Math Problems Vault

Problem 20.1.1

(i) Consider the braid group $\mathcal{B}_{5}$ with the generators $\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ , $\sigma_{4}$ . Simplify

\sigma _{1}^{-1}\sigma _{4}^{-1}\sigma _{3}^{-1}\sigma _{2}\sigma _{3}\sigma _{2}\sigma _{3}^{-1}\sigma _{4}\sigma _{1} .

(ii) Consider the braid group $\mathcal{B}_{3}$ and with the generators $\sigma_{1}$ and $\sigma_{2}$ . Let $a = \sigma_{1}\sigma_{2}\sigma_{1}$ , $b = \sigma_{1}\sigma_{2}$ . Show that $a^{2} = b^{3}$ .

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Problem 20.1.2

(i)(ii)

Consider the braid group $\mathcal{B}_{3}$ . A faithful representation for the generators $\sigma_{1}$ and $\sigma_{2}$ is

\sigma _{1} = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}, \qquad \sigma _{2} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} .

Both are elements of the group $SL(2,\mathbb {Z})$ and the Lie group $SL(2,\mathbb {R})$ .

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(i)

Find the inverse of $\sigma_{1}$ and $\sigma_{2}$ . Find $\sigma_{1}\sigma_{2}$ and $\sigma_{1}^{-1}\sigma_{2}$ .

(ii)

Is $\sigma_{1}\sigma_{2}\sigma_{1} = \sigma_{2}\sigma_{1}\sigma_{2}$ ?

Problem 20.1.3

Consider the unitary $2 \times 2$ matrices

A(\theta ) = \begin{pmatrix} e^{-i\theta } & 0 \\ 0 & e^{i\theta } \end{pmatrix}, \qquad B(\theta ) = \begin{pmatrix} \cos (\theta ) & -i\sin (\theta ) \\ -i\sin (\theta ) & \cos (\theta ) \end{pmatrix}

where $\theta \in \mathbb {R}$ . Find the conditions on $\theta_{1}, \theta_{2}, \theta_{3} \in \mathbb {R}$ such that (braid-like relation)

A(\theta _{1})B(\theta _{2})A(\theta _{3}) = B(\theta _{3})A(\theta _{2})B(\theta _{1}) .

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Problem 20.1.4

Can one find $2 \times 2$ matrices $A$ and $B$ with $\left[A,B\right] \neq 0_{2}$ and satisfying the braid-like relation $ABBA = BAAB$ ?

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Problem 20.1.5

Do the $2 \times 2$ unitary matrices

A = \begin{pmatrix} e^{-i\pi /4} & 0 \\ 0 & ie^{-i\pi /4} \end{pmatrix}, \qquad B = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}

satisfy the braid-like relation $ABA = BAB$ ?

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Problem 20.1.6

The $3 \times 3$ matrices $g_{1}$ and $g_{2}$ play a role for the matrix representation of the braid group $\mathcal{B}_{4}$

g_{1} = \begin{pmatrix} -t & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & -t^{-1} \end{pmatrix}, \qquad g = \begin{pmatrix} 1-t & -t^{-1} & t^{-1} \\ 1-t^{2} & -t^{-1} & 0 \\ 1 & -t^{-1} & 0 \end{pmatrix}

as generators. Let $g_{2} = g_{1}g^{-1}$ . Find the eigenvalues and eigenvectors of $g_{1}$ , $g_{2}$ .

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Problem 20.1.7

Find the conditions on $a,b,c,d,e,f \in \mathbb {R}$ such that

\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \begin{pmatrix} d & e \\ 0 & f \end{pmatrix} \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} = \begin{pmatrix} d & e \\ 0 & f \end{pmatrix} \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \begin{pmatrix} d & e \\ 0 & f \end{pmatrix} .

Find nontrivial solutions to these conditions.

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Problem 20.1.8

(i)(ii)

Let $A$ , $B$ be $n \times n$ matrices with $ABA = BAB$ .

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(i)

Show that $(A \otimes A)(B \otimes B)(A \otimes A) = (B \otimes B)(A \otimes A)(B \otimes B)$ .

(ii)

Show that $(A \oplus A)(B \oplus B)(A \oplus A) = (B \oplus B)(A \oplus A)(B \oplus B)$ .

Problem 20.1.9

(i)(ii)

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(i)

Show that the matrices

S_{1} = \begin{pmatrix} -t & 1 \\ 0 & 1 \end{pmatrix}, \qquad S_{2} = \begin{pmatrix} 1 & 0 \\ t & -t \end{pmatrix}

satisfy the braid-like relation $S_{1}S_{2}S_{1} = S_{2}S_{1}S_{2}$ .

