Each lecture one problem is assigned. In addition there will be approximately four longer assignments.
Homework Problems
Each problem is graded on a scale from 0 (you didn't hand it in) to 2 (it is substantially correct).
On each homework set you can earn at most 6 points (that is, it suffices to select 3 to hand in.)
Homework 1 Due Wed Jan 26:
Show that if \(M\subseteq\mathbb{R}^m\) is a \(k\) dimensional manifold and \(N\subseteq\mathbb{R}^n\) is an \(l\) dimensional manifold then \(M\times N\subseteq\mathbb{R}^{m+n}\) is naturally a \((k+l)\) dimensional manifold.
Let \(SL_2(\mathbb{R})\) denote the set of \(2\times 2\) matrices of determinant one. Prove that \(SL_2(\mathbb{R})\) is a 3-dimensional manifold. Can you generalize to \(SL_3(\mathbb{R})\)? \(SL_n(\mathbb{R})\)?
Show that (path) connectedness on a manifold is an equivalence relation.
Show that the Lie group \(SO(3)\) of all 3x3 orthogonal matrices of determinant one, is connected. (Hint: It may help to first show that any element has an eigenvector with eigenvalue one.)
Let \(M\) be a connected manifold and \(x_0\in M\) a base point. Show that the set of homotopy classes of loops \(\pi_1(M,x_0)\) is a group with respect to the operation \([\gamma][\delta]=[\gamma\ast\delta]\).
Homework 2, due Wed Feb 2:
Prove that the tangent space \(T_pM\) does not depend on the choice of coordinate system around \(p\).
Show that \(Sp(n)\subseteq SL(2n,\mathbb{R})\).
Show that if \(x\) and \(y\) are \(n\times n\) matrices such that \(xy=yx\) then \(\mathrm{exp}(x)\mathrm{exp}(y)=\mathrm{exp}(x+y)\).
Show that if \(G=GL(n,\mathbb{R})\) (which is an open subset of \(\mathbb{R}^{n^2}\), hence the tangent space at every point can be canonically identified with \(\mathbb{R}^{n^2}\)), then \((L_g)_\ast v = gv\) where the product in the right hand side is usual product of square matrices.
Homework 3, due Wed Feb 9:
Show that a closed Lie subgroup is actually a Lie group.
Show that the matrix \(\begin{bmatrix}-1&1\\0&-1\end{bmatrix}\in SL(2,\mathbb{R})\) is not in the image of the exponential map. (Hint: What are the eigenvalues of \(exp(x), x\in sl(2,\mathbb{R})\)?)
Show that if \(\gamma:I\to G\) is a smooth map to a Lie group \(G\) from an open interval \(I\subseteq\mathbb{R}\) containing zero, such that \(\gamma(s+t)=\gamma(s)\gamma(t)\) for all \(s,t\in I\) such that \(s+t\in I\) then there exists a unique one-parameter subgroup \(\tilde{\gamma}:\mathbb{R}\to G\) whose restriction to \(I\) equals \(\gamma\).
Suppose that \(\gamma:\mathbb{R}\to Sp(n)\) is a one-parameter subgroup. Use that \(\gamma(t)\in Sp(n)\) for all \(t\), and differentiate the identity that \(\gamma(t)\) satisfies. Plug in \(t=0\) to get an identity that the tangent vector \(\dot{\gamma}(0)\) must satisfy. Use this information to describe the symplectic Lie algebra \(sp(n)\) and find a basis for \(sp(n)\). You may assume \(n=2\) if it helps.
Show that if \(X\) and \(Y\) are left invariant vector fields, then the vector field \([X,Y]\) is also left invariant.
Assignment 1, due Wed Feb 16:
The first longer assignment consists of an exploration of the Lie groups \(SU(2)\) and \(SO(3)\).
Solve Problems 2.7, 2.8, 2.9 and 2.10 in Kirillov, Jr.
Homework 4, due Wed Feb 23:
Show that, in characteristic zero, the Lie algebra \(\mathfrak{sl}_n\) is simple.
Let \(x\) be a linear transformation of a finite-dimensional vector space over an algebraically closed field. Show that if \(x=d+n\) is the Jordan decomposition of \(x\), then \(\mathrm{ad}\;x=\mathrm{ad}\;d+\mathrm{ad}\;n\) is the Jordan decomposition for \(\mathrm{ad}\;x\).
Show that if \(\mathfrak{g}\) is a finite-dimensional Lie algebra such that there is a vector space decomposition \(\mathfrak{g}=I_1\oplus I_2\oplus\cdots I_n\) where each \(I_j\) is a simple (as a Lie algebra) ideal of \(\mathfrak{g}\), then \(\mathfrak{g}\) is semi-simple (i.e. has no nonzero solvable ideals).
Assigment 2, due Wed March 9:
(details to be posted)
Lecture Summary
Manifolds and Lie Groups
Lecture 1: Definition of manifolds and (real and complex) Lie groups. Implicit function theorem. Examples. Further reading: Spivak, Calculus on Manifolds. In particular Chapter 5.
Lecture 2: Connected manifolds and Lie groups. Discrete Lie groups. Open submanifolds. The connected component at the identity element. G/G0 is discrete.
Lecture 3: Fundamental group of a manifold. Simply connected manifolds and Lie groups. Universal cover of a manifold and Lie group. Further reading: Hatcher, Algebraic Topology. Chapter 1.1 and 1.3. In particular Theorem 1.38 and the paragraph that follows.
Lecture 4: Tangent spaces and vector fields on manifolds, derivative (differential) of a morphism, left-invariant vector fields, Lie algebra (as a vector space) of a Lie group.
Lecture 8: One-parameter subgroups, the exponential map.
Lecture 9: Vector fields as derivations on smooth function. The bracket on vector fields and on the tangent space of a Lie group at the identity.
Lie Algebras and their Structure
Read in Book: Baker-Campbell-Hausdorff formula. Fundamental theorems 3.40,3.41,3.42. Equivalence of categories (connected simply-connected Lie groups)/(finite-dimensional Lie algebras).
Lecture 10. Definition of Lie algebras, homomorphisms. Abelian Lie algebras. Lie subalgebras and Lie ideals. Products.
Lecture 11: Solvable Lie algebras and Lie's Theorem.
Lecture 12: Nilpotent Lie algebras and Engel's theorem.
Lecture 13: The radical. Semisimple and reductive Lie algebras.
Lecture 14: Jordan decomposition and Cartan's First Criterion.
Lecture 15: Cartan's Second Criterion. Characterization of semi-simple Lie algebras.
Lecture 16: Cohomology and Whitehead's Lemmas
Lecture 17: Weyl's Theorem on complete reducibility
Lecture 18: Cartan subalgebras
Lecture 19: Representations of sl(2,C)
Lecture 20: Root space decomposition
Lecture 21: Structure of semisimple Lie algebras
Lecture 22: Root systems
Lecture 23: Positive roots - simple roots - sp(4)
Lecture 24: Cartan matrices and Dynkin diagrams
Lecture 25: Serre's Theorem
Representation Theory
Lecture 26: Subrepresentations, direct sum, tensor product, and the dual of representations. Invariants.
Lecture 27: Irreducible and completely reducible representations. Intertwining operators.
Lecture 28: Schur's Lemma. Unitary representations. The Haar measure on a compact real Lie group.
Unitarizability of representations of compact real Lie groups. Characters. Matrix coefficients. Hilbert space of square-integrable functions on G.
Orthogonality of matrix coefficients. Peter-Weyl Theorem.
The universal enveloping algebra
The Poincare-Birkhoff-Witt theorem
Highest weight theory
Statement on Free Expression
Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.