Lie Algebras Representations- Bibliography

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1 Lie Algebras Representations- Bibliography J. E. Humphreys Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer 1980 Alex. Kirillov An Introduction to Lie Groups and Lie Algebras, Cambridge Univ. Press 2008 R. W. CARTER Lie Algebras of Finite and Affine Type, Cambridge Univ. Press 2005 K. Erdmann and M. J. Wildon Introduction to Lie Algebras, Spinger 2006 Hans Samelson, Notes on Lie Algebras B. C. Hall Lie Groups, Lie Algebras and Representations, Grad. Texts in Maths. Springer 2003 Andreas Čap, Lie Algebras and Representation Theory, Univ. Wien, Lecture Notes 2003 Alberto Elduque, Lie algebras, Univ. Zaragoza, Lecture Notes 2005

2 F = Field of characteristic 0 (ex R, C), α, β,... F and x, y,... A Definition of an Algebra A A is a F-vector space with an additional distributive binary operation or product A A (x, y) m m(x, y) = x y A F-vector space αx = xα, (α + β)x = αx + βx, α(x + y) = αx + αy distributivity (x + y) z = x z + y z, x (y + z) = x y + x z α(x y) = (αx) y = x (αy) = (x y)α associative algebra (x y) z = x (y z), NON associative algebra (x y) z x (y z) commutative algebra x y = y x, NON commutative algebra x y y x

3 Definitions F = Field of characteristic 0 (ex R, C), g = F-Algebra with a product or bracket or commutator α, β,... F and x, y,... g g g (x, y) [x, y] g Definition :Lie algebra Axioms [αx + βy, z] = α [x, z] + β [y, z] (L-i) bi-linearity: [x, αy + βz] = α [x, y] + β [x, z] (L-ii) anti-commutativity: [x, y] = [y, x] [x, x] = 0 (L-iii) Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 or Leibnitz rule: [x, [y, z]] = [[x, y], z] + [y, [x, z]]

4 Examples of Lie algebras (i) g = { x, y,...} real vector space R 3 [ x, y] x y (ii) A associative F- algebra A a Lie algebra with commutator [A, B] = AB BA (iii) V = F-vector space, dim V = n < b bilinear form on V b : V V (v, w) b (u, v) F b(αu+βv, w) = αb(u, w)+βb(v, w), b(u, αv+βw) = αb(u, v)+βb(u, w) o (V, b) = Orthogonal Lie algebra the set of all T End V b(u, Tv) + b(tu, v) = 0 [T 1, T 2 ] = T 1 T 2 T 2 T 1 T 1 and T 2 o (V, b) [T 1, T 2 ] o (V, b)

5 (iv) Angular Momentum in Quantum Mechanics operators on L 1, L 2, L 3 analytic (holomorphic) f (x, y, z) functions L 1 = y z z y, L 2 = z x x z, g = span {L 1, L 2, L 3 } L 3 = x y y x [L i, L j ] L i L j L j L i [L 1, L 2 ] = L 3, [L 2, L 3 ] = L 1, [L 3, L 1 ] = L 2

6 (v) A Poisson algebra is a vector space over a field F equipped with two bilinear products, and {, }, having the following properties: (a) The product forms an associative (commutative) F-algebra. (b) The product {, }, called the Poisson bracket, forms a Lie algebra, and so it is anti-symmetric, and obeys the Jacobi identity. (c) The Poisson bracket acts as a derivation of the associative product, so that for any three elements x, y and z in the algebra, one has {x, y z} = {x, y} z + y {x, z} Example: Poisson algebra on a Poisson manifold M is a manifold, C (M) the smooth /analytic (complex) functions on the manifold. ω ij (x) = ω ji (x), {f, g} = ω ij (x) f j x i x g ij ( ) ω ij ω jk ω ki ω km + ω im + ω jm = 0 x m x m x m m

7 Structure Constants g = span (e 1, e 2,..., e n ) = Fe 1 + Fe Fe n n [e i, e j ] = cij k e k cij k e k, cij k structure constants k=1 (L-i) bi-linearity (L-ii) anti-commutativity: cij k = cji k (L-iii) Jacobi identity: ci m j cl m k + cm j k cl m i + cm k i cl m j = 0 summation on color indices

8 gl(v ) algebra V = F-vector space, dim V = n < general linear algebra gl(v ) gl(v ) End(V ) = endomorphisms on V dim gl(v ) = n 2 gl(v ) is a Lie algebra with commutator [A, B] = AB BA gl(c n ) gl(n, C) M n (C) complex n n matrices Standard basis Standard basis for gl(n, C) = matrices E ij (1 in the (i, j) position, 0 elsewhere) [E ij, E kl ] = δ jk E il δ il E kj

9 General Linear Group GL(n, F) =invertible n n matrices M GL(n, F) det M 0 GL(n, C) is an analytic manifold of dim n 2 AND a group. The group multiplication is a continuous application G G G The group inversion is a continuous application G G Definition of the Lie algebra from the Lie group g(n, C) = T I (GL(n, C)) X (t) GL(n, C) X (t) smooth trajectory = { X (0) = x gl(n, C) } X (0) = I Definition of a Lie group from a Lie algebra { } {x gl(n, C)} = X (t) = e xt t n = n! x n GL(n, F) n=0

10 Jordan decomposition A gl(v ) A = A s + A n, [A s, A n ] = A s A n A n A s = 0 A s diagonal matrix (or semisimple) A n nilpotent matrix (A n ) m = 0, the decomposition is unique x = x s + x n e x = k=0 x k k! e x = e (xs+xn) = e xs e xn λ x s diagonal matrix = 0 λ λ n e λ e x s diagonal matrix = 0 e λ e λn m 1 e xn = k=0 x k n k!

11 Lie algebra of the (Lie) group of isometries V F-vector space, b : V V C bilinear form O(V, b)=group of isometries on V i.e. O(V, b) M : V V b(mu, Mv) = b(u, v) lin } M(t) smooth trajectory O(V, b) M(0) = I M (0) = T o(v, b) = Lie algebra = b(tu, v) + b(u, Tv) = 0 = T tr b + b T = 0 group of isometries O(V, b) Lie algebra o(v, b) = T I (O(V, b)) The Lie algebra o(v, b) is the tangent space of the manifold O(V, b) at the unity I

12 Matrix Formulation for bilinear forms a 1 a 2 V a a =., n b 2 a, b = a i b i = (a 1, a 2,..., a n ) i=1. a n b n M 11 M 12 M 1n M 21 M 22 M 2n End(V ) M M = Ma, b = a, M tr b M n1 M n2 M nn b 1 bilinear form b B matrix n n b (x, y) = x, By M O(V, b) b (Mx, My) + b (x, y) M tr BM = B M tr bm = b T o(v, b) b (Tx, y) + b (x, Ty) = 0 T tr B + BT = 0 n n T tr b + bt = 0