(ii)

Show that the matrices $S_{1} \otimes S_{1}$ and $S_{2} \otimes S_{2}$ satisfy the braid-like relation

(S_{1} \otimes S_{1})(S_{2} \otimes S_{2})(S_{1} \otimes S_{1}) = (S_{2} \otimes S_{2})(S_{1} \otimes S_{1})(S_{2} \otimes S_{2}) .

Problem 20.1.10

(i)(ii)

Let $T$ be an $n \times n$ matrix and $R$ be an $n^{2} \times n^{2}$ matrix. Consider the equation

R(T \otimes T) = (T \otimes T)R .

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(i)

Let $n = 2$ and

T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .

Find all $4 \times 4$ matrices $R$ which satisfy the equation.

(ii)

Let $n = 2$ and

R = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

Find all $T$ which satisfy the equation.

Problem 20.1.11

Find all $4 \times 4$ matrices $A$ , $B$ with $\left[A,B\right] \neq 0_{4}$ satisfying the conditions

ABA = BAB, \qquad ABBA = I_{4} .

The first condition is the braid relation and the second condition $ABBA = I_{4}$ runs under Dirac game.

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Problem 20.1.12

(i)(ii)

The free group, $\Gamma_{2}$ , with two generators $g_{1}$ and $g_{2}$ admits the matrix representation

g_{1} = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}, \qquad g_{2} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} .

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(i)

Find the inverse of the matrices.

(ii)

Calculate $g_{1}g_{2}g_{1}$ and $g_{2}g_{1}g_{2}$ . Discuss.

Problem 20.1.13

Let $A$ , $B$ be $2 \times 2$ matrices and

R_{12} = A \otimes B \otimes I_{2}, \qquad R_{13} = A \otimes I_{2} \otimes B, \qquad R_{23} = I_{2} \otimes A \otimes B .

Find the conditions on $A$ and $B$ such that $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$ .

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Problem 20.1.14

Let $I_{2}$ be the $2 \times 2$ unit matrix. Let $R$ be a $4 \times 4$ matrix over $\mathbb {C}$ . Then $R$ satisfies the Yang-Baxter equation if

(R \otimes I_{2})(I_{2} \otimes R)(R \otimes I_{2}) = (I_{2} \otimes R)(R \otimes I_{2})(I_{2} \otimes R) .

Use computer algebra to show that

R = \begin{pmatrix} 1/\sqrt{2} & 0 & 0 & 1/\sqrt{2} \\ 0 & 1/\sqrt{2} & -1/\sqrt{2} & 0 \\ 0 & 1/\sqrt{2} & 1/\sqrt{2} & 0 \\ -1/\sqrt{2} & 0 & 0 & 1/\sqrt{2} \end{pmatrix}

satisfy the Yang-Baxter equation.

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Problem 20.1.15

Show that the $2 \times 2$ matrix

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

satisfies the braid-like relation

(A \otimes I_{3})(I_{3} \otimes A)(A \otimes I_{3}) = (I_{3} \otimes A)(A \otimes I_{3})(I_{3} \otimes A) .

Show that the $3 \times 3$ matrix

B = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}

satisfies the braid-like relation

(B \otimes I_{2})(I_{2} \otimes B)(B \otimes I_{2}) = (I_{2} \otimes B)(B \otimes I_{2})(I_{2} \otimes B) .

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Problem 20.1.16

Consider the $2 \times 2$ matrices $I_{2}$ and $\sigma_{1}$ and the $4 \times 4$ matrices

R = \frac{1}{2}(I_{2} \otimes I_{2} + I_{2} \otimes \sigma _{1} + \sigma _{1} \otimes I_{2} - \sigma _{1} \otimes \sigma _{1}), \qquad F = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

Calculate the matrix $\widetilde{R} = FR$ . Find the determinant of $\widetilde{R}$ . Does the $4 \times 4$ matrix $\widetilde{R}$ satisfy the braid-like relation

(I_{2} \otimes \widetilde{R})(\widetilde{R} \otimes I_{2})(I_{2} \otimes \widetilde{R}) = (\widetilde{R} \otimes I_{2})(I_{2} \otimes \widetilde{R})(\widetilde{R} \otimes I_{2}) \, ?

Consider the four Bell states

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix} .

Apply the matrix $\widetilde{R}$ to the Bell states. Discuss.

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Problem 20.1.17

(i)(ii)(iii)

Let

B(1) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

and $B(k)$ the $2^{k} \times 2^{k}$ matrix recursively defined by

B(k) := \begin{pmatrix} B(k-1) & B(k-1) \\ 0 & B(k-1) \end{pmatrix} = B(1) \otimes B(k-1), \qquad k = 2,3,\ldots .

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(i)

Find $B(1)^{-1}$ .

(ii)

Find $B(k)^{-1}$ for $k = 2,3,\ldots$ .

