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

Home , Complexification, Complex conjugate

1) Consider the Lie Group SU(2), the group of 2 × 2 complex matrices A T with A A = I and det(A) = 1. The underlying set is

z −w |z|2 + |w|2 = 1 (1) w z

with the standard S3 topology. The usual basis for su(2) is 0 i 0 −1 i 0 X = Y = Z = (2) i 0 1 0 0 −i

(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 A −B (3) BA

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 A −B (4) B A

a) Show that the realiﬁcation of complex matrices and complexiﬁca- tion of quaternionic matrices are algebra homomorphisms. In the

1 quaternionic 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 T Q that satisfy Q 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 0 I J = . (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 semi- simple 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 7→ 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) is an element of V2 g∗⊗g, which can actually be considered an element of V2 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.

2 4) 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

0 M i h π g 0 (8)

is an exact sequence of Lie algebras and so that the bracket in h is given by

[x, i(m)] = i (π(x).m) (9) 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 dif- ferentiable 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 = Lpi U, and let πi : Ui → Ω, given by

−1 πi exp ◦ L −1 (10) , pi be 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 6= j and the intersection is non-empty. Then both πi, πj : Uij → Ω. Using the Cambpell-Baker-Hausdorﬀ −1 formula, show that the transition maps πi ◦ πj are analytic. c) Indicate that the Lie algebra structure (itself determined by the dif- ferentiable structure of G) uniquely determines the analytic structure.