MATH 620X Lie Algebras and their Representations

Spring 2022

Syllabus

Instructor

Time and Place

Texts and Resources

Examination

Each lecture one problem is assigned. In addition there will be approximately four longer assignments.

Homework Problems

Homework 1 Due Wed Jan 26:

  1. 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.
  2. 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 \(SL_3(\mathbb{R})\)? \(SL_n(\mathbb{R})\)?
  3. Show that (path) connectedness on a manifold is an equivalence relation.
  4. 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.)
  5. 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:

  1. Prove that the tangent space \(T_pM\) does not depend on the choice of coordinate system around \(p\).
  2. Show that \(Sp(n)\subseteq SL(2n,\mathbb{R})\).
  3. Show that if \(x\) and \(y\) are \(n\times n\) matrices such that \(xy=yx\) then \(\mathrm{exp}(x)\mathrm{exp}(y)=\mathrm{exp}(x+y)\).
  4. 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:

  1. Show that a closed Lie subgroup is actually a Lie group.
  2. 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 \(exp(x), x\in sl(2,\mathbb{R})\)?)
  3. 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 \(\gamma\).
  4. 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.
  5. 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 \(SO(3)\).

Homework 4, due Wed Feb 23:

  1. Show that, in characteristic zero, the Lie algebra \(\mathfrak{sl}_n\) is simple.
  2. 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 \(\mathrm{ad}\;x\).
  3. Show that if \(\mathfrak{g}\) is a finite-dimensional Lie algebra such that there is a vector space decomposition \(\mathfrak{g}=I_1\oplus I_2\oplus\cdots\oplus I_n\) 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).

Homework 5, due Wed March 2:

  1. A Lie algebra is reductive if its radical coincides with the center. Show that any reductive Lie algebra has a non-degenerate invariant symmetric bilinear form.
  2. Let \(\mathfrak{g}\) be any Lie algebra. Show that \(H^1(\mathfrak{g},\mathfrak{g})\) is isomorphic to the space of outer derivations on \(\mathfrak{g}\), defined as the quotient \(\mathrm{Der}(\mathfrak{g})/(\mathrm{ad}\;\mathfrak{g})\).
  3. Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra with a non-degenerate invariant symmetric bilinear form, \(V\) and \(W\) two representations of \(\mathfrak{g}\), and \(T:V\to W\) an intertwining operator. Show that \(T\circ C_V=C_W\circ T\), where \(C_V\) and \(C_W\) are the respective Casimir operators.

Homework 6, due Wed March 9:

  1. Prove that if \(D\) is a derivation on a Lie algebra \(\mathfrak{g}\), then \(\big(D-(\lambda+\mu)\big)^N\big([x,y]\big)=\sum_{k=0}^N\binom{N}{k}\big[(D-\lambda)^{N-k}(x),\,(D-\mu)^k(y)\big]\) for all \(x,y\in\mathfrak{g}\) and all \(\lambda,\mu\in\Bbbk\).
  2. Verify that if \(V\) and \(W\) are representations of a Lie algebra \(\mathfrak{g}\), then \(\mathrm{Hom}(V,W)\) becomes a representation of \(\mathfrak{g}\) by defining \((x.f)(v)=x.(f(v))-f(x.v)\) for all \(x\in\mathfrak{g}, f\in\mathrm{Hom}(V,W), v\in V\).
  3. Let \(I\) be an abelian ideal of a Lie algebra \(\mathfrak{g}\) and let \(\mathfrak{h}=\mathfrak{g}/I\). Let \(s:\mathfrak{h}\to\mathfrak{g}\) be any linear map such that \(x=s(x)+I\) for all \(x\in\mathfrak{h}\). Define \(f:\mathfrak{h}\times\mathfrak{h}\to \mathfrak{g}\) by \(f(x,y)=[s(x),s(y)]-s([x,y])\). Prove that the image of \(f\) is contained in \(I\) and that \(f\in Z^2(\mathfrak{h}, I)\).

