for X, Y g C, v t C, that is, ad(v) is a skew-hermitian transformation. Thus for any v t C, ad(v) is diagonalizable. Hence, t C is a Cartan subalgebra of g C. Definition 13. A Lie aubalgebra h is called a Cartan subalgebra of a complex Lie algebra g if (1) h is maximal abelian subalgebra, (2) ad(h) is diagonalizable for any H h SU(n) G = SU(n), T is the maximal torus defined by T = {diag [e iθ 1,, e iθ n ] θ θ n = 0} su(n) = {X gl(n, C) X + X = O, tr X = 0} A basis of g is given by 1 U jk = 1, U i jk = i, E l = i i. Then g C is spanned (over R) by

13 An introduction to the geometry of homogeneous spaces 133 Hence g C = sl(n, C) and U jk, U jk, E l, iu jk, iu jk, ie l. t = {diag[iθ 1,, iθ n ] θ k R, θ θ n = 0}, t C = {diag[λ 1,, λ n ] λ k C, λ λ n = 0}. For v = diag[λ 1,, λ n ] t C, if we put E jk = 1 then we have ad(v)e jk = [v, E jk ] = (λ j λ k )E jk. Let x j : t C C be a linear form defined by x j (diag[λ 1,, λ n ]) = λ j, then we have ad(v)e jk = [v, E jk ] = (x j x k )(v)e jk. Thus if we put α = x j x k (t C ), ad(v)x = α(v)x for X g C α = CE jk. The root system of g = su(n) relative to t is = {±(x j x k ) 1 j < k n}, and we have the decomposition g C = t C + α, g C α, where g C α = {X g C ad(v)x = α(v)x for v t C } is the root space corresponding to a root α. (dim C g C α = 1). In usual, we denote by h a Cartan subalgebra, so in our case, h = t C. and then is just it. We take a basis {v 1,, v l } of h R as h R := {v h α(v) R ( α )} v j := diag[0,, 1,, 0] it

14 134 Takashi Koda and put {λ 1,, λ l } its dual basis of h R, λ j = x j. We consider the lexicographic order in h R. The root system is a union of the positive root system and the negative root system. If we put = + α j = x j x j+1 (1 j n 1 = l) then they are the simple roots and α + may be written as a Z 0 -coefficients linear combination of simple roots. The restriction to h of the Killing form B of g C is nondegenerate, and we denote it by (, ). For λ h, we define v λ h by and we define an inner product on h by We put (v λ, v) = λ(v) for v h, (λ, µ) := (v λ, v µ ). (, ) R := (, ) hr, (, ) R := (, ) h R, which are positive definite. For convenience, we denote these simply by (, ). For α, β (α ±β), we consider the sequence β pα,, β α, β, β + α,, β + qα where β (p + 1)α and β + (q + 1)α. {β + mα p m q} is called the α-sequence containing β. We may see that In particular, if p = 0, i.e., β α, Therefore for simple roots α j, α k 2(β, α) (α, α) = p q Z. 2(β, α) (α, α) 0. 2(α j, α k ) (α j, α j ) 0.

15 An introduction to the geometry of homogeneous spaces 135 On the other hand where θ is the angle of α j and α k. 2(α j, α k ) 2(α j, α k ) (α j, α j ) (α k, α k ) = 4 cos2 θ, cos 2 θ = 0, 1 4, 1 2, 3 4 (or 1). Since π/2 θ < π, we have four cases. (1) θ = π 2, (2) θ = 2π 3, α j = α k (3) θ = 3π 4, α j : α k = 1 : 2 or 2 : 1 (4) θ = 5π 6, α j : α k = 1 : 3 or 3 : 1 Then we draw a diagram as follows: (1) we do not join α j and α k (2) join by single line (3) join by double line ==> (arrow from longer root to shorter one) (4) join by triple line > (arrow from longer root to shorter one) Thus the Dynkin diagram has vertices {α 1,, α l } and edges (1) (4). The highest root µ (that is, µ is a positive root such that µ α for any α ) can be written as µ = m 1 α m l α l in the Dynkin diagram, the integers m k are written on each α k. The highest root µ of su(n) is µ = x 1 x n = α α l, m 1 = = m l = 1 2 Geometry of homogeneous spaces 2.1 Homogeneous spaces Definition 14. A C manifold M is homogeneous if a Lie group G acts transitively on M, i.e., (1) There exists a C map : G M M, (a, x) ax such that (i) a(bx) = (ab)x (ii) ex = x for any x M (2) For any x, y M, there exists an element a G such that y = ax