13 Classical Lie Algebras A l or sl(l + 1, C) or special linear algebra Def: (l + 1) (l + 1) complex matrices T with Tr T = 0 Dimension = dim(a l ) = (l + 1) 2 1 = l(l + 2) Standard Basis: E ij, i j, i, j = 1, 2..., l + 1. H i = E ii E i+1 i+1

14 C l or sp(2l, C) or symplectic algebra Def: (2l) (2l) complex matrices T with Tr T = 0 leaving infinitesimally invariant the bilinear form ( ) 0l I b l I l 0 l U, V C 2l } b(u, TV ) + b(tu, V ) = 0 T sp(2l, C) ( ) { M N N T = P M tr tr = N, P tr = P bt + T tr b = 0 btb 1 = T tr

15 B l or o(2l + 1, C) or (odd) orthogonal algebra Def: (2l + 1) (2l + 1) complex matrices T with Tr T = 0 leaving infinitesimally invariant the bilinear form b 0 tr 0 l I l 0 tr I l 0 l 0 = (0, 0,..., 0) U, V C 2l T o(2l + 1, C) T = 0 b c c tr M N b tr P M tr } b(u, TV ) + b(tu, V ) = 0 { N tr = N, P tr = P

16 D l or o(2l, C) or (even) orthogonal algebra Def: (2l) (2l) complex matrices T with Tr T = 0 leaving infinitesimally invariant the bilinear form ( ) 0l I b l I l U, V C 2l } b(u, TV ) + b(tu, V ) = 0 T o(2l, C) ( ) { M N N T = P M tr tr = N, P tr = P 0 l

17 Subalgebras Definition (Lie subalgebra) h Lie subalgebra g (i) h vector subspace of g (ii) [h, h] h Examples of gl(n, C) subalgebras Diagonal matrices d(n, C)= matrices with diagonal elements only [d(n, C), d(n, C)] = 0 n d(n, C) Upper triangular matrices t(n, C), Strictly Upper triangular matrices n(n, C), n(n, C) t(n, C) [t(n, C), t(n, C)] n(n, C), [n(n, C), n(n, C)] n(n, C) [n(n, C), t(n, C)] n(n, C) ex. sl(2, C) Lie subalgebra of sl(3, C)

18 sl(2, C) Algebra sl(2, C) = span (h, x, y) [h, x] = 2x [h, y] = 2y [x, y] = h ex. 2 2 matrices with trace 0 [ ] 1 0 h = x = 0 1 [ ] y = [ ] ex. first derivative operators acting on polynomials of order n h = z z n 2, x = z2 z nz, y = z

19 sl(3, C) Algebra sl(3, C) = span (h 1, h 2, x 1, y 1, x 2, y 2, x 3, y 3 ) ex. 3 3 matrices with trace 0 h 1 = h 2 = [h 1, h 2 ] = 0 [h 1, x 1 ] = 2x 1, [h 1, y 1 ] = 2y 1, [x 1, y 1 ] = h 1, [h 1, x 2 ] = x 2, [h 1, y 2 ] = y 2, [h 1, x 3 ] = x 3, [h 1, y 3 ] = y 3, [h 2, x 2 ] = 2x 2, [h 2, y 2 ] = 2y 2, [x 2, y 2 ] = h 2, [h 2, x 1 ] = x 1, [h 2, y 1 ] = y 1, [h 2, x 3 ] = x 3, [h 2, y 3 ] = y 3, [x 1, x 2 ] = x 3, [x 1, x 3 ] = 0, [x 2, x 3 ] = 0 [y 1, y 2 ] = y 3, [y 1, y 3 ] = 0, [y 2, y 3 ] = 0 [x 1, y 2 ] = 0, [x 1, y 3 ] = y 2, [x 2, y 1 ] = 0, [x 2, y 3 ] = y 1, [x 3, y 1 ] = x 2, [x 3, y 2 ] = x 1, [x 3, y 3 ] = h 1 + h 2 x 1 = y 1 = x 2 = y 2 = x 3 = y 3 =

20 Ideals Definition (Ideal) Ideal I is a Lie subalgebra such that [I, g] I Center Z(g) = {z g : [z, g] = 0} Prop: The center is an ideal. Prop: The derived algebra Dg = [g, g] is an ideal. Prop: If I, J are ideals I + J, [I, J ] and I J are ideals. Theorem I ideal of g g/i {x = x + I : x g} is a Lie algebra [x, y] [x, y] = [x, y] + I

21 Simple Ideals Def: g is simple g has only trivial ideals and [g, g] {0} (trivial ideals of g are the ideals {0} and g) Def: g is abelian [g, g] = {0} Prop: Prop: Prop: g/ [g, g] is abelian g is simple Lie algebra Z(g) = 0 and g = [g, g]. The Classical Lie Algebras A l, B l, C l, D l are simple Lie Algebras

22 Direct Sum of Lie Algebras g 1, g 1 Lie algebras direct sum g 1 g 2 = g 1 g 2 with the following structure: α(x 1, x 2 ) + β(y 1, y 2 ) (αx 1 + βy 1, αx 2 + βy 2 ) g 1 g 2 vector space commutator definition: [(x 1, x 2 ), (y 1, y 2 )] ([x 1, y 1 ], [x 2, y 2 ]) Prop: g 1 g 2 is a Lie algebra } (g 1, 0) g { 1 iso (g1, 0) (0, g 2 ) = {(0, 0)} g 1 g2 = 0 (0, g 2 ) g 2 [(g 1, 0), (0, g 2 )] = {(0, 0)} [g 1, g 2 ] = 0 iso Prop: g 1 and g 2 are ideals of g 1 g 2 Prop: a, b ideals of g a b = {0} a + b = g { a b iso } { } a + b a b = g

23 Lie homomorphisms homomorphism: g { φ (i) φ linear g (ii) φ ([x, y]) = [φ(x), φ(y)] monomorphism: Ker φ = {0}, epimorphism: Im φ = g isomorphism: mono+ epi, automorphism: iso+ {g = g } Prop: Ker φ is an ideal of g Prop: Im φ = φ(g) is Lie subalgebra of g Theorem I ideal of g canonical map g π(x) = x = x + I π g/i { canonical map is a Lie epimorphism }

24 Linear homomorphism theorem Theorem V, U, W linear spaces f : V U and g : V W linear maps { f f epi V U Ker f Ker g g ψ W } {! ψ : g = ψ f Ker ψ = f (Ker g) } Corollary f f V U V U g g ψ ψ 1 W W { } { f epi, g epi, Ker f = Ker g! ψ iso : g = ψ f, U iso W }

25 Lie homomorphism theorem Theorem g, h, l Lie spaces, φ : g epi h and ρ : g l Lie homomorphisms φ g h ρ ψ l { φ Lie-epi, Ker φ Ker ρ } {! ψ : Lie-homo ρ = ψ φ } Corollary φ φ g h g h ρ ρ ψ ψ 1 l l { } { } φ Lie-epi, ρ Lie-epi, Ker φ = Ker ρ! ψ : Lie-iso ρ = ψ φ, h iso l