(iii)

Let $A(k)$ be the $2^{k} \times 2^{k}$ anti-diagonal matrix defined by $A_{ij}(k) = 1$ if $i = 2^{k}+1-j$ and $0$ otherwise with $j = 1,2,\ldots ,2^{k}$ . Let

C(k) = A(k)B(k)^{-1}A(k) .

Show that the matrices $B(k)$ and $C(k)$ have the braid-like relation

B(k)C(k)B(k) = C(k)B(k)C(k) .

Problem 20.1.18

Let

B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \qquad C = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} .

Then $BCB = CBC$ . Is

(B \otimes B)(C \otimes C)(B \otimes B) = (C \otimes C)(B \otimes B)(C \otimes C) \, ?

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Problem 20.1.19

(i)(ii)(iii)

Let $n \geq 2$ . Let $u,v \in \mathbb {C}$ ( $u$ , $v$ are called the spectral parameters). Consider the $n^{2} \times n^{2}$ matrices $R(u)$ over $\mathbb {C}$ with the entries $(r_{k\ell }^{ij})(u)$ with $i,j,k,\ell = 1,\ldots ,n$ and the entries are smooth functions of $u$ . Let $\left\{ \overrightarrow {e}_{i} : i = 1,\ldots ,n\right\}$ be the standard basis in the vector space $\mathbb {C}^{n}$ . Let $\otimes$ be the Kronecker product. Then

\left\{ \overrightarrow {e}_{i} \otimes \overrightarrow {e}_{j} : i,j = 1,\ldots ,n\right\}

is the standard basis in the vector space $\mathbb {C}^{n} \otimes \mathbb {C}^{n}$ . One defines

R(\overrightarrow {e}_{i} \otimes \overrightarrow {e}_{j})(u) := \sum _{\alpha =1}^{n}\sum _{\beta =1}^{n} r_{ij}^{\alpha \beta }(u)\overrightarrow {e}_{\alpha } \otimes \overrightarrow {e}_{\beta }, \qquad i,j = 1,\ldots ,n . \tag {1}

Thus

R(u) = \sum _{i,j=1}^{n}\sum _{\alpha ,\beta =1}^{n} r_{ij}^{\alpha \beta }(u)(\overrightarrow {e}_{\alpha } \otimes \overrightarrow {e}_{\beta })(\overrightarrow {e}_{i} \otimes \overrightarrow {e}_{j})^{T} .

For $n=2$ one finds that the matrix $R(u)$ is given by

R(u) = \begin{pmatrix} r_{11}^{11}(u) & r_{12}^{11}(u) & r_{21}^{11}(u) & r_{22}^{11}(u) \\ r_{11}^{12}(u) & r_{12}^{12}(u) & r_{21}^{12}(u) & r_{22}^{12}(u) \\ r_{11}^{21}(u) & r_{12}^{21}(u) & r_{21}^{21}(u) & r_{22}^{21}(u) \\ r_{11}^{22}(u) & r_{12}^{22}(u) & r_{21}^{22}(u) & r_{22}^{22}(u) \end{pmatrix} .

One defines the three $n^{3} \times n^{3}$ matrices

R^{12}(u) := R(u) \otimes I_{n}, \qquad R^{23}(u) := I_{n} \otimes R(u)

and

R^{13}(\overrightarrow {e}_{i} \otimes \overrightarrow {e}_{j} \otimes \overrightarrow {e}_{k})(u) := \sum _{\alpha =1}^{n}\sum _{\beta =1}^{n} r_{ik}^{\alpha \beta }(u)\overrightarrow {e}_{\alpha } \otimes \overrightarrow {e}_{j} \otimes \overrightarrow {e}_{\beta } .

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(i)

Write down the matrices $R^{12}(u)$ , $R^{23}(u)$ and $R^{13}(u)$ for $n=2$ and find the Yang-Baxter relation

R^{12}(u)R^{13}(u+v)R^{23}(v) = R^{23}(v)R^{13}(u+v)R^{12}(u) .

Thus there are $2^{6} = 64$ equations for $2^{4} = 16$ unknowns.

(ii)

Show that the matrix

R(u) = \begin{pmatrix} 1+u & 0 & 0 & 0 \\ 0 & u & 1 & 0 \\ 0 & 1 & u & 0 \\ 0 & 0 & 0 & 1+u \end{pmatrix}

satisfies the Yang-Baxter equation.

(iii)

Let

P = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

This matrix is called the swap matrix with $P(\overrightarrow {x} \otimes \overrightarrow {y}) = \overrightarrow {y} \otimes \overrightarrow {x}$ where $\overrightarrow {x},\overrightarrow {y} \in \mathbb {C}^{2}$ . We define $\widetilde{R}(u) := PR(u)$ . Then the Yang-Baxter equation can be written as

(\widetilde{R}(v) \otimes I_{2})(I_{2} \otimes \widetilde{R}(u+v))(\widetilde{R}(u) \otimes I_{2}) = (I_{2} \otimes \widetilde{R}(u))(\widetilde{R}(u+v) \otimes I_{2})(I_{2} \otimes \widetilde{R}(v)) .

