Lie groups and Lie algebras: Fortnight 1 Questions

Grading

Remember, you'll need to do at least 4, 4, or 5 questions to get a C, B, or A respectively. For a B or an A, this will need to include 2 (respectively 3) α questions.

β questions

Answer as many as you want.

1. Exp and eigenvectors

(Watch the videos about matrix exponentiation and its properties first)

  1. Suppose that M is a matrix and v is an eigenvector of M with eigenvalue λ . Show that v is an eigenvector of exp ( M ) with eigenvalue e λ .

  2. For a vector v = ( x , y , z ) , define K v := ( 0 - z y z 0 - x - y x 0 ) . Find the 0 -eigenspace of K v and deduce that v is fixed by exp ( θ K v ) for any θ 𝐑 .

2. Exp and rotations

Depends on the previous question

  1. Given vectors v , w 𝐑 3 , show that K v w = v × w .

  2. Given a unit vector u , pick vectors v and w such that u , v , w is a right-handed orthonormal basis of 𝐑 3 (so u × v = w and u × w = - v ). Calculate exp ( θ K u ) v and exp ( θ K u ) w and deduce that exp ( θ K u ) is a rotation. What are the axis and angle of rotation?

3. Jacobi identity

(Technically no prerequisites, but the video about abstract Lie algebras is related.)

  1. Prove the Jacobi identity [ X , [ Y , Z ] ] + [ Y , [ Z , X ] ] + [ Z , [ X , Y ] ] = 0 for matrices X , Y , Z where [ A , B ] := A B - B A .

  2. Given a matrix X , define ad X : 𝔤 𝔩 ( n , 𝐑 ) 𝔤 𝔩 ( n , 𝐑 ) by ad X ( Y ) := [ X , Y ] . Prove that [ ad X , ad Y ] = ad [ X , Y ] . (Hint: evaluate both sides on some Z 𝔤 𝔩 ( n , 𝐑 ) and use the Jacobi identity.)

4. Unitary group

(Watch the video about orthogonal matrices first)

The unitary group U ( n ) is the group of complex n -by- n matrices M such that M M = I . Here, M is the conjugate-transpose of M , for example ( a b c d ) = ( a ¯ c ¯ b ¯ d ¯ ) . Show that the Lie algebra 𝔲 ( n ) of U ( n ) is the space of n -by- n skew-Hermitian matrices { M : M = - M } .

5. Example: surjective exp

(You need to remember Jordan normal form and watch the first video about matrix exponentiation).

We will show that exp : 𝔤 𝔩 ( 2 , 𝐂 ) G L ( 2 , 𝐂 ) is surjective.

  1. Calculate exp ( a b 0 a ) , and hence find a logarithm for the matrix ( λ 1 0 λ ) for any λ 0 𝐂 .

  2. If M G L ( 2 , 𝐂 ) , let N = P - 1 M P be its Jordan normal form. Prove that N = exp ( X ) for some X 𝔤 𝔩 ( 2 , 𝐂 ) and deduce that M = exp ( Y ) for some Y 𝔤 𝔩 ( 2 , 𝐂 ) .

α questions

Answer as many as you want. You will need to do well on at least 2 to get a B- and at least 3 to get an A-.

6. BCH formula

(Watch the video about the BCH formula first. WARNING: This question gets messy before it gets better.)

Show that the third-order term in the Baker-Campbell-Hausdorff formula is 1 12 [ X , [ X , Y ] ] - 1 12 [ Y , [ X , Y ] ] .

7. det(exp) = exp(tr), part 1

(Technically, this has no prerequisites, but the video about S L ( 2 , 𝐂 ) is related)

Let H 𝔤 𝔩 ( n , 𝐑 ) be a matrix.

  1. Show that det ( I + t H ) = 1 + t Tr ( H ) + 𝒪 ( t 2 ) where Tr ( H ) = H 11 + H 22 + + H n n is the trace of H . Hint: Write it out in full, i.e. det ( I + t H ) = det ( 1 + t H 11 t H 12 t H 1 n t H 21 1 + t H 22 t H n 1 t H n 2 1 + t H n n )

  2. Deduce that if M is invertible then det ( M + t H ) = det ( M ) + t det ( M ) Tr ( M - 1 H ) + 𝒪 ( t 2 ) .

  3. If A ( t ) is a path of matrices, use the previous parts to show that d d t det ( A ( t ) ) = det ( A ( t ) ) Tr ( A ( t ) - 1 d A d t ( t ) ) (Hint: Use the Taylor expansion of A ( t ) to find the Taylor expansion of det ( A ( t ) ) and drop terms of higher order.)

Note: By a "path of matrices" I basically mean a matrix A ( t ) whose entries depend on t . You may assume that this dependence is sufficiently differentiable to run your argument.

