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

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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

  • ꢀ ꢁ
  • ꢂ
  • ꢄ

ꢃ

ꢃꢃ

z −w

  • 2
  • 2

|z| + |w| = 1

(1) (2)

  • w
  • z

3

with the standard S topology. The usual basis for su(2) is

  • ꢁ
  • ꢂ
  • ꢁ
  • ꢂ
  • ꢁ
  • ꢂ

0 i

i 0

0 −1

i

0

  • X =
  • Y =
  • Z =

  • 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

  • ꢁ
  • ꢂ

A −B

(3)

  • B
  • A

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

  • ꢁ
  • ꢂ

A −B

(4)

  • B
  • A

a) Show that the realification of complex matrices and complexification 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
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

  • ꢁ
  • ꢂ

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 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)

V

is an element of 2 g∗⊗g, which can actually be considered an element

V

of 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.

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

  • i
  • π

(8)
0

M

  • h
  • g

0is 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 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

i

πi : Ui → Ω, given by

πi , exp−1 ◦ Lp

(10)

−1

i

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 = 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

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