26 First isomorphism theorem Theorem (First isomorphism theorem) φ : g g Lie-homomorphism, π g g/ker φ φ φ φ(g) =! φ : g/ker φ φ(g) g Lie-iso φ(g) iso g/ker φ

27 Noether Theorems Theorem K, L ideals of g K L (i) (ii) L/K ideal of g/k (g/k) / (L/K) iso (g/l) Theorem ( Noether Theorem (2nd isomorphism theor.)) { } { } K, L (K + L) /L K/ (K L) ideals of g iso Parallelogram Low: K + L K L K L

28 Derivations A an F algebra (not necessary associative) with product A A (a, b) a b A bilinear mapping, is a derivation of A if is a linear map A A satisfying the Leibnitz property: Der (A) = all derivations on A Prop: (A B) = A ( (B)) + ( (A)) B Der (A) is a Lie Algebra gl(a) with commutator [, ] (A) ( (A)) ( (A))

29 Derivations on a Lie algebra g a F Lie algebra, is a Lie derivation of g if is a linear map: g x (x) g satisfying the Leibnitz property : ([x, y]) = [ (x), y] + [x, (y)] Der (g) = all derivations on g Prop: Der (g) is a Lie Algebra gl(g) with commutator [, ] (x) ( (x)) ( (x)) Prop: δ Der(g) δ n ([x, y]) = n ( n [ k) δ k (x), δ n k (y) ] Prop: Proposition k=0 δ Der(g) e δ ([x, y]) = [ e δ (x), e δ (y) ] e δ Aut(g) φ(t) smooth monoparametric familly in Aut(g) and φ(0) = Id φ (0) Der(g)

30 Adjoint Representation Definition (Adjoint Representation) Let x g and ad x : g g : ad x (y) = [x, y] Prop: ad [x, y] = [ad x, ad y ] Prop: ad g {ad x } Lie-subalgebra of Der (g) x g Definition: Inner Derivations=ad g Definition Outer Derivations= Der (g) \ad g { } x g Prop: δ Der(g) { [ad x, δ] = ad δ(x) } ad g ideal of Der(g) Prop: τ Aut (g) τ ([x, y]) = [τ(x), τ(y)] ad τ(x) = τ ad x τ 1

31 Matrix Form of the adjoint representation g = span (e 1, e 2,..., e n ) = Ce 1 + Ce Ce n n [e i, e j ] = cij k e k, cij k structure constants k=1 Antisymmetry: c k ij = c k ji Jacobi identity c m i j cl m k + cm j k cl m i + cm k i cl m j = 0 e l = e l g e l.e p = e l.e p = δ l p P i gl(g) : e k.p i e j = (P i ) k j = c k ij n [P i, P j ] = cij k P k k=1 P i = ad ei

32 Representations, Modules V F-vector space, u, v,... V, α, β,... F Definition Representation (ρ, V ) g x ρ ρ(x) gl(v ) V v ρ(x) ρ(x)v V ρ(x) Lie homomorphism ρ(α x + β y) = α ρ(x) + β ρ(y) ρ ([x, y]) = [ρ(x), ρ(y)] = = ρ(x) ρ(y) ρ(y) ρ(x) V = C n ρ(x) M n (C) = gl(v ) = gl (C, n) W V invariant (stable) subspace ρ(g)w W (ρ, W ) is submodule (ρ, W ) (ρ, V )

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34 Theorem W invariant subspace of V (ρ, W ) (ρ, V ) (ρ, V )! induced representation (ρ, V /W ) ρ(g)w W =! ρ(x) : V /W V /W = ρ(x) V V π V /W ρ(x) π V /W ρ(x) π = π ρ(x) Ker ρ = C ρ (g) = {x g : ρ(x)v = {0}} is an ideal Ker ad = C ad (g) = Z(g)= center of g

35 Reducible and Irreducible representation (ρ, V ) irreducible/simple representation (irrep) (non trivial) invariant subspaces ρ(g)w W W = {0} or V (ρ, W ) (ρ, V ) W = {0} or V (ρ, V ) reducible/semisimple representation V = V 1 V 2, (ρ, V 1 ) and (ρ, V 2 ) submodules V = V 1 V 2 ρ(g)v 1 V 1, ρ(g)v 2 V 2 trivial representation V = F dim V = 1

36 Induced Representation (ρ, W ) (ρ, V ) ρ(x) V V π V /W ρ(x) π V /W ρ(x) is the induced representation ρ(x) π = π ρ(x)

37 Jordan Hölder decomposition (ρ, W ) (ρ, V ) and (ρ, V /W ) not irrep nor trivial ( ρ, U ) (ρ, V /W ) W U = π 1 (U) V ρ(x) V V π V /W ρ(x) π V /W ρ(x) π = π ρ(x) (π ρ(x)) (U) = (ρ(x) π) (U) = ρ(x) ( U ) U = π(u) ρ(x)(u) U (ρ, W ) (ρ, U) (ρ, V ) Theorem (Jordan Hölder decomposition) (ρ, V ) representation V = V 0 V 1 V 2 V m = {0}, (ρ, V i ) (ρ, V i+1 ) (ρ Vi /Vi+1, V i /V i+1 ) irrep or trivial

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39 Direct sum of representations (ρ 1, V 1 ), (ρ 2, V 2 ) representations of g Def: Direct sum of representations (ρ 1 ρ 2, V 1 V 2 ) (ρ 1 ρ 2 ) (x) : V 1 V 2 V 1 V 2 V 1 V 2 (v 1, v 2 ) (ρ 1 (x)v 1, ρ 2 (x)v 2 ) V 1 V 2 V 1 V 2 v 1 + v 2 ρ 1 (x)v 1 + ρ 2 (x)v 2 V 1 V 2 1b.pdf

40 Theorem Jordan- Hölder Any representation of a simple Lie algebra is a direct sum of simple representations or trivial representations

41 = The important is to study the simple representations! Theorem (Jordan Hölder decomposition) (ρ, V ) representation V = V 0 V 1 V 2 V m = {0}, (ρ, V i ) (ρ, V i+1 ) (ρ Vi /Vi+1, V i /V i+1 ) irrep or trivial V = V m 1 V m 2 /V m 1 V 1 /V 2 V /V 1 ρ = ρ m 1 ρ m 2 ρ 1 ρ 0 ρ m 1 = ρ Vm 1 trivial, ρ i = ρ Vi /V i+1 simple

42 Tensor Product of linear spaces- Universal Definition F field, F vector spaces A, B Definition Tensor Product A F B is a F-vector space AND a canonical bilinear homomorphism : A B A B, with the universal property. Every F-bilinear form φ : A B C, lifts to a unique homomorphism φ : A B C, such that φ(a, b) = φ(a b) for all a A, b B. Diagramatically: A B A B φ bilinear C! φ

