# Spring 2022

### Time and Place

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

### Examination

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:

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 $$SO(3)$$.

• Solve Problems 2.7, 2.8, 2.9 and 2.10 in Kirillov, Jr.

#### 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) $$\sum_n m_{2n}=\dim V$$ and $$\sum_n m_{2n+1}=\dim V$$.
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 $$C=U(\mathfrak{g})^\mathfrak{h}=U(\mathfrak{g})=\{u\in U(\mathfrak{g})\mid uh=hu\,\forall h\in\mathfrak{h}\}$$ 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.

Lecture 27: Simple Roots

Lecture 28: Cartan Matrices and Dynkin Diagrams

Lecture 29: Classification of Root Systems

Lecture 30: 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.

Lecture 31: Universal enveloping algebra

Lecture 32: The PBW Theorem

Lecture 33: Highest Weight Theory

Lecture 34: Verma Modules

Lecture 35: Classification of Finite-Dimensional Irreducible Representations

Lecture 36: Examples

Lecture 37: Central Characters

Lecture 38: Kac-Moody Algebras

Lecture 39: Hopf Algebras and Quantum Groups

Lecture 40: Hopf algebras contd.

Lecture 41: Tensor Products of Modules

Lecture 42: Tensor Categories and Knot Invariants

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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