16 136 Takashi Koda For an arbitraryly fixed point x 0 M, the closed subgroup H := {a G ax 0 = x 0 } is called an isotropy subgroup of G at x 0. Then M is diffeomorphic to the coset space G/H. M = G/H, x ah, where, a G is an element of G s.t. x = ax 0. Conversely, if H is a closed subgroup of a Lie group G, then the coset space G/H becomes a C manifold, and G acts on M by G G/H G/H, and the map τ g : G/H G/H, is called a left translation by g. (g, ah) (ga)h, ah (ga)h τ g π = π L g 2.2 G-invariant Riemannian metrics Let M = G/H be a homogeneous space of a Lie group G and a closed subgroup H. A Riemannian metric, on M is G-invariant if (τ g ) X, (τ g ) Y = X, Y for g G for any X, Y T M. Then τ g is an isometry of the Riemannian manifold (M,, ). Definition 15. A homogeneous space M = G/H is reductive if there exists a decomposition such that g = h + m (direct sum) Ad(H)m m, i.e., Ad(h)m m ( h H). If g has an Ad(H)-invariant inner product (, ), then the orthogonal complement m := h satisfies the above property. π : G M = G/H is the natural projection, o = eh G/H. π e : T e G T o M onto linear

17 An introduction to the geometry of homogeneous spaces 137 induces an isomorphism For any X gh, Y gh T gh M T o M = T e G/ ker π e = g/h = m X gh, Y gh gh = (τ g 1) gh X gh, (τ g 1) gh Y gh eh = ((τ g 1) gh X gh, (τ g 1) gh Y gh ). Let G/H be a reductive homogeneous space, g = h m (Ad(H)-invariant decomposition). For any h H π(i h (g)) = τ h (π(g)) Hence for any X m g π e (Ad(h)X) = (τ h ) e X (g G). (π e : m = T o M). Then a G-invariant Riemannian metric, on M = G/H induces an inner product (, ) on m. (X, Y ) = X, Y o Then (, ) is Ad(H)-invariant. (Ad(h)X, Ad(h)Y ) = (X, Y ), for h H, X, Y m. Hence, we have the one-to-one correspondence:, : G-inv. Riem. metric on G/H (, ) : Ad(H)-inv. inner product on m In the sequel, we use the same notation, for an Ad(H)-inv. inner product on m. Definition 16. A reductive homogeneous space M = G/H with a Riemannian metric, is naturally reductive if there exists an Ad(H)-invariant decomposition g = h + m (direct sum), Ad(H)m m such that [X, Y ] m, Z + Y, [X, Z] m = 0 for any X, Y, Z m, where [X, Y ] m denotes the m-component of [X, Y ].

18 138 Takashi Koda Remark. Accoring to the direct sum decomposition g = h m, [X, Y ] = [X, Y ] h + [X, Y ] m Definition 17. A homogeneous space M = G/H with a Riemannian metric, is normal homogeneous if there exists a bi-invariant Riemannian metric (, ) on G such that, if we identify m := h (with respect to (, )) with T o M, then, = (, ) m m. Remark. If M = G/H is normal homogeneous, M is naturally reductive. Example If the Kiling form B of a compact Lie group G is negative definite (then G is semisimple), using the inner product B on g, G/H is normal homogeneous. 2.3 Riemannian connection and the canonical connection Let G/H be a homogeneous space. For X g, γ(t) = τ exp tx (gh) is a curve in M through a point gh. The vector field X defined by XgH := γ (0) = d dt τ exp tx (gh) t=0 satisfies at the origin o = eh Xo = d dt π(exp tx) = π e (X). t=0 If G/H is reductive, the isomorphism π e : m = T o M may be expressed as m X Xo T o M. Proposition 2.1. For X, Y g, [X, Y ] = [X, Y ]. Let G/H be a reductive homogeneous space with a G-invariant Riemannian metric,. We denote by the Riemannian connection of (M/H,, ).

19 An introduction to the geometry of homogeneous spaces 139 Proposition 2.2. For any X g, the vector field X is a Killing vector field on G/H, i.e., X Y, Z = [X, Y ], Z + Y, [X, Z] for any vector fields Y, Z X(G/H). Since we have X Y, Z = X Y, Z + Y, X Z, Y X, Z + Y, Z X = 0. Proposition 2.3. Let (G/H,, ) be a naturally reductive homogeneous space. Then for v m = T o M. v X = [X, v] if X h 1 2 [X, v] m if X m Definition 18. Let (G/H,, ) be a naturally reductive homogeneous space. Then there exists a metric connection D with torsion tensor T and the curvature tensor B such that DT = 0, DB = 0. D is called the canonical connection. and The canonical connection D is given by for X, Y, Z m. { D v X [X, v] if X h = [X, v] m if X m D X Y = X Y + 1 T (X, Y ), 2 T (X, Y ) = [X, Y ] m, B(X, Y )Z = [[X, Y ] h, Z],

