Lie Groups and Matrix Lie Groups

We will identify the set of all real \(n\times n\) matrices with \(\mathbb {R}^{n^2}\) as usual. Show that \[ O(n)=\{M\mid M\text { is an orthogonal }n\times n\text { matrix}\} \] is a submanifold of \(\mathbb {R}^{n^2}\).

Let \(\alpha :G\times M\to M\) be an action of a Lie group \(G\) on a \(C^\infty \) manifold \(M\) such that \(\alpha \) is a continuous function. Prove or find a counterexample: if \(\alpha |_{G\times \{x\}}:G\to M\) is a \(C^\infty \) immersion for every \(x\in M\), then the function \(\alpha \) is \(C^\infty \).

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(SL(2,\mathbb {R})\subset \mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(SL(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{\mathrm {id}}SL(2,\mathbb {R})\) as a subspace of \(T_{\mathrm {id}}\mathcal {M}(2)=\mathbb {R}^4\).

Let \(G\) be an \(n\)-dimensional Lie group. Show that the tangent bundle \(TG\) is isomorphic as a bundle to the product bundle \(G\times \mathbb {R}^n\).

Let \(G_0\) and \(G_1\) be Lie groups. Then the product \(G_0\times G_1\) is a Lie group and we let \(p_i:G_0\times G_1\to G_i\) be the projection to each coordinate. Let \(X\) be a vector field on \(G_0\times G_1\) and \(Y_0\) and \(Y_1\) vector fields on \(G_0\) and \(G_1\) such that \(X\) and the \(Y_i\) are \(p_i\)-related. Show that \(X\) if left-invariant if and only if both \(Y_{0}\) and \(Y_{1}\) are left-invariant.

Recall that the orthogonal group \(O(n)\) is

where \(I\) is the identity matrix. Identifying the set \(\mathcal {M}_{n\times n}\) of all real \(n\times n\) matrices with \(\mathbb {R}^{n^2}\), show that \(O(n)\) is a submanifold of \(\mathcal {M}_{n\times n}\) and compute the tangent space to \(O(n)\) at \(I\) as a linear subspace of \(T_I\mathcal {M}_{n\times n}\), which is identified with \(\mathcal {M}_{n\times n}\).

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(SL(2,\mathbb {R})\subset \) \(\mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(SL(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{\mathrm {id}}SL(2,\mathbb {R})\) as a subspace of \(T_{\mathrm {id}}\mathcal {M}(2)=\mathbb {R}^4\).

Let \(\phi : G \to H\) be a Lie group homomorphism between Lie groups \(G\) and \(H\). Let \(X\) and \(Y\) be \(\phi \)-related vectors fields on \(G\) and \(H\), respectively, and assume that \(Y\) is left invariant. Is \(X\) left invariant? Either prove this or provide a counterexample.

Let \(\mathcal {M}(n)\) denote the space of all \(n\times n\) matrices with real entries, iden- tified with \(\mathbb {R}^{n^2}\), and let \(S(n)\) be the subspace of symmetric matrices. Consider the smooth map \(F:\mathcal {M}(n)\to S(n)\) defined by \(f(A)=AA^{\top }\), where \(A^{\top }\) is the transpose of \(A\).

1. Show that the identity matrix \(I\) is a regular value of \(f\) .

2. Deduce that the orthogonal group

   \[ O(n)=\{A\in \mathcal {M}(n)\mid AA^{\top }=I\} \]

   is a submanifold of \(\mathcal {M}(n)\). Carefully state any theorems you are citing.

Consider the Lie subgroup \(Z\) of the Heisenberg group \(H\). Find all connected two dimensional Lie subgroups of \(H\) that contain \(Z\). Describe them as both submanifolds of \(H\) and as groups. \[ H= \left \{ \begin {pmatrix} 1 & x & z\\ 0 & 1 & y\\ 0 & 0 & 1 \end {pmatrix} : x,y,z\in \mathbb {R} \right \},\qquad Z= \left \{ \begin {pmatrix} 1 & 0 & z\\ 0 & 1 & 0\\ 0 & 0 & 1 \end {pmatrix} : z\in \mathbb {R} \right \} \]

Consider the Lie Group \(G=\Ogrp (n)=\{A\in \mathcal {M}_n:A^TA=I\}\) where \(\mathcal {M}_n\cong \mathbb {R}^{n^2}\) is the collection of real \(n\times n\) matrices and \(A^T\) is the transpose of \(A\).