Assigment 2, due Wed March 23:

As always, assume \(\Bbbk\) has characteristic zero.

  1. For a non-negative integer \(n\), let \(V(n)=\mathrm{span} \{x_1^{n-k}x_2^k\}_{k=0}^n\) be the vector space of all degree \(n\) homogeneous polynomials expressions in two variables \(x_1\) and \(x_2\), equipped with \(\rho:\mathfrak{gl}(2,\Bbbk)\to\mathfrak{gl}(V(n))\) given by \(\rho(E_{ij})p(x_1,x_2)=x_i\frac{\partial}{\partial x_j}p(x_1,x_2)\). Show that \(\rho\) is a Lie algebra homomorphism, making \(V(n)\) a representation of \(\mathfrak{gl}(2,\Bbbk)\).
  2. Prove that \(V(n)\) is an irreducible representation of \(\mathfrak{gl}(2,\Bbbk)\). (Hint: First show that if \(U\) is a nonzero subrepresentation of \(V(n)\) then \(x_1^n\in U\).)
  3. Let \(\langle x,y\rangle=\mathrm{Tr}(xy)\) be the trace form on \(\mathfrak{gl}(2,\Bbbk)\). For a basis of your choice, find the corresponding dual basis for \(\mathfrak{gl}(2,\Bbbk)\).
  4. Describe the action of the Casimir operator \(C_{V(n)}\) on \(V(n)\).
  5. Now replace \(n\) by an arbitrary element \(\lambda\in\Bbbk\) and let \(V(\lambda)=\mathrm{span}\{x_1^{\lambda-k}x_2^k\}_{k=0}^\infty\) be an infinite-dimensional vector space with basis being formal expressions \(x_1^{\lambda-k}x_2^k\), \(k=0,1,\ldots\). Using the same formula for \(\rho(E_{ij})\), convince yourself that \(V(\lambda)\) is still a (not necessarily irreducible) representation of \(\mathfrak{gl}(2,\Bbbk)\) (it is a so called contragredient Verma module) and find the action of the Casimir operator \(C_{V(\lambda)}\).

Homework 7, due Wed March 30:

  1. Find the set of roots of the Lie algebra \(\mathfrak{g}=\mathfrak{so}(5,\mathbb{C})\). Use the Cartan subalgebra consisting of all anti-diagonal matrices which belong to \(\mathfrak{g}\).
  2. Prove that if \(\alpha\) is a root of a semisimple Lie algebra \(\mathfrak{g}\) with CSA \(\mathfrak{h}\) and \(H_\alpha\in\mathfrak{h}\) is defined by \(\kappa(H_\alpha,h)=\alpha(h)\) for all \(h\in\mathfrak{h}\), then \(\kappa(H_\alpha,H_\alpha)\neq 0\). (Here \(\kappa\) denotes the Killing form but any NISBF can be used.)
  3. Exercise 4.11 in book: Let \(V\) be a finite-dimensional representation of \(\mathfrak{sl}(2,\mathbb{C})\) and let \(V\cong\bigoplus_{n=0}^\infty V(n)^{\oplus m_n}\) be the decomposition of \(V\) into irreducible representations (Weyl's Theorem). Here \(V(n)\) denotes the f.d. irrep of dimension \(n+1\) and \(m_n\) are non-negative integers (at most finitely many nonzero) called multiplicities indicating how many times \(V(n)\) occurs in \(V\). For \(\mu\in\mathbb{C}\), let \(V[\mu]\) denote the weight space of weight \(\mu\) (i.e. eigenspace of \(h\) with eigenvalue \(\mu\)). Prove that (a) \(m_n=V[n]-V[n+2]\), and (b) \(\sum_n m_{2n}=\dim V[0]\) and \(\sum_n m_{2n+1}=\dim V[1]\).
  4. Let \(R\subset E\) be any root system and suppose that \(\alpha\) and \(\beta\) are non-parallel elements of \(R\). Let \(E_2=\mathbb{R}\alpha\oplus\mathbb{R}\beta\). Prove that \(R\cap E_2\) is a root system in the Euclidean vector space \(E_2\).