Show that $R(u)$ given in (ii) satisfies this equation. Note that without the spectral parameters $u$ , $v$ we obtain Artin's braid relation.

Problem 20.1.20

Let $q=4$ and $w := \exp (2\pi i/q)$ , i.e. $w^{4}=1$ and $w=i$ , $w^{2}=-1$ , $w^{3}=-i$ . Let $D_{1}$ and $D_{2}$ be the diagonal matrices

D_{1} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & w & 0 & 0 \\ 0 & 0 & w^{2} & 0 \\ 0 & 0 & 0 & w^{3} \end{pmatrix}, \qquad D_{2} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & w^{3} & 0 & 0 \\ 0 & 0 & w^{2} & 0 \\ 0 & 0 & 0 & w \end{pmatrix}

and $P$ the permutation matrix

P = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix} .

Consider the $64 \times 64$ matrices

S_{1} = D_{1} \otimes D_{2} \otimes I_{4}, \quad S_{2} = I_{4} \otimes P \otimes I_{4}, \quad S_{3} = I_{4} \otimes D_{1} \otimes D_{2}, \quad S_{4} = I_{4} \otimes I_{4} \otimes P .

Show that

S_{j}^{q} = I_{64} \qquad j = 1,2,3,4

S_{j}S_{j+1}S_{j}^{-1}S_{j+1}^{-1} = wI_{64} \qquad j = 1,2,3

\left[S_{1},S_{3}\right] = 0_{64}, \quad \left[S_{1},S_{4}\right] = 0_{64}, \quad \left[S_{2},S_{4}\right] = 0_{64}

where $0_{64}$ is the $64 \times 64$ zero matrix.

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Problem 20.1.21

Consider the Pauli spin matrices $\sigma_{0} = I_{2}$ , $\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ . Find the $4 \times 4$ matrix

R(u) = \sum _{j=0}^{3} w_{j}(u)\sigma _{j} \otimes \sigma _{j}

with

\begin{align} w_{0}(u) & = \sinh (u+\eta /2), \\ w_{1}(u) & = w_{2}(u) = \sinh (\eta /2)\cosh (\eta /2), \\ w_{3}(u) & = \sinh (\eta /2)\cosh (u+\eta /2) \end{align}

where $\eta$ is a fixed parameter.

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Problem 20.1.22

Let $\mathcal{B}_{n}$ denote the braid group on $n$ strands. $\mathcal{B}_{n}$ is generated by the elementary braids $\left\{ b_{1}, b_{2}, \ldots , b_{n-1}\right\}$ with the braid relations

b_{j}b_{j+1}b_{j} = b_{j+1}b_{j}b_{j+1}, \qquad 1 \leq j < n-1,

b_{j}b_{k} = b_{k}b_{j}, \qquad \left|j-k\right| \geq 2 .

Let $\left\{ \overrightarrow {e}_{1}, \overrightarrow {e}_{2}, \ldots , \overrightarrow {e}_{n}\right\}$ denote the standard basis in $\mathbb {R}^{n}$ . Then $\overrightarrow {u} \in \mathbb {R}^{n}$ can be written as

\overrightarrow {u} = \sum _{k=1}^{n} c_{k}\overrightarrow {e}_{k}, \qquad c_{1}, c_{2}, \ldots , c_{n} \in \mathbb {R} .

Consider the linear operators $B_{j}$ ( $\alpha ,\beta ,\gamma ,\delta \in \mathbb {R}$ and $\alpha ,\gamma \neq 0$ ) defined by

B_{j}\overrightarrow {u} := c_{1}\overrightarrow {e}_{1} + \cdots + (\alpha c_{j+1}+\beta )\overrightarrow {e}_{j} + (\gamma c_{j+1}+\delta )\overrightarrow {e}_{j+1} + \cdots + c_{n}\overrightarrow {e}_{n}

and the corresponding inverse operation

B_{j}^{-1}\overrightarrow {u} := c_{1}\overrightarrow {e}_{1} + \cdots + \frac{1}{\gamma }(c_{j+1}-\delta )\overrightarrow {e}_{j} + \frac{1}{\alpha }(\gamma c_{j}-\beta )\overrightarrow {e}_{j+1} + \cdots + c_{n}\overrightarrow {e}_{n} .

Use computer algebra to show that $B_{1}, B_{2}, \ldots , B_{n-1}$ satisfy the braid condition

B_{j}B_{j+1}B_{j}\overrightarrow {u} = B_{j+1}B_{j}B_{j+1}\overrightarrow {u}

if $\gamma \beta +\delta = \alpha \delta +\beta$ .

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