King’s College London
University Of London
This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board.
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Candidate No: ...... Desk No: ......
MSc Examination
CMMS01 Lie groups and Lie algebras
Summer 2007
Time Allowed: Two Hours
This paper consists of two sections, Section A and Section B. Section A contributes half the total marks for the paper. Answer all questions in Section A. All questions in Section B carry equal marks, but if more than two are attempted, then only the best two will count.
NO calculators are permitted.
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2007 c King’s College London - 2 - Section A CMMS01
SECTION A
A 1. (16 points) (i) State the definition of a matrix Lie group. (ii) State the definition of the Lie algebra of a matrix Lie group. (iii) Give (without proof) the Lie algebra o(n) of the matrix Lie group O(n) = {M ∈ Mat(n, R)|M tM = 1}. (iv) Prove that if A ∈ o(n) then indeed exp(sA) ∈ O(n) for all s ∈ R. (You may use properties of the matrix exponential like exp(A)t = exp(At), etc., without proof.)
(v) Prove conversely that if exp(sA) ∈ O(n) for all s ∈ R, then A ∈ o(n).
A 2. (20 points) (i) State the definition of the property ‘simple’ of a Lie algebra. (ii) Let (V,R) be a representation of a Lie algebra g. State the definition of an invariant subspace of (V,R) and the definition of an irreducible representation. (iii) Show that if g is a simple Lie algebra, then the adjoint representation of g is irreducible. Hint: Suppose the adjoint representation is not irreducible and show that then g has a proper ideal. (iv) Let (V,R) be a representation of a Lie algebra g and let ker(R) = x ∈ g R(x) = 0 .
Show that ker(R) is an ideal of g.
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A 3. (14 points) Consider the two Dynkin diagrams
A3 = ,B3 = i 1 2 3 1 2 3 e e e e e e (i) Using the labelling of the roots indicated above, give the Cartan matrix
A11 A12 A13 A = A21 A22 A23 A31 A32 A33
for A3 and B3. (ii) State the definition of the Cartan matrix Aij via the simple roots α(i).
(2) (3) (iii) Give the angle between the roots α and α for A3 and B3. Hint: Consider the appropriate product of two entries of the Cartan matrix.
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SECTION B
B 4. (i) State the definition of a Cartan subalgebra h of a finite-dimensional semi- simple complex Lie algebra g.
(ii) Show that the set of diagonal matrices h in sl(n, C) is a Cartan subalgebra. Hint: Recall that Ekl is the n×n-matrix with entries (Ekl)ij = δikδjl and that EijEkl = δjkEil. • Note that the elements of h together with {Eij | i 6= j} span sl(n, C). Pn • Show that for H = k=1 λkEkk one has adH Eij = (λi − λj)Eij. (iii) Show that the Killing form of the two diagonal matrices G, H ∈ h with
n n X X G = λkEkk ,H = µkEkk k=1 k=1 Pn is given by κ(G, H) = 2n k=1 λkµk. Hint: Let Ti be a basis of h. Write
P ∗ P ∗ κ(x, y) = i Ti [x, [y, Ti]] + i6=j Eij [x, [y, Eij]]
and show first that κ(Ekk, Ell) = 2nδkl − 2.
B 5. Consider the following two subsets of Mat(2, C), n a 0 o n a b o s = a, b ∈ , s = a, b ∈ . 1 0 b C 2 0 0 C
(i) Show that s1 and s2 are Lie subalgebras of Mat(2, C).
(ii) Show that neither s1 nor s2 are simple. (iii) Let g be a complex Lie algebra of dimension 2. Show that g is isomorphic
to either s1 or to s2. Hint: Distinguish two cases: g abelian and g not abelian. If g is not abelian, show that there is a nonzero Z ∈ g such that for any two X,Y ∈ g the Lie bracket [X,Y ] is proportional to Z. Conclude that one can find a basis {T,Z} of g such that [T,Z] = Z.
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B 6. (i) Give matrices H,E,F ∈ sl(2, C) such that [H,E] = 2E,[H,F ] = −2F and [E,F ] = H. Verify your answer for the case [H,E]. (ii) Let P (x, y) be the complex vector space of polynomials in x and y, i.e.
P (x, y) = span xmyn m, n = 0, 1, 2,... . C
∂ ∂ m n m−1 n Denote by ∂x the differentiation with respect to x, e.g. ∂x x y = mx y , etc. Define
∂ ∂ ∂ ∂ R(H) = x ∂x − y ∂y ,R(E) = x ∂y ,R(F ) = y ∂x .
Show that (P (x, y),R) is a representation of sl(2, C). (iii) Let P (x, y)d the subspace of P (x, y) consisting of all polynomials of degree d, i.e. P (x, y)d = span xmyd−m m = 0, 1, . . . , d . C Show that P (x, y)d is an invariant subspace of P (x, y).
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