8. det(exp) = exp(tr), part 2

(Depends on the previous question, the properties of the exponential map, and for the final part the definition of the Lie algebra of a matrix group)

  1. Using the formula for d d t det ( A ( t ) ) from the previous question, show that if ϕ ( t ) := det ( exp ( t H ) ) then ϕ satisfies the differential equation d ϕ d t = ϕ ( t ) Tr ( H ) .

  2. Assuming that differential equations have unique solutions (with specified initial conditions), prove that det ( exp ( t H ) ) = exp ( t Tr ( H ) ) .

  3. Using this formula, show that the Lie algebra 𝔰 𝔩 ( n , 𝐑 ) of S L ( n , 𝐑 ) = { M G L ( n , 𝐑 ) : det ( M ) = 1 } is the space of matrices with trace zero.

9. Example: non-surjective exp

(You need to remember Jordan normal form. Recall that 𝔰 𝔩 ( 2 , 𝐑 ) is the group of 2-by-2 tracefree matrices and S L ( 2 , 𝐑 ) is the group of 2-by-2 matrices with determinant 1. I also highly recommend trying the previous question first).

We will show that exp : 𝔰 𝔩 ( 2 , 𝐑 ) S L ( 2 , 𝐑 ) is not surjective.

  1. Given B 𝔰 𝔩 ( 2 , 𝐑 ) , show that its Jordan normal form (when considered as a complex matrix) is one of ( λ 0 0 - λ ) , λ 𝐑  or  i 𝐑 , ( 0 1 0 0 ) .

  2. Deduce that one of the following must be true about exp ( B ) :

    1. both its eigenvalues are positive;

    2. both its eigenvalues are unit complex numbers;

    3. both its eigenvalues are equal to 1.

  3. By exhibiting a matrix in S L ( 2 , 𝐑 ) whose eigenvalues satisfy none of these conditions, deduce that exp : 𝔰 𝔩 ( 2 , 𝐑 ) S L ( 2 , 𝐑 ) is not surjective.

10. Simultaneous strict upper-triangularisability

(Just uses good ol'-fashioned linear algebra. It will become relevant if you choose to do a project about Lie's theorem or Engel's theorem on solvable (respectively nilpotent) Lie algebras.)

Recall that a linear map N is called nilpotent if N k = 0 for some k . In this question, you may use the fact that, if N is a nilpotent linear map then there is some basis with respect to which its matrix is strictly upper-triangular.

Let V be a vector space and suppose that M : V V and N : V V are linear maps which are both nilpotent and which commute with one another: M N = N M .

  1. Show that if v M j V then N v M j V . We will write N j : M j V M j V for the restriction N | M j V .

  2. Suppose we have picked a basis of M j V such that the matrix of N j is strictly upper-triangular. Show that we can extend this to get a basis of M j - 1 V for which the matrix of N j - 1 is strictly upper-triangular. (Hint: Pick a complement C for M j V M j - 1 V and consider the map C C induced by N ).

  3. Use this to prove by induction that we can find a basis of V making both M and N simultaneously upper-triangular.

11. Simultaneous upper triangularisability

(Uses Q.10).

Let V be a complex vector space and X : V V be a linear map. Recall that the generalised λ -eigenspace of a linear map X : V V is the subspace V λ := { v V : ( X - λ I ) k v = 0  for some  k } . In this question, you may assume that V = λ V λ : that is, the generalised eigenspaces span V and V λ V μ = 0 unless λ = μ .

Now let X : V V and Y : V V be linear maps and suppose they commute with one another.

  1. Show that Y preserves the generalised eigenspaces of X . Let Y λ : V λ V λ be the restriction Y | V λ .

  2. Let V λ , μ be the μ -generalised eigenspace of Y λ . Show that it is preserved by X .

  3. Deduce that both X and Y are block-diagonal with respect to the splitting V = λ , μ V λ , μ .

  4. Show that we can pick a basis for V λ , μ such that the corresponding blocks of X and Y are upper-triangular. (Hint: Consider X - λ I and Y - μ I and use the previous question about strict upper triangularisability.)

12. Technicality about local coordinates

(Depends on an optional video about invertibility of exp : 𝔤 G )

By mimicking the proof that exp is locally invertible, prove the following claim (which was used in this optional video to prove that exp : 𝔤 G is locally invertible for a matrix group G ):

If 𝔤 𝔤 𝔩 ( n , 𝐑 ) is a Lie subalgebra and W 𝔤 𝔩 ( n , 𝐑 ) is a vector space complement for 𝔤 , show that the map F : 𝔤 W G L ( n , 𝐑 ) defined by F ( v , w ) = exp ( v ) exp ( w ) is locally invertible.