Math Problems Vault

Problem 73.1

(a)(b)(c)

Let $F$ be the field $\mathbb {R}$ or $\mathbb {C}$ .

Prove that:

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(a)

for any matrix $A \in \mathbf{M}_{n}(F)$ , the map $P_{A}: t \mapsto e^{t A}(t \in F)$ is a differentiable matrix representation of the additive group of the field $F$ ;

(b)

every differentiable matrix representation $P$ of the additive group of the field $F$ has the form $P_{A}$ , where $A=P^{\prime }(0)$ ;

(c)

the representations $P_{A}$ and $P_{B}$ are equivalent if and only if the matrices $A$ and $B$ are similar.

Problem 73.2

Prove that $P$ is a matrix representation of the additive group of the field $\mathbb {R}$ , and find a matrix $A$ such that $P=P_{A}$ , if:

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(a)

$P(t)=\left(\begin{array}{rr}\cos t & -\sin t \\ \sin t & \cos t\end{array}\right)$ ;

(b)

$P(t)=\left(\begin{array}{ll}\operatorname {ch} t & \operatorname {sh} t \\ \operatorname {sh} t & \operatorname {ch} t\end{array}\right)$ ;

(c)

$P(t)=\left(\begin{array}{ll}e^{t} & 0 \\ 0 & 1\end{array}\right)$ ;

(d)

$P(t)=\left(\begin{array}{cc}e^{t} & 0 \\ 0 & e^{-t}\end{array}\right)$ ;

(e)

$P(t)=\left(\begin{array}{ll}1 & t \\ 0 & 1\end{array}\right)$ ;

(f)

$P(t)=\left(\begin{array}{cc}1 & e^{t}-1 \\ 0 & e^{t}\end{array}\right)$ .

Problem 73.3

Which of the matrix representations of the group $\mathbb {R}$ from problem 73.2 are equivalent?

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Problem 73.4

In which case are the representations $P_{A}$ and $P_{-A}$ equivalent for $F=\mathbb {C}$ ?

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Problem 73.5

(a)(b)(c)(d)

Find all differentiable complex matrix representations of the groups:

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(a)

$\mathbb {R}_{+}^{*}$ ;

(b)

$\mathbb {R}^{*}$ ;

(c)

$\mathbb {C}^{*}$ ;

(d)

$\mathbf{U}$ (differentiability of the representation with respect to the argument of the complex number $z$ is assumed).

Problem 73.6

Is every complex linear representation of the group $\mathbb {Z}$ obtained by restricting to $\mathbb {Z}$ some representation of the group $\mathbb {C}$ ?

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Problem 73.7

Find all subspaces of the space $\mathbb {C}^{n}$ invariant under the matrix representation $P_{A}$ (see problem 73.1) in the case where the characteristic polynomial of the matrix $A$ has no multiple roots.

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Problem 73.8

Prove that the matrix representation $P_{A}$ (see problem 73.1) is completely reducible if and only if the matrix $A$ is diagonalizable.

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Problem 73.9

Let $R_{n}$ be the space of homogeneous polynomials of degree $n$ in $x, y$ with complex coefficients. For

A=\left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right] \in \mathbf{S L}_{2}(\mathbb {C})

and $f \in R_{n}$ , set

\left(\Phi _{n}(A) f\right)(x, y)=f(a x+c y, b x+d y)

Prove that the restriction of the representation $\Phi_{n}$ to the subgroup $\mathrm{SU}_{2}(\mathbb {C})$ is irreducible.

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Problem 73.10

Let $G=\mathbf{G L}_{2}(\mathbb {C})$ . Call a complex function on $G$ polynomial if it is a polynomial in the matrix entries.

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(a)

Let $t(A)=\operatorname {tr} A, d(A)=\operatorname {det} A$ . Prove that $t$ and $d$ are central polynomial functions on $G$ .

(b)

Prove that any central polynomial function on $G$ is a polynomial in $t$ and $d$ .

(c)

Let $A=\left(a_{i j}\right) \in G$ and $R=\mathbb {C}\left[x, y\right]$ . Denote by $\Psi (A)$ the homomorphism $R \rightarrow R$ for which

\begin{aligned} & \Psi (A): x \mapsto a_{11} x+a_{12} y \\ & \Psi (A): y \mapsto a_{21} x+a_{22} y \end{aligned}

Prove that $\Psi$ is a representation of the group $G$ on the space $R$ , and that the subspaces of homogeneous polynomials of degree $n$ are invariant under the representation $\Psi$ .

(d)

Prove that, for $A \in \mathbf{S L}_{2}(\mathbb {C})$ , the restriction of $\Psi (A)$ to the subspace $R_{n}$ coincides with the operator $\Phi_{n}(A)$ from problem 73.9.

(e)

Let $\chi_{n}$ be the character of the restriction $\left.\Psi \right|_{R_{n}}$ . Prove that

\chi _{n}=t \chi _{n-1}-d \chi _{n-2}

Problem 73.11

(a)(b)(c)

Let $\mathbb {H}$ be the space of complex matrices of the form

X=\left[\begin{smallmatrix} z & w \\ -\bar{w} & \bar{z} \end{smallmatrix}\right]

with the structure of a four-dimensional Euclidean space $(X, X)=\operatorname {det} X$ , and $\mathbb {H}_{0}=\left\{ X \in \mathbb {H} \mid \operatorname {tr} X=0\right\}$ .

Prove that:

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(a)

the map $P: \mathbf{S U}_{2} \rightarrow \mathbf{G L}\left(\mathbb {H}_{0}\right)$ defined by the formula

P(A): X \mapsto A X A^{-1},

is a (real) linear representation of the group $\mathbf{S U}_{2}$ , $\operatorname {Ker} P= \pm E$ , and $\operatorname {Im} P$ consists of all proper orthogonal transformations of the space $H_{0}$ ;

(b)

the map $R: \mathbf{S U}_{2} \times \mathbf{S} \mathbf{U}_{2} \rightarrow \mathbf{G L}(\mathbb {H})$ defined by the formula $R(A, B): X \mapsto A X B^{-1}$ , is a (real) linear representation of the group $\mathbf{S U}_{2} \times \mathbf{S U}_{2}$ , $\operatorname {Ker} R=\left\{ (E, E),(-E,-E)\right\}$ , and $\operatorname {Im} R$ consists of all proper orthogonal transformations of the space $\mathbb {H}_{0}$ ;

(c)

the complexification of the linear representation $P$ is isomorphic to the restriction of the representation $\Phi_{2}$ of the group $\mathbf{S L}_{2}$ from problem 73.9 to the subgroup $\mathbf{S U}_{2}$ .

Problem 73.12

(a)(b)

Let $G$ be a connected solvable topological group and $\rho$ a continuous homomorphism of $G$ into the group of nonsingular linear operators on a finite-dimensional complex space $V$ .

Prove that:

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(a)

$V$ has a nonzero vector that is an eigenvector for all the operators $\rho (g), g \in G$ ;

(b)

$V$ has a basis $e_{1}, \ldots , e_{n}$ such that all the matrices $\rho (g)$ , $g \in G$ , are upper triangular in this basis.

Problem 73.13

Let $F$ be an algebraically closed field and $G$ a solvable group of nonsingular linear operators on a finite-dimensional vector space $V$ over $F$ . Prove that there exist a basis $e_{1}, \ldots , e_{n}$ of $V$ and a normal subgroup $N$ of $G$ of finite index (depending only on $n$ ) such that $N$ consists of upper triangular matrices.

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