20 140 Takashi Koda 2.4 Curvature tensor Theorem 2.4. Let (G/H,, ) be a naturally reductive homogeneous space. Then the Riemannian curvature tensor R is given by (R(X, Y )Z ) o = [[X, Y ] h, Z] 1 2 [[X, Y ] m, Z] m for X, Y, Z m. Then we have 1 4 [[Y, Z] m, X] m 1 4 [[Z, X] m, Y ] m (R(X, Y )Y ) o = [[X, Y ] h, Y ] 1 4 [[X, Y ] m, Y ] m and the sectional curvature is given by R(X, Y )Y, X o = 1 4 [X, Y ] m, [X, Y ] m + [[X, Y ] h, X] m, Y. If (G/H,, ) is normal homogeneous R(X, Y )Y, X o = 1 4 [X, Y ] m, [X, Y ] m + [X, Y ] h, [X, Y ] h Symmetric spaces Definition 19. A connected Riemannian manifold (M, g) is called a Riemannian (globally) symmetric space if p M, s p I(M, g) s.t. s p 2 = id., p is an isolated fixed point of s p s p is the geodesic symmetry: where γ v (t) is the geodesic s.t. s p (γ v (t)) = γ v ( t), γ v (0) = p, γ v (0) = v T p M. The connected Lie group G = I 0 (M, g) (identity component) acts transitively on M, and M becomes a Riemannian homogeneous space G/H, where a closed subgroup H is an isotropy subgroup at p 0 M. The map σ : G G, a s p0 a s p0 is an involutive automorphism of G, and G σ 0 H G σ,

21 An introduction to the geometry of homogeneous spaces 141 where G σ = {a G σ(a) = a}, G σ 0 is the identity component of G σ. Such a pair (G, H) is called a symmetric pair. Examples M = G/H G H S n SO(n + 1) SO(n) sphere Gr k (R n ) O(n) O(k) O(n k) real Grassmann mfd. Gr k (C n ) U(n) U(k) U(n k) complex Grassmann mfd. The Lie algebra g may be decomposed into ±1-engenspaces of the differential dσ : g g. g = h m, dσ h = id, dσ m = id Then we see that [h, h] h, [h, m] m, [m, m] h and hence the canonical connection D coincides with the Riemannian connection, thus for X, Y, Z m. R = 0, R(X, Y )Z = [[X, Y ], Z] 2.6 The root system of a symmetric space Let M = G/H be a Riemannian symmetric space. Then as before, we have the Ad(H)-invariant decomposition g = h m. We take a maximal abelian subspace a m. For any H a, we consider the linear maps ad(h) : g g, which are simultaneously diagonalizable. If there exists X g such that ad(h)x = α(h)x for all H a for some nonzero linear form α a, α is called a root of M relative to a. The set Σ of all roots of M relative to a is called a restricted root system of M. Σ can be obtained as follows: take a Cartan subalgebra h C of g C such that h C a, let be the root system of g C relative to h C. Then Σ coinsides with {α a α } (as well as their multiplicities).

22 142 Takashi Koda 3 Homogeneous structures When does a Riemmanian manifold (M, g) become a Riemannian homogeneous manifold? W. Ambrose and I. M. Singer characterized it by the existense of some tensor field T of type (1, 2) on M, which is called a homogeneous structure. Theorem 3.1. ([1]) Let (M, g) be a connected, simply connected, complete Riemannian manifold. (M, g) is a Riemannian homogeneous space if and only if there exists a tensor field T of type (1, 2) on M satisfying (1) g(t (X)Y, Z) + g(y, T (X)Z) = 0 (2) X R = T (X) R (3) X T = T (X) T where and R denotes the Riemannian connection and the Riemannian curvature tensor of (M, g), respectively. := T is called a A S connection. (1) (3) are equivalent to g = 0, R = 0, T = 0. Remark. Here, we consider the tensor field T : X(M) X(M) X(M) of type (1, 2) as, for X X(M), T (X) : X(M) X(M), Y T (X)Y and (T (X) R)(Y, Z)W := T (X)(R(Y, Z)W ) R(T (X)Y, Z)W R(Y, T (X)Z)W R(Y, Z)T (X)W (T (X) T )(Y )Z := T (X)T (Y )Z T (T (X)Y )Z T (Y )T (X)Z About the homogeneity of almost Hermitian manifold (M, g, J), (J is an almost complex structure; J 2 = I), the following result was shown by K. Sekigawa. Theorem 3.2. ([6]) Let (M, g, J) be a connected, simply connected, complete almost Hermitian manifold. M is a homogeneous almost Hermitian manifold if and only if there exists a tensor field T of type (1, 2) on M satisfying (1) g(t (X)Y, Z) + g(y, T (X)Z) = 0 (2) X R = T (X) R (3) X T = T (X) T (4) X J = T (X) J

