Math Problems Vault

Problem 2.44 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

Consider the Lie group $U(n)$ of the unitary $n \times n$ matrices. Let $\mathbb {T}^{n}$ be the $n$ -torus of diagonal unitary $n \times n$ matrices. Obviously $\mathbb {T}^{n}$ is a subgroup of $U(n)$ . Describe the homogeneous space $U(n)/\mathbb {T}^{n}$ .

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Problem 2.53 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

Consider the Lie group $O(n, \mathbb {R})$ . Show that the $n$ -sphere

S^{n} := \left\{ (x_{1}, \ldots , x_{n}, x_{n+1}) : x_{1}^{2}+\cdots +x_{n+1}^{2} = 1\right\}

is diffeomorphic to $O(n+1, \mathbb {R})/O(n, \mathbb {R})$ .

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Problem 2.54 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

Consider the linear action of $SU(n+1)$ on the vector space $\mathbb {C}^{n+1}$ . This maps complex lines into complex lines and therefore passes to an action on the complex projective space $\mathbb {C} P^{n}$ which is defined to be the vector space of all such lines in $\mathbb {C}^{n+1}$ . Let $n = 1$ , i.e. we consider $SU(2)$ . Find an element $A$ in $SU(2)$ such that

A\frac{1}{\sqrt{2}} \begin{pmatrix} i \\ -i \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} .

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Problem 2.75 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

(i)(ii)(iii)

Let

S^{n} := \left\{ (x_{1}, x_{2}, \ldots , x_{n+1}) \in \mathbb {R}^{n+1} : x_{1}^{2}+x_{2}^{2}+\cdots +x_{n+1}^{2} = 1\right\} .

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(i)

Show that $S^{3}$ can be considered as a subset of $\mathbb {C}^{2}$ ( $\mathbb {C}^{2} \cong \mathbb {R}^{4}$ )

S^{3} = \left\{ (z_{1}, z_{2}) \in \mathbb {C}^{2} : \left|z_{1}\right|^{2}+\left|z_{2}\right|^{2} = 1\right\} .

(ii)

The Hopf map $\pi : S^{3} \to S^{2}$ is defined by

\pi (z_{1}, z_{2}) := (\bar{z}_{1}z_{2}+\bar{z}_{2}z_{1}, \ -i\bar{z}_{1}z_{2}+i\bar{z}_{2}z_{1}, \ \left|z_{1}\right|^{2}-\left|z_{2}\right|^{2}) .

Find the parameterization of $S^{3}$ , i.e. find $z_{1}(\theta ,\phi )$ , $z_{2}(\theta ,\phi )$ and thus show that indeed $\pi$ maps $S^{3}$ onto $S^{2}$ .

(iii)

Show that $\pi (z_{1}, z_{2}) = \pi (z_{1}', z_{2}')$ if and only if $z_{j}' = e^{i\alpha }z_{j}$ ( $j=1,2$ ) and $\alpha \in \mathbb {R}$ .

Problem 2.91 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

(i)(ii)(iii)

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(i)

Let $t \in \mathbb {R}$ . Show that the matrix

g_{t} = \begin{pmatrix} \cosh t & i\sinh t \\ -i\sinh t & \cosh t \end{pmatrix}

has an inverse and find $g_{t}^{-1}$ . $g_{t}$ is an element of $SO(1,1)$ .

(ii)

The symmetric space $SL(2, \mathbb {C})/SU(2)$ can be identified with the space $H^{3}$ of positive $2 \times 2$ matrices with determinant $1$

A = \begin{pmatrix} a+b & x+iy \\ x-iy & a-b \end{pmatrix}, \qquad a,b,c,d \in \mathbb {R}

i.e. $\det A = a^{2}-b^{2}-x^{2}-y^{2} = 1$ . The group action is defined by

g_{t} \cdot A := g_{t}Ag_{t}^{*} .

Find $g_{t} \cdot A$ .

(iii)

Show that on the hyperboloid $a^{2} = 1+b^{2}+x^{2}+y^{2}$ , the matrix $g_{t}$ only changes the coordinates $y$ and $a$ .

Problem 2.104 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Section 2.2 "Supplementary Problems"

Let $n \geq 1$ . We define the $(2n+1)$ -sphere

S^{2n+1} := \left\{ \overrightarrow {z} \in \mathbb {C}^{n+1} : \sum _{j=1}^{n+1} \left|z_{j}\right|^{2} = 1\right\} .

Show that

S^{2n+1} \cong U(n+1)/U(n) \cong SU(n+1)/SU(n) .

For $n=1$ one has $S^{3} \cong U(2)/U(1) \cong SU(2)$ .

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Homogeneous spaces of Lie groups

References