Math 651 Homework 1 - Algebras and Groups Due 2/22/2013

Math 651 Groups Homework 1 - Algebras and Due 2/22/2013 1) Consider the Lie Group SU(2), the group of 2 × 2 complex matrices A with AT A = I and det(A) = 1. The underlying set is <ul style="display: flex;"><li style="flex:1">ꢀ ꢁ </li><li style="flex:1">ꢂ</li><li style="flex:1">ꢄ</li></ul>ꢃꢃꢃz −w <ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul>|z| + |w| = 1 (1) (2) <ul style="display: flex;"><li style="flex:1">w</li><li style="flex:1">z</li></ul>3with the standard S topology. The usual basis for su(2) is <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>0 i i 0 0 −1 i0<ul style="display: flex;"><li style="flex:1">X = </li><li style="flex:1">Y = </li><li style="flex:1">Z = </li></ul><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">0</li><li style="flex:1">0 −i </li></ul>(which are each i times the Pauli matrices: X = iσx, etc.). a) Show this is the algebra of purely imaginary quaternions, under the commutator bracket. b) Extend X to a left-invariant field and Y to a right-invariant field, and show by computation that the Lie bracket between them is zero. c) Extending X, Y , Z to global left-invariant vector fields, give SU(2) the metric g(X, X) = g(Y, Y ) = g(Z, Z) = 1 and all other inner products zero. Show this is a bi-invariant metric. d) Pick ꢀ > 0 and set g(X, X) = ꢀ2, leaving g(Y, Y ) = g(Z, Z) = 1. Show this is left-invariant but not bi-invariant. √2) The realification of an n×n complex matrix A+ −1B is its assignment it to the 2n × 2n matrix <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>A −B (3) <ul style="display: flex;"><li style="flex:1">B</li><li style="flex:1">A</li></ul>Any n × n quaternionic matrix can be written A + Bk where A and B are complex matrices. Its complexification is the 2n × 2n complex matrix <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>A −B (4) <ul style="display: flex;"><li style="flex:1">B</li><li style="flex:1">A</li></ul>a) Show that the realification of complex matrices and complexification of quaternionic matrices are algebra homomorphisms. In the 1quaternionic case, show that quaternionic conjugation corresponds to complex conjugation and the transposition of the off-diagonal blocks (but not the transpositions of the blocks themselves), and that quaternionic transposition corresponds to transposition of the blocks themselves (so that quaternionic conjugate-transpose is the same as complex conjugate-transpose of the complexified matrix). b) The matrix groups Sp(n) are the n × n quaternion-valued matrices Q that satisfy QT Q = I. Show that Sp(1) ≈ SU(2). c) We defined the the algebras sp(2n, R) and sp(2n, C) to be the 2 × 2 real (resp. complex) matrices X that satisfy XT J + JX = 0 (5) where <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>0I<ul style="display: flex;"><li style="flex:1">J = </li><li style="flex:1">.</li></ul>(6) −I 0 The Lie algebra sp(n) of the group Sp(n) defined above is necessarily a real Lie algebra. Via the complexification process, show that sp(n)⊗C ≈ sp(2n, C) (the real algebras sp(2n, R) and sp(n) are real forms of the complex algebra sp(2n, C)). 3) Let g be a finite-dimensional Lie algebra over C with radical r, and assume r is abelian. In this problem we will use the vanishing of H2 for semisimple algebras to prove that g = h⊕r. As a side note, this is the essential step in Levi’s theorem (that g = h n r whether or not r is abelian). a) Let g = g/r and show that r is a g-module. In the future, denote the canonical projection g → g simply with a bar: x → x¯ ∈ g. b) Let σ : g → g be any vector-space splitting. This means that when x ∈ g we have σ(x¯) = x¯. Show that g(x, y) , σ([x, y]) − [σ(x), σ(y)] (7) Vis an element of 2 g∗⊗g, which can actually be considered an element Vof 2 g∗ ⊗ r. c) Show that the map g is closed, so g represents a class in H2(g, r). d) By the vanishing of H2(g, r), we know g is exact. Show that this results in a map η : g → η so that g(x, y) = η([x, y]), and that σ −η : g → g is a Lie algebra monomorphism, so provides the desired Lie algebra compliment to r. 24) In class we stated that H2(g, M) could be identified with the isomorphism classes of abelien extensions of g by M. We proved half of this assertion: that given a representative of a class in H2, we could construct such a Lie algebra extension, and if a different representative in the same class were chosen, the two extensions, though not the same, were isomorphic. Now assume M is a g-module and <ul style="display: flex;"><li style="flex:1">i</li><li style="flex:1">π</li></ul>(8) 0M<ul style="display: flex;"><li style="flex:1">h</li><li style="flex:1">g</li></ul>0is an exact sequence of Lie algebras and so that the bracket in h is given by <ul style="display: flex;"><li style="flex:1">[x, i(m)] = i (π(x).m) </li><li style="flex:1">(9) </li></ul>for m ∈ M, and that [i(m), i(n)] = 0. Prove there is class in H2(g, M) that provides this extension. 5) Let G be a connected Lie algebra, meaning a manifold with a given differentiable structure, along with a group structure. Prove that G has a canonical analytic structure. You may proceed as you wish, but here is one practicable approach: Let Ω ⊂ g be a connected domain containing 0 which is small enough that exp : Ω → U is a diffeomorphism, and let {pi}i∈Λ (where Λ ⊆ Z) be a set of points in G. Let Ui = Lp U, and let iπi : Ui → Ω, given by πi , exp−1 ◦ Lp (10) −1 ibe the charts. Then show the following: a) The pi can be chosen so that {Ωi}i∈Λ is a locally finite covering of G b) Set Uij = Ui ∩ Uj whenever i = j and the intersection is non-empty. Then both πi, πj : Uij → Ω. Using the Cambpell-Baker-Hausdorff formula, show that the transition maps πi ◦ πj−1 are analytic. c) Indicate that the Lie algebra structure (itself determined by the differentiable structure of G) uniquely determines the analytic structure. 3

Math 651 Homework 1 - Algebras and Groups Due 2/22/2013

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