23 An introduction to the geometry of homogeneous spaces 143 An almost contact metric manifold (M, φ, ξ, η, g) is a C manifold M with an almost contact metric structure (φ, ξ, η, g) such that (1) φ is a tensor field of type (1, 1), (2) ξ X(M) is called a characteristic vector field, (3) η is a 1-form, (4) g is a Riemannian metric satisfying φ 2 X = X + η(x)ξ, η(ξ) = 1, g(x, ξ) = η(x), g(φx, φy ) = g(x, Y ) η(x)η(y ) About the homogeneity of (M, φ, ξ, η, g), the following result was shown. Theorem 3.3. ([5]) Let (M, φ, ξ, η, g) be a connected, simply connected, complete almost contact metric manifold. M is a homogeneous almost contact metric manifold if and only if there exists a skew-symmetric tensor field T of type (1, 2) on M satisfying (1) g(t (X)Y, Z) + g(y, T (X)Z) = 0 (2) X R = T (X) R (3) X T = T (X) T (4) X η = η T (X) (5) X φ = T (X) φ References [1] W. Ambrose and I. M. Singer, On homogeneous Riemannian manifolds, Duke Math. J., 25 (1958), [2] A. Arvanitoyeorgos, An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, Amer. Math. Sci., [3] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, [4] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Interscience, New York, 1963 and [5] T. Koda and Y. Watanabe, Homogeneous almost contact Riemannian manifolds and infinitesimal models, Bollettino U. M. I. (7) 11-B (1997), Suppl. fasc. 2, [6] K. Sekigawa, Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J., 7 (1978),

24 144 Takashi Koda [7] W. Ziller, The Jaacobi equation on naturally reductive compact Riemannian homogeneous spaces, Comment. Math. Helvetici, 52(1977),

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

More information

Symmetric Spaces Toolkit

Symmetric Spaces Toolkit Symmetric Spaces Toolkit SFB/TR12 Langeoog, Nov. 1st 7th 2007 H. Sebert, S. Mandt Contents 1 Lie Groups and Lie Algebras 2 1.1 Matrix Lie Groups........................ 2 1.2 Lie Group Homomorphisms...................

More information

These notes are incomplete they will be updated regularly.

These notes are incomplete they will be updated regularly. These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras 2 1.1. Definition

More information

THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

More information

1: Lie groups Matix groups, Lie algebras

1: Lie groups Matix groups, Lie algebras Lie Groups and Bundles 2014/15 BGSMath 1: Lie groups Matix groups, Lie algebras 1. Prove that O(n) is Lie group and that its tangent space at I O(n) is isomorphic to the space so(n) of skew-symmetric matrices

More information

Notation. For any Lie group G, we set G 0 to be the connected component of the identity.

Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence

More information

8. COMPACT LIE GROUPS AND REPRESENTATIONS

8. COMPACT LIE GROUPS AND REPRESENTATIONS 8. COMPACT LIE GROUPS AND REPRESENTATIONS. Abelian Lie groups.. Theorem. Assume G is a Lie group and g its Lie algebra. Then G 0 is abelian iff g is abelian... Proof.. Let 0 U g and e V G small (symmetric)

More information

SEMISIMPLE LIE GROUPS

SEMISIMPLE LIE GROUPS SEMISIMPLE LIE GROUPS BRIAN COLLIER 1. Outiline The goal is to talk about semisimple Lie groups, mainly noncompact real semisimple Lie groups. This is a very broad subject so we will do our best to be

More information

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive

More information

LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups

LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are

More information

1 v >, which will be G-invariant by construction.

1 v >, which will be G-invariant by construction. 1. Riemannian symmetric spaces Definition 1.1. A (globally, Riemannian) symmetric space is a Riemannian manifold (X, g) such that for all x X, there exists an isometry s x Iso(X, g) such that s x (x) =

More information

Killing Vector Fields of Constant Length on Riemannian Normal Homogeneous Spaces

Killing Vector Fields of Constant Length on Riemannian Normal Homogeneous Spaces Killing Vector Fields of Constant Length on Riemannian Normal Homogeneous Spaces Ming Xu & Joseph A. Wolf Abstract Killing vector fields of constant length correspond to isometries of constant displacement.

More information

Left-invariant Einstein metrics

Left-invariant Einstein metrics on Lie groups August 28, 2012 Differential Geometry seminar Department of Mathematics The Ohio State University these notes are posted at http://www.math.ohio-state.edu/ derdzinski.1/beamer/linv.pdf LEFT-INVARIANT

More information

Differential Geometry, Lie Groups, and Symmetric Spaces

Differential Geometry, Lie Groups, and Symmetric Spaces Differential Geometry, Lie Groups, and Symmetric Spaces Sigurdur Helgason Graduate Studies in Mathematics Volume 34 nsffvjl American Mathematical Society l Providence, Rhode Island PREFACE PREFACE TO THE

More information

Symmetries, Fields and Particles. Examples 1.

Symmetries, Fields and Particles. Examples 1. Symmetries, Fields and Particles. Examples 1. 1. O(n) consists of n n real matrices M satisfying M T M = I. Check that O(n) is a group. U(n) consists of n n complex matrices U satisfying U U = I. Check

More information

Problems in Linear Algebra and Representation Theory

Problems in Linear Algebra and Representation Theory Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific

More information

Linear Algebra. Workbook

Linear Algebra. Workbook Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx

More information

Notes on Lie Algebras

Notes on Lie Algebras NEW MEXICO TECH (October 23, 2010) DRAFT Notes on Lie Algebras Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA E-mail: iavramid@nmt.edu 1

More information

X-RAY TRANSFORM ON DAMEK-RICCI SPACES. (Communicated by Jan Boman)