43 Tensor product - Constructive Definition Definition Tensor product A F B can be constructed by taking the free F-vector space generated by all formal symbols a b, a A, b B, and quotienting by the bilinear relations: (a 1 + a 2 ) b = a 1 b + a 2 b, a (b 1 + b 2 ) = a b 1 + a b 2, r(a b) = (ra) b = a (rb) a 1, a 2 A, b B, a A, b 1, b 2 B, r F

44 Examples: U and V linear spaces u = u 1 u 2. u n U, v = v 1 u 2. v m V u v = u 1 v 1 u 1 v 2. u 1 v m u 2 v 1 u 2 v 2. u 2 v m.. u n v 1 u n v 2. u n v m U V

45 U End(U), V End(V ) u 11 u 12 u 1p u 21 u 22 u 2p U =.... u q1 u q2 u qp V = v 11 v 12 v 1m v 21 v 22 v 2m.... v n1 v n2 v nm U V = u 11 V u 12 V u 1p V u 21 V u 22 V u 2p V.... u q1 V u q2 V u qp V U V End (U V ) (U V) (u v) Uu Vv

46 Tensor product of representations (ρ 1, V 1 ), (ρ 2, V 2 ) repsesentations of the Lie algebra g. ) L Tensor product representation (ρ 1 ρ2, V 1 V 2 ) L (ρ 1 ρ2 (x) : V 1 V 2 V 1 V 2 ) L v 1 v 2 (ρ 1 ρ2 (x) (v 1 v 2 ) (ρ 1 L ρ2 ) (x) (v 1 v 2 ) = ρ 1 (x)v 1 v 2 + v 1 ρ 2 (x)v 2 (ρ 1 L ρ2 ) (x) = ρ 1 (x) Id 2 + Id 1 ρ 2 (x)

47 Dual representation V dual space of V, V u : V v u.v C (ρ, V ) representation of g! dual representation ( ρ D, V ) ρ D (x) : V u ρ D (x)u V ρ D (x) = ρ (x) ρ D (x)u.v = u.ρ(x)v ρ D (αx + βy) = αρ D (x) + βρ D (y) ρ D ([x, y]) = [ ρ D (x), ρ D (y) ]

48 sl(2, C) representations sl(2, C) = span (h, x, y) [h, x] = 2x [h, y] = 2y [x, y] = h (ρ, V ) irrep of sl(2, C) Jordan decomposition V = V λ, λ eigenvalue of ρ(h), λ V λ = Ker (ρ(h) λi) n λ, v V λ ρ(h)v = λv, V λ Vµ = {0} If v eigenvector of ρ(h) with eigenvalue λ v V λ ρ(x)v eigenvector with eigenvalue λ + 2 ρ(x)v V λ+2 ρ(y)v eigenvector with eigenvalue λ 2 ρ(y)v V λ 2 ρ(h)v = λv ρ(h)ρ(x)v = (λ + 2)ρ(x)v and ρ(h)ρ(y)v = (λ 2)ρ(y)v v 0 V, v eigenvector of ρ(h) such that ρ(x)v 0 = 0 If (ρ, V ) is a finite dimensional representation of sl(2, C) then for every z sl(2, C) Trρ(z) = 0

49 Let ρ(h)v 0 = µv 0 and ρ(x)v 0 = 0 n N : ρ(y) n+1 v 0 = 0 ρ(x)v 0 = 0, v k = 1 k! (ρ(y))k v 0 ρ(h)v k = (µ 2k)v k ρ(y)v k = (k + 1)v k+1 ρ(x)v k = (µ k + 1)v k 1 W = span (v 0, v 1,..., v n ) and ρ(z)w W z sl(2, C) ρ(h) = (ρ, W ) submodule of (ρ, V ) µ µ µ µ 2n Tr ρ(h) = 0 µ = n

50 Theorem (ρ, V ) irrep of sl(2, C) V = span (v 0, v 1,..., v n ) ρ(x)v 0 = 0, v k = 1 k! (ρ(y))k v 0 ρ(h)v k = (n 2k)v k ρ(y)v k = (k + 1)v k+1 ρ(x)v k = (n k + 1)v k 1 ρ(x)v k = µ k v k µ k = n 2k, µ k are the weights of the representation. µ k = n, n 2, n 4,..., n + 4, n + 2, n n is the highest weight of the irrep

51 dim V = n + 1 Eigenvalues n, n + 2,..., n 2, n irreducible representation (ρ, V ) ρ(x)v 0 = 0, v k = 1 k! (ρ(y))k v 0 ρ(y) n+1 v 0 = 0 ρ(h)v k = (n 2k)v k ρ(y)v k = (k + 1)v k+1 ρ(x)v k = (n k + 1)v k 1 α = α = α = g = [g, g] Weyl Theorem Any representation V = n k n V n, V n irrep Theorem For any finite dimensional representation (ρ, V ) the decomposition in direct sum of irreps is unique. The eigenvalues of ρ(h) are integers.

52 Weyl Theorem sl(2) Casimir definition (ρ, V ) a representation of g = sl(2, C). The Casimir (operator) K ρ corresponding to this representation is an element in End V such that K ρ = ρ 2 (h) + 2 (ρ(x)ρ(y) + ρ(y)ρ(x)) [K ρ, ρ(g)] = {0} V = λ V λ, λ are eigenvalues of K ρ (ρ, V λ ) is submodule of (ρ, V ). Any representation (module) of g = sl(2, C) contains a submodule (ρ, V n ), which is an irreducible representation. The eigenvalues λ of the Casimir (operator) are given by the formula λ = n(n + 2), where n = 0, 1, 2, 3,.... Let V λ the submodule corresponding to the eigenvalue λ = n(n + 2) of the Casimir. If v V λ then we can found an irreducible submodule V n of V λ and v V n. Therefore V λ = V n V n V }{{ n = k } n V n k ntimes

53 sl(3, C) sl(3, C) = span (h 1, h 2, x 1, y 1, x 2, y 2, x 3, y 3 ) [h 1, h 2 ] = 0 [h 1, x 1 ] = 2x 1, [h 1, y 1 ] = 2y 1, [x 1, y 1 ] = h 1, [h 1, x 2 ] = x 2, [h 1, y 2 ] = y 2, [h 1, x 3 ] = x 3, [h 1, y 3 ] = y 3, [h 2, x 2 ] = 2x 2, [h 2, y 2 ] = 2y 2, [x 2, y 2 ] = h 2, [h 2, x 1 ] = x 1, [h 2, y 1 ] = y 1, [h 2, x 3 ] = x 3, [h 2, y 3 ] = y 3, [x 1, x 2 ] = x 3, [x 1, x 3 ] = 0, [x 2, x 3 ] = 0 [y 1, y 2 ] = y 3, [y 1, y 3 ] = 0, [y 2, y 3 ] = 0 [x 1, y 2 ] = 0, [x 1, y 3 ] = y 2, [x 2, y 1 ] = 0, [x 2, y 3 ] = y 1, [x 3, y 1 ] = x 2, [x 3, y 2 ] = x 1, [x 3, y 3 ] = h 1 + h 2 sl(2, C) subalgebras: span(h 1, x, 1, y 1 ), span(h 2, x, 2, y 2 ), span(h 1 + h2, x, 3, y 3 )