Homework 8, due Wed April 6:

  1. Let \(R\subset E\) be a root system. Show that the following three statements are equivalent: (i) There are nonempty root systems \(R_1,R_2\subset E\) such that \(R_1\perp R_2\) (i.e. \((\beta_1,\beta_2)=0\forall\beta_i\in R_i\)), and \(R\) is the disjoint union of \(R_1\) and \(R_2\); (ii) Relative to any polarization, the set of simple roots \(\Pi=\{\alpha_1,\ldots,\alpha_r\}\) of \(R\) can be decomposed as a nontrivial disjoint union \(\Pi=\Pi_1\sqcup\Pi_2\) such that \(\Pi_1\perp\Pi_2\); (iii) The Cartan matrix \(A=(a_{ij})_{1\le i,j\le r}\), \(a_{ij}=n_{\alpha_i,\alpha_j}\) is, after conjugating by a permutation matrix if necessary, a block matrix with at least two blocks. A root system satisfying either of these conditions is called reducible. Otherwise \(R\) is called irreducible.
  2. Let \(\mathfrak{g}\) be a semisimple Lie algebra. Show that \(\mathfrak{g}\) is simple if and only if its root system is irreducible.
  3. Find and prove a corrected version of Exercise 7.3 in the book which says "Let \(S=\{v_i\}_i\) be a subset of a Euclidean space \(E\) such that \((v_i, v_j)\le 0\) for all \(i\neq j\). Show that \(S\) is a linearly independent set."
  4. Read about the Lie algebra \(\mathfrak{sp}(4)\) in the lecture notes. Show that its root system is isomorphic to the root system you found for \(\mathfrak{so}(5)\).

Homework 9, due Fri April 15:

  1. Show that \(U(\mathfrak{g}_1\times\mathfrak{g}_2)\cong U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)\) for any Lie algebras \(\mathfrak{g}_i\).
  2. Show that if \(x\in\mathfrak{g}\) then \(\mathrm{ad}\;x\) extends to a derivation on \(U(\mathfrak{g})\). Furthermore, show that if \(\mathfrak{g}\) is semisimple and \(\alpha\) is a root and \(x\in\mathfrak{g}_\alpha\) then \(\mathrm{ad}\;x\) is locally nilpotent on \(U(\mathfrak{g})\) (that is, for any \(u\in U(\mathfrak{g})\) there exists \(N>0\) such that \((\mathrm{ad}\;x)^Nu=0\)). Conclude that \(\exp(\mathrm{ad}\; x)\) is a well-defined automorphism of \(U(\mathfrak{g})\).
  3. Let \(\mathfrak{g}\) be a semisimple Lie algebra with Cartan subalgebra \(\mathfrak{h}\). Show that there exists an anti-automorphism \(\tau\) of \(\mathfrak{g}\) (i.e. a linear bijection such that \(\tau([x,y])=[\tau(y),\tau(x)]\)) which satisfies \(\tau(\mathfrak{g}_\alpha)=\mathfrak{g}_{-\alpha}\) for all roots \(\alpha\) and is the identity on \(\mathfrak{h}\).
  4. Let \(\mathcal{D}_1\) and \(\mathcal{D}_2\) be Dynkin diagrams and \(\theta\) be a morphism of Dynkin diagrams (an injective map from the vertex set of \(\mathcal{D}_1\) to the vertex set of \(\mathcal{D}_2\) such that if \(ij\) is an edge then \(\theta(i)\theta(j)\) is an edge of the same type, including multiplicity and orientation). Show that \(\theta\) induces a Lie algebra homomorphism from \(\mathfrak{g}(\mathcal{D}_1)\to\mathfrak{g}(\mathcal{D}_2)\).