X-RAY TRANSFORM ON DAMEK-RICCI SPACES. (Communicated by Jan Boman) Volume X, No. 0X, 00X, X XX Web site: http://www.aimsciences.org X-RAY TRANSFORM ON DAMEK-RICCI SPACES To Jan Boman on his seventy-fifth birthday. François Rouvière Laboratoire J.A. Dieudonné Université

More information

Spring 2018 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 3

Spring 2018 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 3 Spring 2018 CIS 610 Advanced Geometric Methods in Computer Science Jean Gallier Homework 3 March 20; Due April 5, 2018 Problem B1 (80). This problem is from Knapp, Lie Groups Beyond an Introduction, Introduction,

More information

THE ISOPARAMETRIC STORY

THE ISOPARAMETRIC STORY THE ISOPARAMETRIC STORY QUO-SHIN CHI This is a slightly revised and updated version of the notes for the summer mini-course on isoparametric hypersurfaces I gave at National Taiwan University, June 25-July

More information

REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES

REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES MAKIKO SUMI TANAKA 1. Introduction This article is based on the collaboration with Tadashi Nagano. In the first part of this article we briefly review basic

More information

Exercises Lie groups

Exercises Lie groups Exercises Lie groups E.P. van den Ban Spring 2009 Exercise 1. Let G be a group, equipped with the structure of a C -manifold. Let µ : G G G, (x, y) xy be the multiplication map. We assume that µ is smooth,

More information

Topics in Representation Theory: Roots and Complex Structures

Topics in Representation Theory: Roots and Complex Structures Topics in Representation Theory: Roots and Complex Structures 1 More About Roots To recap our story so far: we began by identifying an important abelian subgroup of G, the maximal torus T. By restriction

More information

ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES. 1. Introduction

ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES. 1. Introduction ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES ANDREAS ČAP AND KARIN MELNICK Abstract. We use the general theory developed in our article [1] in the setting of parabolic

More information

Metrics, Connections, and Curvature on Lie Groups

Metrics, Connections, and Curvature on Lie Groups Chapter 17 Metrics, Connections, and Curvature on Lie Groups 17.1 Left (resp. Right) Invariant Metrics Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian

More information

REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

More information

Group Theory - QMII 2017

Group Theory - QMII 2017 Group Theory - QMII 017 Reminder Last time we said that a group element of a matrix lie group can be written as an exponent: U = e iαaxa, a = 1,..., N. We called X a the generators, we have N of them,

More information

The Cartan Decomposition of a Complex Semisimple Lie Algebra

The Cartan Decomposition of a Complex Semisimple Lie Algebra The Cartan Decomposition of a Complex Semisimple Lie Algebra Shawn Baland University of Colorado, Boulder November 29, 2007 Definition Let k be a field. A k-algebra is a k-vector space A equipped with

More information

Clifford Algebras and Spin Groups

Clifford Algebras and Spin Groups Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of

More information

Austere pseudo-riemannian submanifolds and s-representations

Austere pseudo-riemannian submanifolds and s-representations Austere pseudo-riemannian submanifolds and s-representations Tokyo University of Science, D3 Submanifold Geometry and Lie Theoretic Methods (30 October, 2008) The notion of an austere submanifold in a

More information

1 Smooth manifolds and Lie groups

1 Smooth manifolds and Lie groups An undergraduate approach to Lie theory Slide 1 Andrew Baker, Glasgow Glasgow, 12/11/1999 1 Smooth manifolds and Lie groups A continuous g : V 1 V 2 with V k R m k open is called smooth if it is infinitely

More information

CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =

CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.

More information

2-STEP NILPOTENT LIE GROUPS ARISING FROM SEMISIMPLE MODULES

2-STEP NILPOTENT LIE GROUPS ARISING FROM SEMISIMPLE MODULES 2-STEP NILPOTENT LIE GROUPS ARISING FROM SEMISIMPLE MODULES PATRICK EBERLEIN UNIVERSITY OF NORTH CAROLINA AT CHAPEL HILL Abstract Let G 0 denote a compact semisimple Lie algebra and U a finite dimensional

More information

A 2 G 2 A 1 A 1. (3) A double edge pointing from α i to α j if α i, α j are not perpendicular and α i 2 = 2 α j 2

A 2 G 2 A 1 A 1. (3) A double edge pointing from α i to α j if α i, α j are not perpendicular and α i 2 = 2 α j 2 46 MATH 223A NOTES 2011 LIE ALGEBRAS 11. Classification of semisimple Lie algebras I will explain how the Cartan matrix and Dynkin diagrams describe root systems. Then I will go through the classification

More information

Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs UNIVERSITY OF TOKYO

Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs UNIVERSITY OF TOKYO UTMS 2011 8 April 22, 2011 Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs by Toshiyuki Kobayashi and Yoshiki Oshima T UNIVERSITY OF TOKYO GRADUATE SCHOOL OF

More information

Part III Symmetries, Fields and Particles

Part III Symmetries, Fields and Particles Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often