54 Weights-Roots (ρ, V ) is a representation of sl(3, C) Def: µ = (m 1, m 2 ) is a weight v V : ρ(h 1 )v = m 1 v, ρ(h 2 )v = m 2 v, v is a weightvector Prop: Every representation of sl(3, C) has at leat one weight Prop: µ = (m 1, m 2 ) Z 2

55 Def: If (ρ, V ) = (ad, g) weight α is called root α = (a 1, a 2 ) ad h1 z = a 1 z, ad h2 z = a 2 z z = x 1, x 2, x 3, y 1, y 2, y 3 root α rootvector z α α 1 (2, 1) x 1 α 2 ( 1, 2) x 2 α 1 + α 2 (1, 1) x 3 α 1 ( 2, 1) y 1 α 2 (1, 2) y 2 α 1 α 2 ( 1, 1) y 3 Def: α 1 and α 2 are the positive simple roots

56 Highest weight Def: h = span(h 1, h 2 ) is the maximal abelian subalgebra of sl(3, C), the Cartan subalgebra. Theorem The weights µ = (m 1, m 2 ) and the roots α = (a 1, a 2 ) are elements of the h µ(h i ) = m i, α(h i ) = a i ρ(h)v = µ(h)v ρ(h)ρ(z α )v = (µ(h) + α(h)) ρ(z α )v Def: µ 1 µ 2 µ 1 µ 2 = aα 1 + bα 2, a 0 and b 0 Def: µ 0 is highest weight if for all weights µ µ 0 µ

57 Highest Weight Cyclic Representation Definition (ρ, V ) is a highest weight cyclic representation with highest weight µ 0 iff 1 v V ρ(h)v = µ 0 (h)v, v is a cyclic vector 2 ρ(x 1 )v = ρ(x 2 )v = ρ(x 3 )v = 0 3 if (ρ, W ) is submodule of (ρ, V ) and v W W = V

58 Theorem If z 1, z 2,... z n are elements of sl(3, C) then n ( p=1 k 1 +k 2 + +k 8 =p ρ(z n) ρ(z n 1 ) ρ(z 2 ) ρ(z 1 ) = c k1,k 2,...,k 8 ρ k 3 (y 3 )ρ k 2 (y 2 )ρ k 1 (y 1 )ρ k 4 (h 1 )ρ k 5 (h 2 )ρ k 6 (x 3 )ρ k 7 (x 2 )ρ k ) 8 (x 1 ) = c k1,k 2,...,k 8 ρ k 3 (y 3 )ρ k 2 (y 2 )ρ k 1 (y 1 )ρ k 4 (h 1 )ρ k 5 (h 2 )ρ k 6 (x 3 )ρ k 7 (x 2 )ρ k 8 (x 1 ) k 1 +k 2 + +k 8 =n } {{ } order n terms + ( ) order n 1 terms Corollary If z 1, z 2,... z n are elements of sl(3, C) and (ρ, V ) is a highest weight cyclic representation with highest weight µ 0 and cyclic vector v then ρ(z n) ρ(z n 1 ) ρ(z 2 ) ρ(z 1 )v = n ( p=1 l 1 +l 2 +l 3 =p ) d l1,l 2,l 3 ρ l 3 (y 3 )ρ l 2 (y 2 )ρ l 1 (y 1 ) v and ) V = span (ρ l 3 (y 3 )ρ l 2 (y 2 )ρ l 1 (y 1 )v, l i N 0

59 Theorem Every irrep (ρ, V ) of sl(3, C) is a highest weight cyclic representation with highest weight µ 0 = (m 1, m 2 ) and m 1 0, m 2 0 Theorem The irrep (ρ, V ) with highest weight µ 0 is a direct sum of linear subspaces V µ where µ = µ 0 (kα 1 + lα 2 + mα 3 ) = µ (k α 1 + l α 2 ) and k, l, m, k and l are positive integers. Corollary The set of all possible weights µ is a subset of Z 2 R 2 The set of all possible weights corresponds on a discrete lattice in the real plane

60 sl(3, C) Fundamental Representations Representation (1, 0) ρ(x 1 ) = ρ(y 1 ) = ρ(h 1 ) = ρ(x 2 ) = ρ(y 2 ) = ρ(h 2 ) = ρ(x 3 ) = ρ(y 3 ) = The dual of the representation (1, 0) is the representation (0, 1), i.e (1, 0) D = (0, 1) The representation (1, 0) L (0, 1) is the direct sum (0, 0) (1, 1)

61 Weight Lattice for sl(3, C) Let (ρ, V ) the fundamental representation (1, 0) then ρ(sl(3, C)) = iso traceless 3 3 matrices Def: (r, s) def Tr (ρ(r) ρ(s)) (x,{ y) is a non-degenerate } bilinear form on g = sl(3, C) (x, g) = {0} x = 0 Det (x i, x j ) 0, where x k = h, x, y Proposition (.,.) is a non-degenerate bilinear form on h { } ν h h ν h : ν(h) = (h ν, h) (h) h iso Definition µ, ν h (µ, ν) def (h µ, h ν ) (α 1, α 1 ) = (α 2, α 2 )

62 Roots (h α1, h 1 ) = α 1 (h 1 ) = 2, (h α1, h 2 ) = α 1 (h 2 ) = 1 (h α2, h 1 ) = α 2 (h 1 ) = 1, (h α2, h 2 ) = α 2 (h 2 ) = 2 h α1 = h 1, h α2 = h 2 (α 1, α 1 ) = 2, (α 2, α 2 ) = 2, (α 1, α 2 ) = 1 α 1 = α 2 = 2, angle α 1 α 2 = 120 o

63 Fundamental weights µ 1 (h 1 ) = 1, µ 1 (h 2 ) = 0, µ 2 (h 1 ) = 0, µ 2 (h 2 ) = 1 µ 1 = 2 3 α α 2, µ 2 = 1 3 α α 2 2 µ 1 = µ 2 = 3, angle µ 1µ 2 = 60 o µ = m 1 µ 1 + m 2 µ 2, m i N 0

64

65 µ weight v V : ρ(h)v = µ(h)v p, q N o : (ρ, W ) is a sl(2, C) submodule W = span ( ρ q (y i )v, ρ q 1 (y i )v,...,, ρ(y i )v, v, ρ(x i )v,..., ρ p 1 (x i )v, ρ p (x i )v ) Def: If µ is a weight and α i, i = 1, 2, 3 a root, a chain is the set of permitted values of the weights µ qα i, µ (q 1)α i,..., µ + (p 1)α i, µ + pα i The weights of a chain (as vectors) are lying on a line perpendicular to the vertical of the vector α i. This vertical is a symmetry axis of this chain. Prop: The weights of a representation is the union of all chains Theorem (Weyl transform) If µ is a weight and α any root then there is another weight given by the transformation (µ, α) (µ, α) S α (µ) = µ 2 α and 2 (α, α) (α, α) Z