Homework 10, due Wed April 20:

  1. Let \(R\) be a polarized root system and \(Q=\mathbb{Z}R=\mathbb{Z}\Pi\) be the root lattice and \(Q^+=\mathbb{Z}_{\ge 0}\Pi\) be the positive cone consisting of all non-negative integer linear combinations of simple roots. The Kostant partition function \(p:\mathfrak{h}^\ast\to\mathbb{Z}_{\ge 0}\) is defined as follows: \(p(\lambda)\) is the number of ways to write \(\lambda\) as a sum of positive roots (up to reordering of the terms). More precisely, \(p(\lambda)\) is the cardinality of the set \(\{n\in (\mathbb{Z}_{\ge 0})^{R_+}\mid \sum_{\alpha\in R_+} n(\alpha)\alpha=\lambda\}\). Note that \(p(\lambda)=0\) unless \(\lambda\in Q^+\). Show that \(p(\lambda)=\dim U(\mathfrak{n}_+)_\lambda=\dim U(\mathfrak{n}_-)_{-\lambda}\) where \(U(\mathfrak{n}_\pm)\) are regarded as a weight module with respect to the adjoint action of \(\mathfrak{h}\).
  2. Show that if \(V\) is a highest weight module with highest weight \(\lambda\), then \(\dim V_\mu \le p(\lambda-\mu)\) for all \(\mu\in\mathfrak{h}^\ast\).
  3. Define a linear map \(\varphi:U(\mathfrak{g})\to U(\mathfrak{h})\) by requiring \(\varphi(\mathfrak{n}_-U(\mathfrak{g}))=0\), \(\varphi(U(\mathfrak{g})\mathfrak{n}_+)=0\), \(\varphi\big|_{U(\mathfrak{h})}=\mathrm{Id}_{U(\mathfrak{h})}\). Let \(C=U(\mathfrak{g})^\mathfrak{h}=U(\mathfrak{g})[0]=\{u\in U(\mathfrak{g})\mid uh=hu\,\forall h\in\mathfrak{h}\}\) be the centralizer of \(\mathfrak{h}\) in \(U(\mathfrak{g})\). Show that the kernel of \(\varphi\big|_C\) is a two-sided ideal of \(C\). Conclude that \(\varphi\big|_C\) is an algebra homomorphism.
  4. For \(\lambda\in\mathfrak{h}^\ast\), let \(W(\lambda)=\Bbbk_\lambda \otimes_{U(\mathfrak{b}_-)} U(\mathfrak{g})\) be the right-handed analog of the Verma module. The Cartan involution \(\tau\) from previous HW gives a map \(\tilde{\tau}:M(\lambda)\to W(\lambda)\) defined by \(\tilde{\tau}(u\otimes v_\lambda)=v_\lambda\otimes\tau(u)\). Define a bilinear map on \(M(\lambda)\) by \((x,y)=\tilde{\tau}(x)\otimes y\in W(\lambda)\otimes_{U(\mathfrak{g})} M(\lambda)\). Show that the bilinear form is actually scalar valued, and satisfies \((ux,y)=(x,\tau(u)y)\) for all \(u\in U(\mathfrak{g})\).
  5. Prove that the radical \(\{v\in M(\lambda)\mid (v,M(\lambda))=0\}\) coincides with the unique maximal submodule \(N(\lambda)\) of \(M(\lambda)\).

Homework 11, due Wed April 27:

  1. Let \(\mathfrak{g}=\mathfrak{sl}(n)\). Let \(\rho\) be the Weyl vector (half the sum of the positive roots). Verify the so called strange formula: \((\rho,\rho)=\frac{1}{24}\dim \mathfrak{g}\) where \((\cdot,\cdot)\) is the Killing form. (You may use a known relationship between Killing form and trace form, see Exercise 5.2 in Kirillov, Jr.) For more info, see this paper. (There is also a very strange formula, due to Kac, related to affine Lie algebras.)
  2. For each root system of rank two, find the fundamental weights \(\omega_1, \omega_2\) expressed in terms of the simple roots \(\alpha_1, \alpha_2\) (chosen, say, such that \(|\alpha_1|\le |\alpha_2|\)). In each case, draw the root systems and sketch the points of the weight lattice \(P=\mathbb{Z}\omega_1\oplus\mathbb{Z}\omega_2\). Mark the portion consisting of the dominant integral weights \(P_+\).
  3. Let \(V=\mathbb{k}^3\) be the tautological representation of \(\mathfrak{sl}(3,\Bbbk)\). Find the dominant integral weight \(\lambda=k_1\omega_1+k_2\omega_2\) such that \(V\simeq L(\lambda)\). Plot the support of \(V\) in the weight lattice \(P\) of \(\mathfrak{sl}(3,\Bbbk)\).
  4. Let \(V\) be a representation of a Lie algebra \(\mathfrak{g}\). Let \(V\wedge V\) be the subspace of \(V\otimes V\) spanned by the set of anti-symmetric tensors \(\{u\wedge v:=u\otimes v - v\otimes u\mid u,v\in V\}\). Show that \(V\wedge V\) is a representation with respect to the action \(x.(u\wedge v)=(x.u)\wedge v + u\wedge (x.v)\) and that \(\dim V\wedge V=\binom{\dim V}{2}\).
  5. Combining Problems 3 and 4: Let \(V=\mathbb{k}^3\) be the tautological representation of \(\mathfrak{sl}(3,\Bbbk)\). Then \(V\wedge V\) is an irreducible representation (optional: check this). Find the dominant integral weight \(\lambda=k_1\omega_1+k_2\omega_2\) such that \(V\wedge V\simeq L(\lambda)\). Plot the support of \(V\wedge V\) in the weight lattice \(P\) of \(\mathfrak{sl}(3,\Bbbk)\).

Homework 12, due Wed May 4:

  1. Let \(\mathfrak{g}\) be the (untwisted) affine Kac-Moody algebra associated to the Dynkin diagram \(A_1^{(1)}\). Show that there is a (non-trivial) Lie algebra map \(\varphi:\mathfrak{g}\to\widehat{\mathfrak{sl}}_2=\mathfrak{sl}_2\otimes\Bbbk[t,t^{-1}]\oplus\Bbbk c \oplus\Bbbk d\) by defining it on generators and showing the defining relations are preserved. The index set \(I\) is usually chosen as \(\{0,1\}\) and the Cartan datum chosen so that \(\alpha_0(d_1)=1\) and \(\alpha_1(d_1)=0\). Hint: Look for a map such that \(\varphi(e_1)=e\otimes 1\) while \(\varphi(e_0)=f\otimes t\). (In fact the two Lie algebras are isomorphic.)
  2. Let \(W\) be the Weyl group of the Kac-Moody algebra from the previous problem. Show that \(W\) is isomorphic to the semidirect product of the symmetric group \(S_2\) and a free abelian group. (Look at how the two simple reflections act on \(\mathfrak{h}^\ast\).)
  3. Show that if \(B\) is an algebra which is also a coalgebra, then \(\Delta\) and \(\epsilon\) are algebra maps if and only if \(m\) and \(u\) are coalgebra maps.

Homework 13, due Wed May 11:

  1. Let \(U\) be the \(\Bbbk[q,q^{-1}]\) -subalgebra of \(U_q(\mathfrak{sl}_2)\) generated by \(E,F,K,K^{-1}\). Show that there is a \(\Bbbk\) -algebra map \(\varphi:U\to U_\hbar(\mathfrak{sl}_2)\) given by \(\varphi(q^{\pm 1})=\exp(\pm\hbar), \varphi(E)=E, \varphi(F)=F, \varphi(K^{\pm 1})=\exp(\pm\hbar H)\).
  2. Let \(V\) be the \(2\) -dimensional vector space over \(\Bbbk(q)\) with basis \(e_-, e_+\). Show there is a representation of \(U_q(\mathfrak{sl}_2)\) on \(V\) determined by \(K.e_\pm=q^{\pm 1}e_{\pm}, E.e_+=0, E.e_-=e_+, F.e_+=e_-, F.e_-=0\).
  3. Let \(H\) be a Hopf algebra and let \(R\in H\otimes H\). For any two left \(H\) -modules \(V\) and \(W\), define \(c_{V,W}:V\otimes W\to W\otimes V\) by \(c_{V,W}(v\otimes w)=R\cdot (w\otimes v)\). Show that the braid relation (hexagon identity) \((c_{V,W}\otimes \mathrm{Id}_U)\circ(\mathrm{Id}_V\otimes c_{U,W})\circ(c_{U,V}\otimes \mathrm{Id}_W)=(\mathrm{Id}_W\otimes c_{U,V})\circ (c_{U,W}\otimes \mathrm{Id}_V)\circ (\mathrm{Id}_U\otimes c_{V,W})\) holds for all \(U,V,W\) if and only if \(R\) satisfies the quantum Yang-Baxter Equation \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\) where \(R_{12}=R\otimes 1_H\), \(R_{23}=1_H\otimes R\), \(R_{13}=(\mathrm{Id}_H\otimes\tau)(R\otimes 1_H)=(\tau\otimes \mathrm{Id}_H)(1_H\otimes R)\) where \(\tau:x\otimes y\mapsto y\otimes x\). (Hint: Take \(U=V=W=H\) the left regular representation.)
  4. Let \(R=\Bbbk[\hbar]/(\hbar^2)\). Let \(A\) be an associative \(\Bbbk\) -algebra and define an \(R\) -bilinear multiplication \(\ast\) on \(A[\hbar]/(\hbar^2)\) by \(a\ast b = ab+f(a,b)\hbar\) for \(a,b\in A\), where \(f:A\times A\to A\) is some \(R\) -bilinear map. (i) Show that \(\ast\) is associative if and only if \(f(a,b)\) satisfies the Hochschild \(2\) -cocycle identity: \(af(b,c)-f(ab,c)+f(a,bc)-f(a,b)c=0\). (ii) Show that this deformed algebra \(\big(A[\hbar]/(\hbar^2), \ast\big)\) is isomorphic to the usual algebra \(A[\hbar]/(\hbar^2)\) if and only if \(f(a,b)\) satisfies the Hochschild \(2\) -coboundary condition that \(f(a,b)=a\delta(b)-\delta(ab)+\delta(a)b\) for some \(R\) -linear map \(\delta:A\to A\).
  5. Select any problem that you didn't already solve from a previous homework set.

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 5-6: Classical groups

Lecture 7: Open, immersed, embedded submanifolds. Closed Lie subgroups.

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: Consequences for semisimple Lie algebras; The Casimir Operator.

Lecture 17: Chevalley-Eilenberg cohomology

Lecture 18: Proof of Whitehead's First Lemma

Lecture 19: The Abstract Jordan Decomposition

Lecture 20: Weyl's Theorem on Complete Reducibility. Short exact sequences of Lie algebras.

Lecture 21: Proof of Whitehead's Second Lemma and Levi's Theorem

Classification of Semisimple Lie Algebras

Lecture 22: Cartan subalgebras

Lecture 23: Root space decomposition

Lecture 24: Representations of \(\mathfrak{sl}(2)\)

Lecture ??: Discussion of (contragredient) Verma modules and the universal enveloping algebra

Lecture 25: The root system of a semisimple Lie algebra

Lecture 26: Abstract root systems. The root systems of rank at most 2.

Positive roots - simple roots - sp(4)

Cartan matrices and Dynkin diagrams

Serre's Theorem

Representation Theory

Already covered throughout the course so far: Subrepresentations, direct sums, Hom(V,W), invariants. Irreducible and completely reducible representations. Intertwining operators.

Schur's Lemma. Characters. Tensor products and the dual of representations. Clebsch-Gordan decomposition.

The universal enveloping algebra

The Poincare-Birkhoff-Witt theorem

Highest weight theory, Verma modules, category O.

Topics from Lie group representation theory (read yourselves):

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.

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