More information

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

A FUNDAMENTAL THEOREM OF INVARIANT METRICS ON A HOMOGENEOUS SPACE. 1. Introduction

A FUNDAMENTAL THEOREM OF INVARIANT METRICS ON A HOMOGENEOUS SPACE. 1. Introduction A FUNDAMENTAL THEOREM OF INVARIANT METRICS ON A HOMOGENEOUS SPACE ADAM BOWERS 1. Introduction A homogeneous space is a differentiable manifold M upon which a Lie group acts transitively. At the foundation

More information

THE THEOREM OF THE HIGHEST WEIGHT

THE THEOREM OF THE HIGHEST WEIGHT THE THEOREM OF THE HIGHEST WEIGHT ANKE D. POHL Abstract. Incomplete notes of the talk in the IRTG Student Seminar 07.06.06. This is a draft version and thought for internal use only. The Theorem of the

More information

1 Hermitian symmetric spaces: examples and basic properties

1 Hermitian symmetric spaces: examples and basic properties Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................

More information

Introduction to Lie Groups

Introduction to Lie Groups Introduction to Lie Groups MAT 4144/5158 Winter 2015 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike

More information

Representation Theory

Representation Theory Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim

More information

Cartan s Criteria. Math 649, Dan Barbasch. February 26

Cartan s Criteria. Math 649, Dan Barbasch. February 26 Cartan s Criteria Math 649, 2013 Dan Barbasch February 26 Cartan s Criteria REFERENCES: Humphreys, I.2 and I.3. Definition The Cartan-Killing form of a Lie algebra is the bilinear form B(x, y) := Tr(ad

More information

Topics in Representation Theory: SU(n), Weyl Chambers and the Diagram of a Group

Topics in Representation Theory: SU(n), Weyl Chambers and the Diagram of a Group Topics in Representation Theory: SU(n), Weyl hambers and the Diagram of a Group 1 Another Example: G = SU(n) Last time we began analyzing how the maximal torus T of G acts on the adjoint representation,

More information

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1 Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

More information

1 Linear Algebra Problems

1 Linear Algebra Problems Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and

More information

AN INTRODUCTION TO FLAG MANIFOLDS Notes 1 for the Summer School on Combinatorial Models in Geometry and Topology of Flag Manifolds, Regina 2007

AN INTRODUCTION TO FLAG MANIFOLDS Notes 1 for the Summer School on Combinatorial Models in Geometry and Topology of Flag Manifolds, Regina 2007 AN INTRODUCTION TO FLAG MANIFOLDS Notes 1 for the Summer School on Combinatorial Models in Geometry and Topology of Flag Manifolds, Regina 2007 1. The manifold of flags The (complex) full flag manifold

More information

LIE ALGEBRAS: LECTURE 7 11 May 2010

LIE ALGEBRAS: LECTURE 7 11 May 2010 LIE ALGEBRAS: LECTURE 7 11 May 2010 CRYSTAL HOYT 1. sl n (F) is simple Let F be an algebraically closed field with characteristic zero. Let V be a finite dimensional vector space. Recall, that f End(V

More information

DIFFERENTIAL GEOMETRY HW 12

DIFFERENTIAL GEOMETRY HW 12 DIFFERENTIAL GEOMETRY HW 1 CLAY SHONKWILER 3 Find the Lie algebra so(n) of the special orthogonal group SO(n), and the explicit formula for the Lie bracket there. Proof. Since SO(n) is a subgroup of GL(n),

More information

Notes 9: Eli Cartan s theorem on maximal tori.

Notes 9: Eli Cartan s theorem on maximal tori. Notes 9: Eli Cartan s theorem on maximal tori. Version 1.00 still with misprints, hopefully fewer Tori Let T be a torus of dimension n. This means that there is an isomorphism T S 1 S 1... S 1. Recall

More information

On classification of minimal orbits of the Hermann action satisfying Koike s conditions (Joint work with Minoru Yoshida)

On classification of minimal orbits of the Hermann action satisfying Koike s conditions (Joint work with Minoru Yoshida) Proceedings of The 21st International Workshop on Hermitian Symmetric Spaces and Submanifolds 21(2017) 1-2 On classification of minimal orbits of the Hermann action satisfying Koike s conditions (Joint

More information

SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS

SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the

More information

Homogeneous Lorentzian structures on the generalized Heisenberg group

Homogeneous Lorentzian structures on the generalized Heisenberg group Homogeneous Lorentzian structures on the generalized Heisenberg group W. Batat and S. Rahmani Abstract. In [8], all the homogeneous Riemannian structures corresponding to the left-invariant Riemannian

More information

On the Harish-Chandra Embedding

On the Harish-Chandra Embedding On the Harish-Chandra Embedding The purpose of this note is to link the Cartan and the root decompositions. In addition, it explains how we can view a Hermitian symmetric domain as G C /P where P is a

More information

Lie Groups. Andreas Čap

Lie Groups. Andreas Čap Lie Groups Fall Term 2018/19 Andreas Čap Institut für Mathematik, Universität Wien, Oskar Morgenstern Platz 1, A 1090 Wien E-mail address: Andreas.Cap@univie.ac.at Preface Contents Chapter 1. Lie groups

More information

Weyl Group Representations and Unitarity of Spherical Representations.

