MATH 744: Lie Groups I (Fall 2010)

Title: Lie Groups I
Course web site: http://www.math.umd.edu/~jmr/744/
Meeting times: MWF, 11:00am-11:50am (MTH 1313)
Instructor: Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. Office hours (starting the second week of class) are Mondays 1-2 and Wednesdays 1-2, or by appointment. The class will not meet on Friday, September 10, because of Rosh Hashanah.

Text: Brian Hall, Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics, Vol. 222, Springer, ISBN 978-0-387-40122-5. Other good references for certain topics are:

Prerequisite: A good background in undergraduate mathematics (especially linear algebra, abstract algebra, and vector calculus).

Catalog description: An introduction to the fundamentals of Lie groups, including some material on groups of matrices and Lie algebras.

Course Description:

This course will be a basic introduction to Lie group theory, starting from scratch. The subject is quite interdisciplinary, and will blend techniques and ideas from algebra, analysis, geometry, and topology. There will be also be brief mention of applications, for example to differential geometry and physics. I would like to do the basic structure theory, the methods for going back and forth between the group and the Lie algebra, and then some representation theory of compact connected Lie groups, leading to the Weyl character formula and perhaps even the Borel-Weil-Bott Theorem. It will not be possible to cover all of these topics, but we will do as many as time permits, depending on the interests of the class.

Course Requirements:

Homework will be assigned, collected and graded regularly, but there will be no exams. If you want to get an "A" in the course, you need to do most of the homework in a satisfactory way.

Interesting Links:

Homework Assignments

  1. (due Wed., Sept. 8) Hall, section 1.9, problems 6, 7, 8. Also do this additional problem: An element of O(n) is called a reflection if it fixes an (n − 1)-dimensional linear subspace of Rn (pointwise) and acts by −1 on the orthogonal complement of this subspace. Show that every element of O(n) is a product of finitely many reflections, and that its determinant is 1 if there are an even number of reflection factors, −1 if there are an odd number. Hint: first do the case n = 2, using Hall problem #6. For the general case, you can use the fact from linear algebra that every orthogonal matrix is conjugate to one which is a direct sum of 2 × 2 or 1 × 1 orthogonal blocks.
  2. (due Mon., Sept. 27) Hall, section 2.10, problems 20, 21, 25, 30. Hint for #21: The only case that takes work is when the dimension is 2 and there is a non-zero bracket. Show in this case that there is a basis X, Y, such that [X, Y] = Y. Hint for #25: For any connected matrix Lie group G, the image of the exponential map generates the group (algebraically), i.e., any element is a finite product of exponentials. But if the group is commutative, a product of exponentials is itself an exponential.
  3. (due Mon., Oct. 11) Hall, section 3.9, problems 6, 7, 9. Also do this additional problem: Classify all nilpotent Lie algebras g of dimension 3 or 4 over R. You should find that in dimension 3, such a Lie algebra is either abelian or isomorphic to the Heisenberg algebra h (i.e., the strictly upper-triangular 3 × 3 matrices). In dimension 4, it turns out that there are only two nonabelian possibilities (up to isomorphism). If [g, g] is non-zero and central, g is isomorphic to h ⊕ R (direct sum). Otherwise, you should find that g has a basis X, Y, Z, W, with [X, Y] = Z, [X, Z] = W, and with RY + RZ + RW abelian. (Hint for the 3-dimensional case: if g is nilpotent, then its center z is non-zero. The bracket then factors through an antisymmetric bilinear map on g/z. From this one can see that [g, g] is either 0 or one-dimensional. Also, the quotient Lie algebra g/z is nilpotent with dimension at most 2, hence by the previous homework assignment is abelian. That means [g, g] ⊆ z.)
    ♦     Partial solutions for the assignment above
  4. (due Mon., Oct. 25) Hall, section 4.11, problems 2 and 15. Also do the following problems.
    1. Let (π, V) be an irreducible (complex) representation of a group G, and consider the action π⊗1 of G on V ⊗ W, where W is a vector space of dimension r with the trivial action of G (where every group element acts as the identity). Determine the commuting ring of the representation of G on V ⊗ W, and determine all the invariant subspaces.
    2. Let π denote the standard representation of SU(2) on C2. Show that π⊗π decomposes as the direct sum of the trivial 1-dimensional representation and of the (complexified) 3-dimensional adjoint representation of SU(2). What is the decomposition of π⊗π⊗π into irreducibles? (Hint: there are three irreducible summands.)

    ♦ Partial solutions for the assignment above
  5. (due Mon., Nov. 8) Hall, section 5.8, problems 2, 6, 12.
    ♦ Partial solutions for the assignment above
  6. (due Mon., Nov. 22) Problem set on Lie's Theorem and Cartan's Criterion.
    1. Let g be a complex Lie subalgebra of gl(V), V a finite-dimensional complex vector space, and let h be the radical of g. Show that if X is in g, there is a basis for V in which h acts by upper-triangular matrices and [X, h] acts by strictly upper-triangular matrices. (Hint: apply Lie's Theorem to h + CX.) Deduce that if Y and Z lie in h, then TrV(Y[X, Z]) = 0.
    2. Let g be a complex Lie subalgebra of gl(V), V a finite-dimensional complex vector space. Assume that the symmetric bilinear form H(X, Y) = TrV(XY) on g is nondegenerate. Prove that g has abelian radical. (Note that this is not exactly the situation of the usual form of Cartan's criterion since V is not necessarily the adjoint representation. However, you can use the result of (a) to deduce that the radical of g is abelian.)
    3. Now show under the hypotheses of (b) that g is reductive. (Hint: you already know from (b) that the radical is abelian. Show that its orthogonal complement under H is a semisimple ideal s and that g = (rad g) + s.) Sorry, this is not easy just with the hypotheses given, but assume H is also nondegenerate on the radical.
    4. Give an example where g satisfies the hypothesis of (b) but is not semisimple.
    5. Let g be a semisimple complex Lie subalgebra of gl(V), V a finite-dimensional complex vector space. Show that the symmetric bilinear form H(X, Y) = TrV(XY) on g is nondegenerate. (This is one direction of the usual form of Cartan's criterion if V is the adjoint representation.)

    ♦ Solutions for the assignment above
  7. (due Monday, December 6)
    1. Let g = sl(n, C). Verify that g is simple of dimension n2 − 1, with the subalgebra h of diagonal matrices of trace 0 as a Cartan subalgebra. Find all the roots α (there are n2 − n of them) and for each one compute the corresponding element Hα. In what way does this example generalize what we did for SU(3)?
    2. Continuing with Exercise (a) above, show that one gets a system of n − 1 simple roots αi associated to the pairs (i, i + 1), i < n. Compute the Killing form B restricted to h, by computing B(Hαi, Hαj) for all the simple roots with 1 ≤ i, j ≤ n − 1. You should find that the matrix is tridiagonal.

    ♦ Solutions for the assignment above