66 sl(3, C)-Representation (4,0)

67 sl(3, C)-Representation (1,2)

68 sl(3, C)-Representation (2,2)

69 Killing form (ρ, V ) representation of g B ρ (x, y) def Tr (ρ(x) ρ(y)) Prop: B ρ ([x, y], z) = B ρ (x, [y, z]) B ρ (ad y x, z) + B ρ (x, ad y z) = 0 Definition (Killing form) B(x, y) def B ad (x, y) = Tr (ad x ad y ) g = span (e 1, e 2,..., e n ) g = span ( e 1, e 2,..., e n), e i (e j ) = e i.e j = δj i n n B(x, y) = e k.ad x ad y e k = e k. [x, [y, e k ]] k=1 k=1

70 Prop: δ Der(g) B(δ(x), y) + B(x, δ(y)) = 0 Prop: τ Aut(g) ad τx = τ ad x τ 1 B(τx, τy) = B(x, y) Prop: k ideal of g x, y k B(x, y) = B k (x, y) Prop: k ideal of g, k = {x g : B(x, k) = {0}} k is an ideal.

71 Derived Series Def: Derived Series g (0) = g g (1) = D(g) = [g, g] g (2) = D 2 (g) = [ g (1), g (1)] g (n) = D n (g) = [ g (n 1), g (n 1)] g (k) is an ideal of g Definition (Solvable Lie Algebra) g solvable n N : g (n) = {0}

72 Exact sequence - extensions Def: exact sequence, a, b, g Lie spaces a µ g λ b Im µ = Ker λ Def: g extension of b by a 0 a µ 1:1 g λ epi b 0

73 Solvable Lie Algebra Def: Solvable Lie Algebra g solvable n N : g (n) = {0} Prop: g solvable, k Lie-subalgebra k solvable Prop: Prop: Prop: g solvable, φ : g Lie Hom g φ(g) solvable I solvable ideal AND g/i solvable g solvable a and b solvable ideals a + b solvable ideal

74 Theorem Theorem µ a g λ b Im µ = Ker λ a solvable AND b solvable g solvable {0} a µ 1:1 g π g/ker λ λ epi iso Im µ = Ker λ g solvable b {0} a solvable AND b solvable

75 Equivalent definitions Theorem (a) g solvable g (n) = {0} (b) exists g = g 0 g 1 g 2 g n = 0 g i ideals and g i /g i+1 abelian. (c) exists g = h 0 h 1 h 2 h p = 0 h i subalgebras of g h i+1 ideal of h i and h i /h i+1 abelian. (d) exists g = h 0 h subalgebras of g h i h i+1 1 h 2 h q = 0 ideal of h i and dim h i /h i+1 = 1. g solvable Lie algebra m ideal such that dim(g/m) = 1

76 Theorem g solvable ad g t(n, C). There is a basis in g, where ALL the matrices ad x, x g. are upper triangular matrices. λ 1 (x) 0 λ 2 (x) ad x = 0 0 λ 3 (x) λ n (x)

77 Radical Definition Rad(g) = radical = maximal solvable ideal = sum of all solvable ideals Def: Rad(g) = {0} g semi-simple Theorem (Radical Property) Rad (g/rad (g)) = { 0 } g/rad(g) is semisimple Prop: g = g 1 g 2 Rad(g) = Rad(g 1 ) Rad(g 2 )

78 Nilpotent Lie Algebras g 0 = g, g 1 = [g, g], g 2 = [ g, g 1],..., g n = [ g, g n 1] g k ideals Definition g nilpotent g n = 0 g nilpotent g solvable Prop: g nilpotent m : ad x1 ad x2 ad xm = 0 (ad x ) m = 0 Prop: g nilpotent g i ideals g = g 0 g 1 g 2 g r = {0} [g, g i ] g i+1 and dim g i /g i+1 = 1 Prop: g nilpotent basis e 1, e 2,..., e n where ad x is strictly upper triangular Prop: g nilpotent B(x, y) = 0 Prop: g nilpotent Z(g) {0} Prop: g/z(g) nilpotent g nilpotent

79 Engel s Theorem Theorem ( Engel s theorem) g nilpotent n : (ad x ) n = 0 g nilpotent g ad-nilpotent Lemma g gl(v ) and x g x n = 0 { } = (ad x ) 2n = 0

80 Engel s Theorem in Linear Algebra Prop: step #1 g gl(v ) and x g x n = 0 x m (ad x ) m = 0 { } = v V : x g xv = 0 m Lie subalgebra of g g π g/m adx g π adx g/m ( adx ) m = 0 step #2 Induction hypothesis m ideal such that dim g/m = 1 g = Cx 0 + m step #3 Induction hypothesis and g = Cx 0 + m U = {v V : x m xv = 0} {0} v V : x g xv = 0 and gu U

81 Engel s Theorem Prop : { } g gl(v ) and x g x n = = 0 V = V 0 V 1 V 2 V n = 0, V i+1 = gv i (g) n V = 0 x 1 x 2 x 3 x n = 0 V = V 0 V 1 V 2 V n = 0, V i+1 = gv i is a flag Theorem ( Engel s-theorem) Theorem g nilpotent n : (adx) n = 0 g nilpotent exists a basis in g such that all the matrices ad x, x g are strictly upper diagonal. Prop: g nilpotent {x g Tr ad x = 0} B(x, y) = Tr (ad x ad y ) = 0

82 Lie Theorem in Linear Algebra Theorem (Lie Theorem) g solvable Lie subalgebra of gl(v ) v V : x g xv = λ(x)v, λ(x) C step #1 Lemma (Dynkin Lemma) g gl(v ) a ideal λ a : W = {v V : a a av = λ(a)v} step # 2 g and [g, g] {0}, { a ideal gw W } : g = Ce 0 + a step # 3 induction hypothesis on a Lie theorem on g

83 Lie Theorem Lemma g solvable and (ρ, V ) irreducible module dim V = 1 Theorem g solvable and (ρ, V ) a representation exists a basis in V where all the matrices ρ(x), x g are upper diagonal. Theorem g nilpotent and (ρ, V ) a representation exists a basis in V all the matrices ρ(x), x g satisfy the relation (ρ(x) λ(x)i) n = 0, where λ(x) C or the matrices (ρ(x) λ(x)i) are strictly upper diagonal. Prop: g solvable Dg = [g, g] is nilpotent Lie algebra

