ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ Αναπαραστάσεις Αλγεβρών Lie Ενότητα 1: Αναπαραστάσεις Αλγεβρών Lie Κ. Δασκαλογιάννης Τμήμα Μαθηματικών Α.Π.Θ. (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 1 / 131

Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό,όπως εικόνες,που υπόκειται σε άλλου τύπου άδειας χρήσης, η άδεια χρήσης αναφέρεται ρητώς. (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 2 / 131

Χρηματοδότηση Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί στα πλαίσια του εκπαιδευτικού έργου του διδάσκοντα. Το έργο Άνοικτά Ακαδημαικά Μαθήματα στο Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης έχει χρηματοδοτήσει μόνο την αναδιαμόρφωση του εκπαιδευτικού υλικού. Το έργο υλοποιείται στο πλαίσιο του Επιχειρησιακού Προγράμματος Εκπαίδευση και Δια Βίου Μάθηση και συγχρηματοδοτείται απο την Ευρωπαική Ενωση (Ευρωπαικό Κοινωνικό Ταμείο) και από εθνικούς πόρους. (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 3 / 131

Περιεχόμενα Ενότητας Εισαγωγή. Απλές και ημίαπλες Lie άλγεβρες. Περιβάλλουσες άλγεβρες. Δομή Hopf. Αναπαραστάσεις και modules. Παραδείγματα εφαρμογών. (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 4 / 131

Σκοποί Ενότητας Εισαγωγή των μεταπτυχιακών φοιτητών στην μελέτη και στις τεχνικές αλγεβρών μη προσεταιριστικών όπως είναι οι Lie άλγεβρες. (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 5 / 131

Definition of an Algebra 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 6 / 131

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]] (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 7 / 131

Examples of Lie algebras (1) (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) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 8 / 131

Examples of Lie algebras (2) (iv) 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) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 9 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 10 / 131

Poisson algebra (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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 11 / 131

Structure Constants g = span (e 1, e 2,..., e n ) = Fe 1 + Fe 2 + + 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 12 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 13 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 14 / 131

Jordan decomposition 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 = x s diagonal matrix = e x s diagonal matrix = k=0 x k k! e x = e (xs+xn) = e xs e xn λ 1 0 0... 0 0 λ 2 0... 0... 0 0 0 0... λ n m 1 e xn = e λ1 0 0... 0 0 e λ2 0... 0... 0 0 0 0... e λn k=0 x k n k! (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 15 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 16 / 131

Classical Lie Algebras (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 17 / 131

Classical Lie Algebras (2) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 18 / 131

Classical Lie Algebras (3) 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 1 0 0 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 19 / 131

Classical Lie Algebras (4) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 20 / 131

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) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 21 / 131

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 [ 0 1 0 0 ] y = [ 0 0 1 0 ] ex. first derivative operators acting on polynomials of order n h = z z n 2, x = z2 z nz, y = z (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 22 / 131

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 ] = 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 h 1 = h 2 = 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 x 1 = y 1 = 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 x 2 = y 2 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 x 3 = y 3 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 23 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 24 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 25 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 26 / 131

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 } (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 27 / 131

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 W ψ } {! ψ : g = ψ f Ker ψ = f (Ker g) } Corollary f V g f U V g ψ W W { } { f epi, g epi, Ker f = Ker g! ψ iso : g = ψ f, U iso W } ψ 1 U (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 28 / 131

Lie homomorphism theorem Theorem g, h, l Lie spaces, φ : g epi h and ρ : g l Lie homomorphisms g φ h { φ Lie-epi, Ker φ Ker ρ } {! ψ : Lie-homo ρ = ψ φ } ρ l ψ Corollary ρ φ g φ h g ρ ψ l l { } { φ Lie-epi, ρ Lie-epi, Ker φ = Ker ρ! ψ : Lie-iso ρ = ψ φ, h iso l ψ 1 h } (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 29 / 131

First isomorphism theorem Theorem (First isomorphism theorem) φ : g g Lie-homomorphism, φ g φ(g) π φ g/ker φ =! φ : g/ker φ φ(g) g Lie-iso φ(g) iso g/ker φ (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 30 / 131

K L (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 31 / 131 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

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)) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 32 / 131

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) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 33 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 34 / 131

Matrix Form of the adjoint representation g = span (e 1, e 2,..., e n ) = Ce 1 + Ce 2 + + 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 35 / 131

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 ) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 36 / 131

1.pdf (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 37 / 131

Theorem (1) 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 V π V /W = ρ(x) V 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 38 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 39 / 131

Induced Representation (ρ, W ) (ρ, V ) V π V /W ρ(x) V ρ(x) π V /W ρ(x) π = π ρ(x) ρ(x) is the induced representation (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 40 / 131

Jordan Hölder decomposition (1) (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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 41 / 131

Jordan Hölder decomposition (2) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 42 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 43 / 131

Theorem (Jordan- Hölder) Theorem Jordan- Hölder Any representation of a simple Lie algebra is a direct sum of simple representations or trivial representations (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 44 / 131