Weyl Group Representations and Unitarity of Spherical Representations. Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν

More information

Lecture Notes LIE GROUPS. Richard L. Bishop. University of Illinois at Urbana-Champaign

Lecture Notes LIE GROUPS. Richard L. Bishop. University of Illinois at Urbana-Champaign Lecture Notes on LIE GROUPS by Richard L. Bishop University of Illinois at Urbana-Champaign LECTURE NOTES ON LIE GROUPS RICHARD L. BISHOP Contents 1. Introduction. Definition of a Lie Group The course

More information

Exercises Lie groups

Exercises Lie groups Exercises Lie groups E.P. van den Ban Spring 2012 Exercise 1. Let G be a group, equipped with the structure of a C -manifold. Let µ : G G G, (x, y) xy be the multiplication map. We assume that µ is smooth,

More information

Lectures on Lie groups and geometry

Lectures on Lie groups and geometry Lectures on Lie groups and geometry S. K. Donaldson March 25, 2011 Abstract These are the notes of the course given in Autumn 2007 and Spring 2011. Two good books (among many): Adams: Lectures on Lie groups

More information

HOMOGENEOUS VECTOR BUNDLES

HOMOGENEOUS VECTOR BUNDLES HOMOGENEOUS VECTOR BUNDLES DENNIS M. SNOW Contents 1. Root Space Decompositions 2 2. Weights 5 3. Weyl Group 8 4. Irreducible Representations 11 5. Basic Concepts of Vector Bundles 13 6. Line Bundles 17

More information

AN UPPER BOUND FOR SIGNATURES OF IRREDUCIBLE, SELF-DUAL gl(n, C)-REPRESENTATIONS

AN UPPER BOUND FOR SIGNATURES OF IRREDUCIBLE, SELF-DUAL gl(n, C)-REPRESENTATIONS AN UPPER BOUND FOR SIGNATURES OF IRREDUCIBLE, SELF-DUAL gl(n, C)-REPRESENTATIONS CHRISTOPHER XU UROP+ Summer 2018 Mentor: Daniil Kalinov Project suggested by David Vogan Abstract. For every irreducible,

More information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

More information

arxiv: v1 [math.dg] 19 Nov 2009

arxiv: v1 [math.dg] 19 Nov 2009 GLOBAL BEHAVIOR OF THE RICCI FLOW ON HOMOGENEOUS MANIFOLDS WITH TWO ISOTROPY SUMMANDS. arxiv:09.378v [math.dg] 9 Nov 009 LINO GRAMA AND RICARDO MIRANDA MARTINS Abstract. In this paper we study the global

More information

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)

More information

HOMOGENEOUS EINSTEIN METRICS

HOMOGENEOUS EINSTEIN METRICS HOMOGENEOUS EINSTEIN METRICS Megan M. Kerr A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor

More information

About Lie Groups. timothy e. goldberg. October 6, Lie Groups Beginning Details The Exponential Map and Useful Curves...

About Lie Groups. timothy e. goldberg. October 6, Lie Groups Beginning Details The Exponential Map and Useful Curves... About Lie Groups timothy e. goldberg October 6, 2005 Contents 1 Lie Groups 1 1.1 Beginning Details................................. 1 1.2 The Exponential Map and Useful Curves.................... 3 2 The

More information

Simple Lie algebras. Classification and representations. Roots and weights

Simple Lie algebras. Classification and representations. Roots and weights Chapter 3 Simple Lie algebras. Classification and representations. Roots and weights 3.1 Cartan subalgebra. Roots. Canonical form of the algebra We consider a semi-simple (i.e. with no abelian ideal) Lie

More information

Published as: Math. Ann. 295 (1993)

Published as: Math. Ann. 295 (1993) TWISTOR SPACES FOR RIEMANNIAN SYMMETRIC SPACES Francis Burstall, Simone Gutt and John Rawnsley Published as: Math. Ann. 295 (1993) 729 743 Abstract. We determine the structure of the zero-set of the Nijenhuis

More information

Many of the exercises are taken from the books referred at the end of the document.

Many of the exercises are taken from the books referred at the end of the document. Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the

More information

Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky

Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey

More information

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

More information

arxiv:math-ph/ v1 31 May 2000

arxiv:math-ph/ v1 31 May 2000 An Elementary Introduction to Groups and Representations arxiv:math-ph/0005032v1 31 May 2000 Author address: Brian C. Hall University of Notre Dame, Department of Mathematics, Notre Dame IN 46556 USA E-mail

More information

Matrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course

Matrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course Matrix Lie groups and their Lie algebras Mahmood Alaghmandan A project in fulfillment of the requirement for the Lie algebra course Department of Mathematics and Statistics University of Saskatchewan March

More information

Lie algebras. Alexei Skorobogatov. March 20, Basic definitions and examples 2. 2 Theorems of Engel and Lie 4