1. Describe the set of matrices which form the tangent space \(T_PG\) at the point \(P\in G\).
2. Given \(V\in T_IG\) and \(P\in G\), find \(X_V(P)\in T_PG\) where \(X_V\) is the left invariant vector field on \(G\) such that \(X_V(I)=V\). Show that your \(X_V\) is left invariant.
3. Let \(\theta _t^X\) be the flow for the left invariant vector field \(X=X_V\) for some \(V\in T_IG\). Find and verify a formula for \(\theta _t^X(P)\) in terms of \(P\), \(t\) and \(V\). Compute \(\theta _t^X(I)\) in case \(G=O(2)\) and \[ V=\begin {pmatrix}0&1\\-1&0\end {pmatrix}. \]

Find all connected Lie subgroups of \(\mathrm {SL}_2(\mathbb {R})\) that contains the subgroup \[ H= \left \{ \begin {pmatrix} 1 & x\\ 0 & 1 \end {pmatrix} : x\in \mathbb {R} \right \}. \]

Let the inclusion of \(\SO (2)\) into \(\SO (3)\) be induced by the inclusion of \(\mathbb {R}^2\) into \(\mathbb {R}^3\). Find all connected, 2-dimensional Lie subgroups of \(\SO (3)\) that contain \(\SO (2)\).

Let \(G\) be a Lie group with Lie algebra \(\mathfrak {g}\) and exponential map \(\exp :\mathfrak {g}\to G\). For a nonzero \(v\in \mathfrak {g}\), let \(L\) be the line spanned by \(v\). Prove that \(\exp |_L:L\to G\) is an immersion.

Let \(G\) be a Lie group, and let \(H\) be the connected component of \(G\) that contains the identity. Prove that \(H\) is a Lie subgroup of \(G\), and that \(H\) is a normal subgroup of \(G\). (Do not use the theorem that closed subgroups of Lie groups are Lie groups.)

Let \(G\) be a Lie group. Prove that \(G\) is orientable, and that if \(L_g\in \operatorname {Diff}(G)\) is left multiplication by \(g\in G\), then \(L_g\) preserves orientation.

Let \(\Phi :\mathfrak {sl}_2(\mathbb {R})\to \mathbb {R}\) be a Lie algebra homomorphism, where \(\mathfrak {sl}_2(\mathbb {R})\) is the Lie algebra of \(\mathrm {SL}_2(\mathbb {R})\) and \(\mathbb {R}\) is the Lie algebra of the Lie group \(\mathbb {R}\). Prove that \(\Phi =0\).

Let \(G\) be a connected Lie group with Lie algebra \(\mathfrak {g}\) such that \([X,Y]=0\) for all \(X,Y\in \mathfrak {g}\). Prove that \(G\) is abelian.

Show that \(S^n\) has a continuous field of nonzero tangent vectors iff \(n>0\) is odd. Conclude that the even-dimensional spheres are not Lie groups.

Prove that \(S^2\) is not a Lie group.

Show that the Lie algebra of \(\mathrm {SL}_3(\mathbb {R})\) is the space of \(3\times 3\) matrices whose trace equals 0.

Let \(G\) be a Lie group with Lie algebra \(\mathfrak {g}\). If \(\mathfrak {h}\) is a subalgebra of \(\mathfrak {g}\), then prove there is a Lie subgroup \(H\le G\) whose Lie algebra is \(\mathfrak {h}\).

Let \(G\) be an \(n\)-dimensional Lie group. Show that the tangent bundle \(TG\) is diffeomorphic to \(G\times \mathbb {R}^n\).

Prove that the fundamental group of any Lie group is abelian.

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(\mathrm {SL}(2,\mathbb {R})\subset \mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(\mathrm {SL}(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{id}\mathrm {SL}(2,\mathbb {R})\) as a subspace of \(T_{id}\mathcal {M}(2)=\mathbb {R}^4\). Is the line \[ \begin {pmatrix}t&0\\0&1\end {pmatrix},\quad t\in \mathbb {R} \] transverse to \(\mathrm {SL}(2,\mathbb {R})\)? Justify your answer.

Regard \(\SO (n)\) as a subset of the space of \(n\times n\) matrices, which in turn is identified with \(\mathbb {R}^{n^2}\). Show that \(\SO (n)\) is a submanifold and identify the tangent space to \(\SO (n)\) at the identity \(I\) (i.e. the Lie algebra of \(\SO (n)\)).

Let \(A\) be an element of the Lie algebra of \(\mathrm {GL}_n(\mathbb {R})\). What is the left-invariant vector field on \(\mathrm {GL}_n(\mathbb {R})\) that is associated to \(A\)? Show that it is left-invariant.

Let \(G\) be a Lie group, and let \(\widetilde {G}\) be the universal cover of \(G\). Endow \(\widetilde {G}\) with the structure of a Lie group.

Let \(G\) be a Lie group, and \(H\) a closed Lie subgroup. Sketch the proof that \(G/H\) is a manifold.

