- Lectures: MWF 3:20pm – 4:10pm Carver 290
- Office hours: MT 4:10pm – 6:00pm Carver 470

- Main text: Kirillov, Jr., An introduction to Lie Groups and Lie Algebras
- Michael Spivak, Calculus on Manifolds, Ch. 5
- Lecture notes

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

- 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.)

- 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 \([\gamma][\delta]=[\gamma\ast\delta]\).

- 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.

- 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.

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.

- 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 \(\mathfrak{g}=I_1\oplus I_2\oplus\cdots\oplus I_n\) 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).

- 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.
- 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 \(\mathrm{Der}(\mathfrak{g})/(\mathrm{ad}\;\mathfrak{g})\). - 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.

- 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 \(\lambda,\mu\in\Bbbk\).
- 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 \(x\in\mathfrak{g}, f\in\mathrm{Hom}(V,W), v\in V\).
- 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 \(f\in Z^2(\mathfrak{h}, I)\).

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

- 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 \(\mathfrak{gl}(2,\Bbbk)\).
- 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 \(x_1^n\in U\).)
- 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 \(\mathfrak{gl}(2,\Bbbk)\).
- Describe the action of the Casimir operator \(C_{V(n)}\) on \(V(n)\).
- 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 \(C_{V(\lambda)}\).

- 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 \(\mathfrak{g}\).
- 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.)
- 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 \(\sum_n m_{2n+1}=\dim V[1]\). - 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 \(E_2\).

- 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*. - Let \(\mathfrak{g}\) be a semisimple Lie algebra. Show that \(\mathfrak{g}\) is simple if and only if its root system is irreducible.
- 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."
- 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 \(\mathfrak{so}(5)\).

- Show that \(U(\mathfrak{g}_1\times\mathfrak{g}_2)\cong U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)\) for any Lie algebras \(\mathfrak{g}_i\).
- 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 \(U(\mathfrak{g})\).
- 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 \(\mathfrak{h}\).
- 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 \(\mathfrak{g}(\mathcal{D}_1)\to\mathfrak{g}(\mathcal{D}_2)\).

- 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 \(\mathfrak{h}\). - Show that if \(V\) is a highest weight module with highest weight \(\lambda\), then \(\dim V_\mu \le p(\lambda-\mu)\) for all \(\mu\in\mathfrak{h}^\ast\).
- 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.
- 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 \(u\in U(\mathfrak{g})\).
- Prove that the radical \(\{v\in M(\lambda)\mid (v,M(\lambda))=0\}\) coincides with the unique maximal submodule \(N(\lambda)\) of \(M(\lambda)\).

- 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.) - 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 \(P_+\).
- 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 \(\mathfrak{sl}(3,\Bbbk)\).
- 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 \(\dim V\wedge V=\binom{\dim V}{2}\).
- 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 \(\mathfrak{sl}(3,\Bbbk)\).

- 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.) - 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 \(\mathfrak{h}^\ast\).)
- 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.

- 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 \(\varphi(q^{\pm 1})=\exp(\pm\hbar), \varphi(E)=E, \varphi(F)=F, \varphi(K^{\pm 1})=\exp(\pm\hbar H)\).
- 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 \(K.e_\pm=q^{\pm 1}e_{\pm}, E.e_+=0, E.e_-=e_+, F.e_+=e_-, F.e_-=0\).
- 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.) - 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 \(\delta:A\to A\).
- Select any problem that you didn't already solve from a previous homework set.

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/G^{0} 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.

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

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

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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