Lie algebras. Alexei Skorobogatov. March 20, Basic definitions and examples 2. 2 Theorems of Engel and Lie 4 Lie algebras Alexei Skorobogatov March 20, 2017 Introduction For this course you need a very good understanding of linear algebra; a good knowledge of group theory and the representation theory of finite

More information

Lie Groups, Lie Algebras and the Exponential Map

Lie Groups, Lie Algebras and the Exponential Map Chapter 7 Lie Groups, Lie Algebras and the Exponential Map 7.1 Lie Groups and Lie Algebras In Gallier [?], Chapter 14, we defined the notion of a Lie group as a certain type of manifold embedded in R N,

More information

Killing fields of constant length on homogeneous Riemannian manifolds

Killing fields of constant length on homogeneous Riemannian manifolds Killing fields of constant length on homogeneous Riemannian manifolds Southern Mathematical Institute VSC RAS Poland, Bedlewo, 21 October 2015 1 Introduction 2 3 4 Introduction Killing vector fields (simply

More information

Introduction to Lie Groups and Lie Algebras. Alexander Kirillov, Jr.

Introduction to Lie Groups and Lie Algebras. Alexander Kirillov, Jr. Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr. Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA E-mail address: kirillov@math.sunysb.edu URL: http://www.math.sunysb.edu/~kirillov/liegroups/

More information

Symmetric Spaces. Andrew Fiori. Sept McGill University

Symmetric Spaces. Andrew Fiori. Sept McGill University McGill University Sept 2010 What are Hermitian? A Riemannian manifold M is called a Riemannian symmetric space if for each point x M there exists an involution s x which is an isometry of M and a neighbourhood

More information

Reduction of Homogeneous Riemannian structures

Reduction of Homogeneous Riemannian structures Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad

More information

Metrics and Holonomy

Metrics and Holonomy Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it

More information

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem

On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013

More information

SYMMETRIC SPACES. PETER HOLMELIN Master s thesis 2005:E3. Centre for Mathematical Sciences Mathematics

SYMMETRIC SPACES. PETER HOLMELIN Master s thesis 2005:E3. Centre for Mathematical Sciences Mathematics SYMMETRIC SPACES PETER HOLMELIN Master s thesis 2005:E3 Centre for Mathematical Sciences Mathematics CENTRUM SCIENTIARUM MATHEMATICARUM Abstract In this text we study the differential geometry of symmetric

More information

Stratification of 3 3 Matrices

Stratification of 3 3 Matrices Stratification of 3 3 Matrices Meesue Yoo & Clay Shonkwiler March 2, 2006 1 Warmup with 2 2 Matrices { Best matrices of rank 2} = O(2) S 3 ( 2) { Best matrices of rank 1} S 3 (1) 1.1 Viewing O(2) S 3 (

More information

The local geometry of compact homogeneous Lorentz spaces

The local geometry of compact homogeneous Lorentz spaces The local geometry of compact homogeneous Lorentz spaces Felix Günther Abstract In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which

More information

Elementary linear algebra

Elementary linear algebra Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

More information

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

More information

CLASSICAL GROUPS DAVID VOGAN

CLASSICAL GROUPS DAVID VOGAN CLASSICAL GROUPS DAVID VOGAN 1. Orthogonal groups These notes are about classical groups. That term is used in various ways by various people; I ll try to say a little about that as I go along. Basically

More information

Topics in Representation Theory: Roots and Weights

Topics in Representation Theory: Roots and Weights Topics in Representation Theory: Roots and Weights 1 The Representation Ring Last time we defined the maximal torus T and Weyl group W (G, T ) for a compact, connected Lie group G and explained that our

More information

Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification

More information

A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS

A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS by J. Szenthe Abstract. In case of Riemannian manifolds isometric actions admitting submanifolds

More information

CHAPTER 6. Representations of compact groups

CHAPTER 6. Representations of compact groups CHAPTER 6 Representations of compact groups Throughout this chapter, denotes a compact group. 6.1. Examples of compact groups A standard theorem in elementary analysis says that a subset of C m (m a positive

More information

Bott Periodicity. Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel

Bott Periodicity. Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel Bott Periodicity Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel Acknowledgements This paper is being written as a Senior honors thesis. I m indebted

More information

(6) For any finite dimensional real inner product space V there is an involutive automorphism α : C (V) C (V) such that α(v) = v for all v V.

(6) For any finite dimensional real inner product space V there is an involutive automorphism α : C (V) C (V) such that α(v) = v for all v V. 1 Clifford algebras and Lie triple systems Section 1 Preliminaries Good general references for this section are [LM, chapter 1] and [FH, pp. 299-315]. We are especially indebted to D. Shapiro [S2] who

More information

Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups

Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Dennis I. Barrett Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University Grahamstown,

More information

The following definition is fundamental.

The following definition is fundamental. 1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic

More information

1.4 Solvable Lie algebras

1.4 Solvable Lie algebras 1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)

More information

Master Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed

Master Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like

More information

MATH 583A REVIEW SESSION #1

MATH 583A REVIEW SESSION #1 MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),

More information