Important = 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 45 / 131

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 bilinear φ A B! φ C (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 46 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 47 / 131

Examples (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 48 / 131

Examples (2) U End(U), V End(V ) u 11 u 12 u 1n u 21 u 22 u 2n U =.... u n1 u q2 u nn V = v 11 v 12 v 1m v 21 v 22 v 2m.... v m1 v m2 v mm U V = u 11 V u 12 V u 1n V u 21 V u 22 V u 2n V.... u n1 V u n2 V u nn V U V End (U V ) (U V) (u v) Uu Vv (U 1 V 1 ) (U 2 V 2 ) U 1 U 2 V 1 V 2 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 49 / 131

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) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 50 / 131

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) ] (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 51 / 131

sl(2, C) representations (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 52 / 131

sl(2, C) representations (2) 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 0 0... 0 0 µ 2 0... 0 0 0 µ 4... 0......... 0 0 0... µ 2n Tr ρ(h) = 0 µ = n (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 53 / 131

Theorem (2.1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 54 / 131

Theorem (2.2) 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 n 0 0 0 0 0 n 2 0 0 0 ρ(h) =...... 0 0 0 n + 2 0 0 0 0 0 n 0 n 0 0 0 0 0 n 1 0 0 ρ(x) =...... 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ρ(y) =...... 0 0 0 0 0 0 0 n 0 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 55 / 131

Theorem (2.3) 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 = 2-1 +1 α = 2-2 0 +2 α = 2-3 -1 +1 +3 g = [g, g] (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 56 / 131

Weyl Theorem 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. (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 57 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 58 / 131

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 ) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 59 / 131

Weights-Roots (1) (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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 60 / 131

Weights-Roots (2) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 61 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 62 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 63 / 131

Theorem (3) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 64 / 131

Corollary (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 65 / 131

Theorem (4) 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. (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 66 / 131

Corollary (2) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 67 / 131

sl(3, C) Fundamental Representations Representation (1, 0) ρ(x 1 ) = ρ(y 1 ) = ρ(h 1 ) = 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 ρ(x 2 ) = ρ(y 2 ) = ρ(h 2 ) = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 ρ(x 3 ) = ρ(y 3 ) = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 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) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 68 / 131

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) iso h Definition µ, ν h (µ, ν) def (h µ, h ν ) (α 1, α 1 ) = (α 2, α 2 ) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 69 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 70 / 131

Fundamental weights (1) 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 1 + 1 3 α 2, µ 2 = 1 3 α 1 + 2 3 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 71 / 131

Fundamental weights (2) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 72 / 131

Chain (1) µ weight v V : ρ(h)v = µ(h)v p N o : v 0 = ρ p (x i )v 0 and ρ(x i )v 0 = 0, ρ(h)v 0 = (µ(h) + pα i (h))v 0 q N o : v 1 = ρ(y i ) p+q v 0 and ρ(y i )v 1 = ρ(y i ) p+q+1 v 0 = 0, ρ(h)v 1 = (m(h) qα i (h))v 1. W = span ( ρ p+q (y i )v 0, ρ p+q 1 (y i )v 0,..., ρ(y i ) 2 v 0,..., ρ(y i )v 0, v 0 ) (ρ, W ) is a sl(2, C) submodule 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 73 / 131

Theorem (Weyl transform) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 74 / 131

sl(3, C)-Representation (4,0) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 75 / 131

sl(3, C)-Representation (1,2) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 76 / 131

sl(3, C)-Representation (2,2) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 77 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 78 / 131

Propositions (1) 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. (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 79 / 131

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} (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 80 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 81 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 82 / 131

Theorem (5) Theorem µ a g λ b Im µ = Ker λ a solvable AND b solvable g solvable (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 83 / 131

Theorem (6) Theorem {0} a µ 1:1 g π g/ker λ λ epi iso b {0} Im µ = Ker λ g solvable a solvable AND b solvable (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 84 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 85 / 131

Theorem (7) 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) 0 0 0 0 0 λ n (x) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 86 / 131

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 ) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 87 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 88 / 131

Engel s Theorem (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 89 / 131

Engel s Theorem in Linear Algebra Prop: g gl(v ) and x g x n = 0 { } = v V : x g xv = 0 step #1 x m (ad x ) m = 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 90 / 131

Engel s Theorem (2) 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) g nilpotent n : (adx) n = 0 Theorem 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 91 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 92 / 131

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. (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 93 / 131

Propositions (2) Prop: g solvable Dg = [g, g] is nilpotent Lie algebra ( W (ρ, W ) (ρ, V ) ρ(x) = 0 V /W ) Solvable Nilpotent ρ(x) = ρ(x) = λ 1 (x)... 0 λ 2 (x)... 0 0 λ 3 (x)... 0 0 0 λ 4 (x)......... λ(x)... 0 λ(x)... 0 0 λ(x)... 0 0 0 λ(x)......... (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 94 / 131