84 Solvable Nilpotent ( W (ρ, W ) (ρ, V ) ρ(x) = 0 V /W ρ(x) = ρ(x) = λ 1 (x)... 0 λ 2 (x) λ 3 (x) λ 4 (x) λ(x)... 0 λ(x) λ(x) λ(x) )

85 Linear Algebra A End V M n (C) p A (t) = Det(A ti) = characteristic polynomial p A (t) = ( 1) n m (t λ i ) m i m, m i = n, λ i eigenvalues i=1 i=1 i=1 V = m V i, V i = Ker (A λ i I) m i p A (A) = 0 and AV i V i Q i (t) = p A(t) (t λ i ) m If v V i j and j i Q i (A)v = 0 Prop: Chinese remainder theorem Q(t), (Q(0) 0) and P(t), (P(0) 0) polynomials with no common divisors, i.e (Q(t), P(t)) = 1 s(t), s(0) = 0 and r(t) polynomials such that s(t)q(t) + r(t)p(t) = 1 Prop: Prop: s i (t) and r i (t) polynomials such that s i (t)q i (t) + r i (t) (t λ i ) m i = 1 S(t) = k µ i s i (t)q i (t) v V j S(A)v = µ j v i

86 Prop: Jordan decomposition A gl(v ) A = A s + A n, [A s, A n ] = A s A n A n A s = 0 A s diagonal matrix (or semisimple) and v V i A s v = λ i v, A n nilpotent matrix (A n ) n = 0, the decomposition is unique Prop:! p(t) and q(t) polynomials such that A s = p(a) and A n = q(a) E ij n n matrix with zero elements with exception of the ij element, which is equal to 1 [E ij, E kl ] = δ jk E il δ il E kj Prop: A = n λ i E ii The eigenvalues of ad A are equal to λ i λ j i=1 ad A E ij = (λ i λ j ) E ij Prop: ad A = ad As + ad An and (ad A ) s = ad As, (ad A ) n = ad An

87 Cartan Criteria Lemma { g gl(v ) x, y g Tr(xy) = 0 Theorem ( 1st Cartan Criterion) g solvable B Dg = 0 Corollary B(g, g) = {0} g solvable } [g, g] = Dg = g (1) nilpotent B is non degenerate, a ideal of g a = {x g : B (x, a) = {0}} is an ideal. Theorem ( 2nd Cartan Criterion) g semisimple B non degenerate

88 Prop a semi-simple ideal of g g = a a, a is an ideal. Prop: g semisimple, a ideal g = a a Theorem g semisimple g direct sum of simple Lie algebras g = i g i, g i is simple Theorem g semisimple Der(g) = adg Theorem g semisimple g = [g, g] = Dg = g (1) g semisimple, (ρ, V ) a representation ρ(g) sl(v ).

89 Abstract Jordan decomposition Prop: g semisimple Z(g) = {0} Prop: g semisimple g iso ad g Prop: Der(g), = s + n the Jordan decomposition s Der(g) and n Der(g) Theorem (Abstract Jordan Decomposition) g semisimple ad x = (ad x ) s + (ad x ) n Jordan decomposition and x = x s + x n where x s g, (ad x ) s = ad xs and x n g, (ad x ) n = ad xn

90 Toral subalgebra g is semisimple Definition semisimple element x s is semisimple x g : x = x s + x n x s is semisimple element ad xs is diagonalizable on g x s and y s semisimple elements x s + y s is semisimple and [x s, y s ] is semisimple. Definition Toral/ Cartan subalgebra Toral subalgebra h = {x s, x = x s + x n g} i.e the set of all semisimple elements Theorem The toral or Cartan subalgebra is abelian

91 φ is a Lie epimorphism g φ epi g and g simple g is simple and isomorphic to g. φ is a Lie epimorphism g φ epi g and g semisimple g is semisimple. Proposition g semisimple, (ρ, V ) a representation, x = x s + x n ρ(x) = ρ(x s ) + ρ(x n ) and ρ(x s ) = (ρ(x)) s, ρ(x n ) = (ρ(x)) n the Jordan decomposition of ρ(x) ρ(x n ) m = 0 for some m N. There is some basis in V where all the matrices ρ(x s ) are diagonal and all matrices ρ(x n ) are either strictly upper either lower triangular matrices.

92 Roots construction g semi-simple algebra, h = {x s, x g, x = x s + x n } Toral subalgebra or Cartan subalgebra, h = Ch 1 + Ch 2 + Ch l. ad h is a matrix Lie algebra of commuting matrices all the matrices have common eigenvectors Σ i = eigenvalues of h i g = g λi, g λi linear vector space λ i Σ i x g λi ad hi x = λ i x and λ i ν i g λi g νi = {0} x g λ1 g λ2 g λl, h = c 1 h 1 + c 2 h c l h l ad h x = (c 1 λ 1 + c 2 λ c l λ l ) x h = Cµ 1 + Cµ Cµ l, µ i h, µ i (h j ) = δ ij Definition of the roots h λ = λ 1 µ 1 + λ 2 µ λ l µ l c 1 λ 1 + c 2 λ c l λ l = λ(h) λ is a root x g λ = g λ1 g λ2 g λl [h, x] = ad h x = λ(h)x g = g λ, λ µ g λ gµ = {0} λ

93 Roots g semisimple algebra, h = {x s, x g, x = x s + x n } Toral subalgebra ad h is a matrix Lie algebra of commuting matrices all the matrices have common eigenvectors Theorem (Root space) Exists root space h g = h g α α x g α, h h ad h x = [h, x] = α(h)x h is a Lie subalgebra, g α are vector spaces [g λ, g µ ] g λ+µ if λ + µ Prop: λ, µ [g λ, g µ ] = {0} if λ + µ

94 Semisimple roots g semisimple Lie algebra, h Toral/Cartan subalgebra B(g λ, g µ ) = 0 if λ + µ 0 h, h h B(h, h ) = λ n λ λ(h)λ(h ), n λ = dim g λ α, x g α (ad x ) m = 0 Proposition The Killing form is a non degenerate bilinear form on the Cartan subalgebra h {h h, B(h, h) = 0} {h = 0}

95 { α, α(h) = 0 h = 0} {span( ) = h } {α α } {α g α {0}} Killing form B non degenerate on h h φ 1:1 epi t φ h, φ(h) = B(t φ, h) x g α, y g α [x, y] = B(x, y)t α, t α h is unique [g α, g α ] = Ct α, α(t α ) = B(t α, t α ) h α = 2t α B(t α, t α ) S α = Ch α + Cx α + Cy a iso sl(2, C)

96 Prop: dim g α = 1 Prop: α and pα p = 1 S α = Ch α + Cx α + Cy a sl(2, C) iso g = h (Cx α + Cy a ) = h g + g α + g + = Cx α, g = Cy α α + α + Example g = sl (3, C) h = Ch 1 + Ch 2, g + = Cx 1 + Cx 2 + Cx 3, g = Cy 1 + Cy 2 + Cy 3