Show that \(\mathrm {SL}_n(\mathbb {R})\) is a manifold, and that it is a Lie group. (Do not use the theorem that any closed subgroup of a Lie group is a Lie group.)

1. What is a subalgebra of a Lie algebra?
2. Let \(G\) be a Lie group, with Lie algebra \(\mathfrak {g}\). Prove that any Lie subgroup \(H\) of \(G\) corresponds to a subalgebra of \(\mathfrak {g}\).
3. What is a smooth plane field on \(G\)?
4. Give an example of a foliation on a 2-torus that has compact leaves and noncompact leaves.
5. State Frobenius’ theorem.

6. Prove that any subalgebra of \(\mathfrak {g}\) corresponds to a foliation of \(G\).

7. Given a subalgebra of \(\mathfrak {g}\), and the corresponding foliation of \(G\), prove that the leaf containing the identity is a group, and thus that there is a correspondence between connected Lie subgroups of \(G\) and subalgebras of \(\mathfrak {g}\).

Let \(G\) be a Lie group and \(1\in G\) the identity element. Prove that \(\pi _1(G,1)\) is an abelian group.

Give a nowhere vanishing vector field on \(\mathrm {SL}_n(\mathbb {R})\).

Given a Lie group \(G\) with Lie algebra \(\mathfrak {g}\) and some \(v\in \mathfrak {g}\), let \(\{\theta _t^v\}_{t\in \mathbb {R}}\) be the corresponding 1-paramater group of diffeomorphisms of \(G\). Prove that \(\theta _t^v(gh)=g\theta _t^v(h)\) for any \(t\in \mathbb {R}\) and \(g,h\in G\).

Let \(U\) be a connected Lie subgroup of \(\mathrm {GL}_n(\mathbb {R})\). Suppose that every element of \(U\) has all of its entries below the main diagonal equal to 0, and all of its entries on the main diagonal equal to 1. Why is \(U\) diffeomorphic to \(\mathbb {R}^k\) for some \(k\)?

Let \(G\) be a Lie group with Lie algebra \(\mathfrak {g}\). Given a subalgebra \(\mathfrak {h}\subseteq \mathfrak {g}\), prove there is a Lie subgroup \(H\le G\) whose Lie algebra is \(\mathfrak {h}\). (The entire proof with all details would be best. If not, then try to write the main ideas of the proof.)

Let \(G\) be a Lie group with a discrete subgroup \(\Gamma \leq G\). Prove that \(\Gamma \) acts freely and properly discontinuously on \(G\). (It will follow that \(\Gamma \backslash G\) is a smooth manifold, but you don’t have to show this.)

Let \(G\cong \SO (2)\ltimes \mathbb {R}^4\) be the Lie group given by matrices \[ \begin {pmatrix} \cos (\theta )&-\sin (\theta )&x&y\\ \sin (\theta )&\cos (\theta )&z&w\\ 0&0&\cos (\theta )&-\sin (\theta )\\ 0&0&\sin (\theta )&\cos (\theta ) \end {pmatrix} \] where \(\theta ,x,y,z,w\in \mathbb {R}\). Find the linear subspace of \(\mathfrak {gl}_4(\mathbb {R})\) that is the Lie algebra of \(G\).

What is the Lie algebra of \(\mathrm {SO}(n)\)?

Very briefly for each, explain why the following spaces are smooth manifolds: \(S^n\), \(\mathbb {P}^n(\mathbb {R})\), \(T^n\), \(\mathrm {SL}_n(\mathbb {R})\), and \(\mathrm {GL}_n(\mathbb {R})\).

1. Let \(G\) be the Lie subgroup of \(\mathrm {GL}_3(\mathbb {R})\) that consists of the following matrices: \[ \begin {pmatrix} a & b & x\\ c & d & y\\ 0 & 0 & 1 \end {pmatrix} \] where \(ad-bc=1\). Find a basis for the Lie algebra of \(G\), considered as a Lie subalgebra of \(M_{3\times 3}(\mathbb {R})\).
2. Let \(\mathfrak {gl}_2(\mathbb {R})\) be the Lie algebra of \(\mathrm {GL}_2(\mathbb {R})\). For \[ \begin {pmatrix}2&1\\3&-2\end {pmatrix}, \begin {pmatrix}1&3\\0&1\end {pmatrix} \in \mathfrak {gl}_2(\mathbb {R}) \] find \[ \left [ \begin {pmatrix}2&1\\3&-2\end {pmatrix}, \begin {pmatrix}1&3\\0&1\end {pmatrix} \right ]. \]

1. State Sard’s Theorem.
2. Show that any smooth map \(f:\mathrm {SL}_2(\mathbb {R})\to S^4\) is homotopic to a constant map.