Linear Algebra (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 95 / 131

Linear Algebra (2) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 96 / 131

Jordan decomposition (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 97 / 131

Jordan decomposition (2) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 98 / 131

Cartan Criteria (1) Lemma { g gl(v ) x, y g Tr(xy) = 0 } [g, g] = Dg = g (1) nilpotent Theorem ( 1st Cartan Criterion) g solvable B Dg = 0 Corollary B(g, g) = {0} g solvable B is non degenerate, a ideal of g a = {x g : B (x, a) = {0}} is an ideal. (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 99 / 131

Cartan Criteria (2) Theorem ( 2nd Cartan Criterion) g semisimple B non degenerate 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 100 / 131

Cartan Criteria (3) 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 ). (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 101 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 102 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 103 / 131

Lie epimorphism 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. (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 104 / 131

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 2 + + c l h l ad h x = (c 1 λ 1 + c 2 λ 2 + + c l λ l ) x h = Cµ 1 + Cµ 2 + + Cµ l, µ i h, µ i (h j ) = δ ij Definition of the roots h λ = λ 1 µ 1 + λ 2 µ 2 + + λ l µ l c 1 λ 1 + c 2 λ 2 + + 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 = λ g λ, λ µ g λ gµ = {0} (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 105 / 131

Root space 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 106 / 131

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} (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 107 / 131

Killing form-non degenerate bilinear form on the Cartan subalgebra { 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) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 108 / 131

Propositions (3) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 109 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 110 / 131

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) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 111 / 131

Definitions-Propositions 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 : (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 112 / 131

Propositions (4) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 113 / 131

Propositions (5) 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/3 1 1 1 2π/3 1 1 2 π/4 2 1 2 3π/4 2 1 3 π/6 3 1 3 5π/6 3 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 114 / 131

Propositions (6) (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 115 / 131

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 = σ Σ n σ σ, n σ 0, n σ Z (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 116 / 131

Coxeter diagrams (1) 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 ) 0 3 4 (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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 117 / 131

Coxeter diagrams (2) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 118 / 131

Coxeter diagrams (3) Rank 2 root systems Root system A1 A1 Root system A2 Root system B2 Root system G2 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 119 / 131

Dynkin diagrams (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 120 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 121 / 131 ]

Weight ordering (1) 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 122 / 131

Weight ordering (2) If (α, β) > 0 α β If α β (α, β) < 0 If α, β Σ (α, β) < 0 Let µ α span such that µ α (h β ) = (µ α, β) = δ α,β, where α, β Σ µ o = α Σ n α µ α, n α N {0} µ = α Σ m α µ α, m α Z + or Z, µ µ o (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 123 / 131

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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 124 / 131

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. (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 125 / 131

Chain (2) µ weight v V : ρ(h)v = µ(h)v p, q N o : (ρ, W ) is a g submodule 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 (Α.Π.Θ.) Αναπαραστασεις Αλγεβρων Lie 126 / 131

Βιβλιογραφία 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. (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 127 / 131

Σημείωμα Αναφοράς Copyright Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης, Δασκαλογιάννης Κωνσταντίνος. Άναπαραστάσεις Αλγεβρών Lie. Εκδοση: 1.0. Θεσσαλονίκη 2014. Διαθέσιμο από τη δικτυακή διεύθυνση: http://eclass.auth.gr/courses/ocrs435/ (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 128 / 131

Σημείωμα Αδειοδότησης Το παρόν υλικό διατίθεται με τους όρους της άδειας χρήσης Creative Commons Αναφορά, Παρόμοια Διανομή 4.0[1] ή μεταγενέστερη, Διεθνής Εκδοση. Εξαιρούνται τα αυτοτελή έργα τρίτων π.χ. φωτογραφίες, διαγράμματα κ.λ.π., τα οποία εμπεριέχονται σε αυτό και τα οποία αναφέρονται μαζί με τους όρους χρήσης τους στο Σημείωμα Χρήσης Εργων Τρίτων. Ο δικαιούχος μπορεί να παρέχει στον αδειοδόχο ξεχωριστή άδεια να χρησιμοποιεί το έργο για εμπορική χρήση, εφόσον αυτό του ζητηθεί. [1]http://creativecommons.org/licenses/by-sa/4.0/ (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 129 / 131

Διατήρηση Σημειωμάτων Οποιαδήποτε αναπαραγωγή ή διασκευή του υλικού θα πρέπει να συμπεριλαμβάνει: το Σημείωμα Αναφοράς το Σημείωμα Αδειοδότησης τη δήλωση Διατήρησης Σημειωμάτων το Σημείωμα Χρήσης Εργων Τρίτων (εφόσον υπάρχει) μαζί με τους συνοδευόμενους υπερσυνδέσμους. (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 130 / 131

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ Τέλος Ενότητας Επεξεργασία:Αναστασία Γρηγοριάδου Θεσσαλονίκη, Εαρινό Εξάμηνο 2014-2015 (Α.Π.Θ.) Αναπαραστάσεις Αλγεβρών Lie 131 / 131