97 Strings of Roots β, α String of Roots β g β β + α g β+α β α g β α β + 2α g β+2α β 2α g β 2α β + pα g β+pα β qα g β qα β + (p + 1)α β (q + 1)α {β qα, β (q 1)α,..., β α, β, β + α,..., β + (p 1)α, β + pα} g α β = g β qα g β (q 1)α g β α g β g β+α g β+(p 1)α g β+pα ( ) Prop: ad, g α β is an irreducible representation of S a sl(2, C) iso Prop: β, α β(h α ) = q p Z, β β(h α )α

98 Cartan and Cartan-Weyl basis Cartan Basis [E α, E α ] = t α, [h α, E α ] = α(t α )E α, [E α, E β ] = k αβ E α+β, if α + β B(E α, E α ) = 1, B(t α, t β ) = α(t β ) = β(t α ) Cartan Weyl Basis α t α root H α = 2 α(t α ) t α coroot [X α, X α ] = H α, [H α, X α ] = 2X α, [H α, X α ] = 2X α [X α, X β ] = N αβ X α+β, if α + β CX α + CH α +CX α iso sl(2)

99 Def: (α, β) B(t α, t β ) = α(t β ) = β(t α ) = (β, α) Prop: < β, α > 2 (β, α) (α, α) = q p Z Prop: w α (β) β < β, α > α, w α (β) Def: Weyl Transform w α :

100 Prop: β α < β, α >< 0, Prop: β + α < β, α >> 0 Prop: β(h α ) = < β, α > Z, B(h α, h β ) = γ(h α )γ(h β ) Z Prop: Prop: Prop: B(h α, h α ) = γ ( < α, γ >) 2 > 0 γ β and β = m c i α i, α i c i Q i=1 span Q ( ) = E Q, E Q is a Q-euclidean vector space. Prop: (α, β) B(t α, t β ) Q, (α, α) > 0, (α, α) = 0 α = 0

101 Prop: There are not strings with five members < β, α >= 0, ±1, ±2, ±3 (α, β) = α β cos θ < β, α >= 2 β cos θ = 0, ±1, ±2, ±3 α < α, β > < β, α >= 4 cos 2 θ < α, β > < β, α > θ β / α 0 0 π/2 1 1 π/ π/ π/ π/ π/ π/6 3

102

103 Simple Roots =roots is a finite set and span R ( ) =<< >>= E = R l Proposition E is not a finite union of hypersurfaces of dimension l 1 E is not the union of hypersurfaces vertical to any root z E : α (α, z) 0 Definition of positive/negative roots + = {α : (α, z) > 0}, positive roots = {α : (α, z) < 0}, negative roots = +, + = Definition of Simple roots } Σ = {β + : is not the sum of two elements of + Σ Proposition β + β = n σ σ, n σ 0, n σ Z σ Σ

104 Coxeter diagrams α i ɛ i = α i (αi, α i ) = unit vector Σ simple roots Admissible unit roots Def: Admissible unit vectors A = {ɛ 1, ɛ 2,... ɛ n } 1 ɛ i linearly independent vectors 2 i j (ɛ i, ɛ j ) (ɛ i, ɛ j ) 2 = 0, 1, 2, 3, Coxeter diagrams for two points ɛ i ɛ j 4 (ɛ i, ɛ j ) 2 = 0 ɛ i ɛ j 4 (ɛ i, ɛ j ) 2 = 1 ɛ i ɛ j 4 (ɛ i, ɛ j ) 2 = 2 ɛ i ɛ j 4 (ɛ i, ɛ j ) 2 = 3

105 Prop: A = A {ɛ i } admissible Prop: The number of non zero pairs is less than n Prop: The Coxeter graph has not cycles Prop: The maximum number of edges is three Prop: If {ɛ 1, ɛ 2,..., ɛ m } A and 2 (ɛ k, ɛ k+1 ) = 1, i j > 0 (ɛ i, ɛ j ) = 0 ɛ = m ɛ k A = (A {ɛ 1, ɛ 2,..., ɛ m }) {ɛ} admissible k=1

106 Rank 2 root systems Root system A1 A1 Root system A2 Root system B2 Root system G2

107 Dynkin diagrams

108 Serre relations g = h g + g, simple roots Σ = {α 1, α 2,..., α r } x i g αi, y i g αi, B(x i, y i ) = 2 (α i, α i ) h i = h αi α i (h i ) = 2, α i (h j ) Z Serre Relations Cartan Matrix: a ij = α j (h i ) = 2 (α j, α i ) (α i, α i ) 1 [h i, h j ] = 0 2 [h i, x j ] = a ij x j, [h i, y j ] = a ij y j 3 [x i, y j ] = δ ij h i 4 (ad xi ) 1 a ij x j = 0, (ad yi ) 1 a ij y j = 0 [ 2 1 Example sl (3, C) Cartan Matrix = 1 2 ]

109 Weight ordering Theorem The weights µ and the roots α are elements of the h h h, z α g α ρ(h)v = µ(h)v ρ(h)ρ(z α )v = (µ(h) + α(h)) ρ(z α )v Definition of weight ordering µ 1 µ 2 µ 1 µ 2 = α + k α α, k α 0

110 Highest weight Definition of Highest weight µ 0 is highest weight if for all weights mu µ 0 µ Def: (ρ, V ) is a highest weight cyclic representation with highest weight µ 0 iff 1 v V ρ(h)v = µ 0 (h)v, v is a cyclic vector 2 α + ρ(x α )v = 0 3 if (ρ, W ) is submodule of (ρ, V ) and v W W = V

111 Highest weight representation Theorem Every irrep (ρ, V ) of g is a highest weight cyclic representation with highest weight µ 0 and µ o (h α ) N {0}, α + Theorem The irrep (ρ, V ) with highest weight µ 0 is a direct sum of linear subspaces V µ where µ = µ 0 ( α + k α α) and k α are positive integers.

112 µ weight v V : ρ(h)v = µ(h)v p, q N o : (ρ, W ) is a g submodule W = span ( ρ q (y α )v, ρ q 1 (y α )v,...,, ρ(y α )v, v, ρ(x α )v,..., ρ p 1 (x α )v, ρ p (x α )v ) Def: If µ is a weight and α a root, a chain is the set of permitted values of the weights µ qα, µ (q 1)α,..., µ + (p 1)α, µ + pα The weights of a chain (as vectors) are lying on a line perpendicular to the vertical of the vector α. This vertical is a symmetry axis of this chain. Prop: The weights of a representation is the union of all chains Theorem (Weyl transform) If µ is a weight and α any root then there is another weight given by the transformation (µ, α) (µ, α) S α (µ) = µ 2 α and 2 (α, α) (α, α) Z

113 If (α, β) > 0 α β If α β (α, β) < 0 If α, β + (α, β) < 0 Let µ α span such that µ ( α)(h β ) = δ α,β, where α, β + µ o = n α µ α, α + µ = n α N {0} α + m α µ α, m α Z, µ µ o