Let \(\mathrm {SL}(2,\mathbb {R})\) denote the subset of \(M^2\) (\(2\times 2\) real matrices), with determinant equal to one. Prove that \(\mathrm {SL}(2,\mathbb {R})\) is a regular submanifold of \(M^2=\mathbb {R}^4\), and find it’s dimension.

Let \(G\) be an \(n\)-dimensional Lie group, and \(TG\) it’s tangent bundle. Prove that \(TG\) is diffeomorphic to \(G\times \mathbb {R}^n\).

Identify the set of all \(n\times n\) matrices with \(\mathbb {R}^{n^2}\) and let \(O(n)\) be the subset consisting of orthogonal matrices (i.e. those that satisfy \(AA^T=I\)). Show that \(O(n)\) is a submanifold of \(\mathbb {R}^{n^2}\) and compute the tangent space at \(I\).

Let \(\mathcal {M}_{n\times n}\) be the set of all real \(n\times n\) matrices. This set is naturally a manifold, since it can be identified with \(\mathbb {R}^{n^2}\) by choosing an ordering of entries. Let \(O(n)\subset \mathcal {M}_{n\times n}\) be the set of orthogonal matrices, i.e. matrices \(A\) with \(AA^T=I\). Prove that \(O(n)\) is a submanifold of \(\mathcal {M}_{n\times n}\). What is the dimension of \(O(n)\)?

Prove that the Euler characteristic of the orthogonal group \(O(n)\) is 0 for \(n\ge 1\).

Identify the set \(M(n)\) of all \(n\times n\) matrices with real entries with \(\mathbb {R}^{n^2}\) as usual. Show that the set \[ O(n)=\{M\in M(n)\mid MM^t=I\} \] is a submanifold of \(\mathbb {R}^{n^2}\) (here \(M^t\) denotes the transpose of \(M\) and \(I\) is the identity matrix).

Recall that an \(n\times n\) real matrix \(M\) is orthogonal if \[ \langle Mv,Mw\rangle =\langle v,w\rangle \] for all \(v,w\in \mathbb {R}^n\) (and \(\langle \cdot ,\cdot \rangle \) is the inner product). Let \(O(n)\) denote the set of all orthogonal \(n\times n\) matrices, viewed as a subset of \(\mathbb {R}^{n^2}\). Show that \(O(n)\) is a manifold, and compute its dimension.

A. Let \[ S=\{(z_1,\ldots ,z_n)\in \mathbb {C}^n:z_1^2+\cdots +z_n^2=1\}. \] Prove that \(S\) is a smooth (real) submanifold of \(\mathbb {C}^n=\mathbb {R}^{2n}\), and find it’s dimension.

B. Prove that the orthogonal group \(O(n)\) is a smooth submanifold of the space of \(n\times n\) matrices \(M(n)=\mathbb {R}^{n^2}\), and find it’s dimension.

Let \(U(n)\) denote the set of unitary matrices (the complex \(n\times n\) matrices \(A\) which satisfy \(A\overline {A}^{T}=I\)). Show that \(U(n-1)\to U(n)\to S^{2n-1}\) is a fiber bundle. (Hint 1: Find explicit local trivializations. Hint 2: State a general theorem whose conclusion is that a smooth map between closed manifolds is a submersion and verify the hypotheses.)

Deduce that \(\pi _r(U(n))\simeq \pi _r(U(n+1))\) for \(r<2n\). What are \(\pi _1(U(3))\), \(\pi _2(U(3))\), \(\pi _3(U(3))\)?

Consider the set of matrices \[ O(n)=\{A\in \mathcal {M}_n:AA^T=1\} \] where \(\mathcal {M}_n\cong \mathbb {R}^{n^2}\) is the collection of all \(n\times n\) matrices and \(A^T\) means the transposition of \(A\). Show that \(O(n)\) is a smooth submanifold of \(\mathcal {M}_n\). Describe the set of matrices which form the tangent space to \(O(n)\) at the identity matrix.

Let \(M_2(\mathbb {R})\) be the set of all \(2\times 2\) matrices with real coefficients, and identify it with \(\mathbb {R}^4\) in the usual way. Let \(SL(2,\mathbb {R})\) be the subset of \(M_2(\mathbb {R})\) consisting of those matrices with determinant equal to one.

(a) Show \(SL(2,\mathbb {R})\) is a submanifold of \(M_2(\mathbb {R})\).

(b) Exhibit the tangent space to \(SL(2,\mathbb {R})\) at the identity matrix.

\(\mathrm {U}(n)\), the space of complex \(n\times n\) unitary matrices, has the structure of a smooth real manifold.