Math Problems Vault

Problem 20.2.1

Let $n \geq 2$ and $j,k = 1,\ldots ,2n-1$ . Let $Q$ be the $4 \times 4$ matrix

Q = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & s^{4} & 1 & 0 \\ 0 & 1 & s^{-4} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} .

Then $Q^{2} = (s^{4}+s^{-4})Q$ . Consider the $2^{2n}$ matrices

Q_{j} = I_{2} \otimes \cdots \otimes I_{2} \otimes Q \otimes I_{2} \otimes \cdots \otimes I_{2}

where the $4 \times 4$ matrix $Q$ is at the $j$ -th position. These matrices satisfy the relations (Temperley-Lieb algebra)

Q_{j}Q_{j} = (s^{4}+s^{-4})Q_{j}

Q_{j}Q_{j\pm 1}Q_{j} = Q_{j}

Q_{j}Q_{k} = Q_{k}Q_{j} \qquad \text{if} \quad \left|j-k\right| \geq 2 .

These matrices play a role for the $q$ -state Potts model. Consider the case $n=2$ and the $16 \times 16$ matrices

Q_{1} = Q \otimes I_{2} \otimes I_{2}, \qquad Q_{2} = I_{2} \otimes Q \otimes I_{2}, \qquad Q_{3} = I_{2} \otimes I_{2} \otimes Q .

Write computer algebra programs in SymbolicC++ and Maxima which check that these matrices satisfy

Q_{1}Q_{1} = (s^{4}+s^{-4})Q_{1}, \quad Q_{2}Q_{2} = (s^{4}+s^{-4})Q_{2}, \quad Q_{3}Q_{3} = (s^{4}+s^{-4})Q_{3}

Q_{1}Q_{2}Q_{1} = Q_{1}, \quad Q_{2}Q_{3}Q_{2} = Q_{2}, \quad Q_{3}Q_{2}Q_{3} = Q_{3},

Q_{2}Q_{1}Q_{2} = Q_{2}, \quad Q_{1}Q_{3} = Q_{3}Q_{1} .

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Problem 20.2.2

(i)(ii)

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(i)

Find the conditions on the $2 \times 2$ matrices over $\mathbb {C}$ such that $ABA = BAB$ . Find solutions where $AB \neq BA$ .

(ii)

Find the conditions on the $2 \times 2$ matrices $A$ and $B$ such that

A \otimes B \otimes A = B \otimes A \otimes B .

Find solutions where $AB \neq BA$ , i.e. $\left[A,B\right] \neq 0_{2}$ .

Problem 20.2.3

Consider the $4 \times 4$ matrices

A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 1 & 3 & 3 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 3 & 3 & 1 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

Is $AB^{-1}A = B^{-1}AB^{-1}$ ?

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Problem 20.2.4

(i)(ii)

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(i)

Consider the Pauli spin matrix $A = \sigma_{1}$ . Does the matrix $A$ satisfy the braid-like relation

(A \otimes I_{2})(I_{2} \otimes A)(A \otimes I_{2}) = (I_{2} \otimes A)(A \otimes I_{2})(I_{2} \otimes A) \, ?

Does the matrix $A$ satisfy 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) \, ?

Ask the same question for the Pauli spin matrices $\sigma_{2}$ and $\sigma_{3}$ .

(ii)

Let $n \geq 2$ and $m \geq 2$ . Let $I_{n}$ be the $n \times n$ identity matrix. Let $J_{m}$ be counter identity matrix, i.e. the $m \times m$ matrix whose entries are all equal to $0$ except those on the counterdiagonal which are all equal to $1$ . Find the conditions on $n$ and $m$ such that

(J_{n} \otimes I_{m})(I_{m} \otimes J_{n})(J_{n} \otimes I_{m}) = (I_{m} \otimes J_{n})(J_{n} \otimes I_{m})(I_{m} \otimes J_{n}) .

Problem 20.2.5

Let

T = \begin{pmatrix} t_{11} & t_{12} \\ t_{21} & t_{22} \end{pmatrix}

be an element of the Lie group $SL(2,\mathbb {R})$ , i.e. $\operatorname {det}\left(T\right) = 1$ and $t_{jk} \in \mathbb {R}$ . Let

R = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 4 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .

Find all $T \in SL(2,\mathbb {R})$ such that $\left[R, T \otimes T\right] = 0_{4}$ .

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Problem 20.2.6

Let $\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ be the Pauli spin matrices. Consider the $4 \times 4$ matrix

R = a(\lambda ,\mu )\sigma _{1} \otimes \sigma _{1} + b(\lambda ,\mu )(\sigma _{2} \otimes \sigma _{2} + \sigma _{3} \otimes \sigma _{3})

where

a(\lambda ,\mu ) = \frac{1}{4}\frac{\lambda ^{2}+\mu ^{2}}{\lambda ^{2}-\mu ^{2}}, \qquad b(\lambda ,\mu ) = \frac{1}{2}\frac{\lambda \mu }{\lambda ^{2}-\mu ^{2}} .

Does $R$ satisfy the braid-like relation

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

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Problem 20.2.7

Consider the braid group $\mathcal{B}_{3}$ with the generators $\left\{ \sigma_{1},\sigma_{2}\right\}$ and the relation

\sigma _{1}\sigma _{2}\sigma _{1} = \sigma _{2}\sigma _{1}\sigma _{2} .

Let $t \neq 0$ . Show that a matrix representation is given by

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

with the inverse matrices

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

Let $\sigma = \sigma_{1}\sigma_{2}\sigma_{1}^{-1}\sigma_{2}^{-1}\sigma_{1}^{-1}$ . Find $f(t) = \operatorname {det}\left(\sigma - I_{2}\right)$ . Find minima and maxima of $f$ .

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Problem 20.2.8

(i)(ii)

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(i)

Let $a,b,c,d \in \mathbb {R}$ and $a \neq 0$ . Does

R = \begin{pmatrix} a & 0 & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & d & a-cd/a & 0 \\ 0 & 0 & 0 & a \end{pmatrix}

satisfy $(R \otimes I_{4})(I_{4} \otimes R)(R \otimes I_{4}) = (I_{4} \otimes R)(R \otimes I_{4})(I_{4} \otimes R)$ ? We have to check that $R^{2} \otimes R = R \otimes R^{2}$ .

(ii)

Find the condition on the $4 \times 4$ matrix

S = \begin{pmatrix} s_{11} & 0 & 0 & s_{14} \\ 0 & s_{22} & s_{23} & 0 \\ 0 & s_{32} & s_{33} & 0 \\ s_{41} & 0 & 0 & s_{44} \end{pmatrix}

such that $(S \otimes I_{4})(I_{4} \otimes S)(S \otimes I_{4}) = (I_{4} \otimes S)(S \otimes I_{4})(I_{4} \otimes S)$ . We have to check that $S^{2} \otimes S = S \otimes S^{2}$ .

Problem 20.2.9

(i)(ii)

Let $R$ be a $4 \times 4$ matrix written as

R = \begin{pmatrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{pmatrix}

where $R_{12}, R_{21}, R_{22}$ are the $2 \times 2$ block matrices. Suppose that $R$ satisfies the equation

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

Consider the $8 \times 8$ matrix

\widetilde{R} = R \star R = \begin{pmatrix} R_{11} & 0_{2} & 0_{2} & R_{12} \\ 0_{2} & R_{11} & R_{12} & 0_{2} \\ 0_{2} & R_{21} & R_{22} & 0_{2} \\ R_{12} & 0_{2} & 0_{2} & R_{22} \end{pmatrix} .

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(i)

Does $\widetilde{R}$ satisfy $(\widetilde{R} \otimes I_{2})(I_{2} \otimes \widetilde{R})(\widetilde{R} \otimes I_{2}) = (I_{2} \otimes \widetilde{R})(\widetilde{R} \otimes I_{2})(I_{2} \otimes \widetilde{R})$ ?

(ii)

Does $\widetilde{R}$ satisfy $(\widetilde{R} \otimes I_{4})(I_{4} \otimes \widetilde{R})(\widetilde{R} \otimes I_{4}) = (I_{4} \otimes \widetilde{R})(\widetilde{R} \otimes I_{4})(I_{4} \otimes \widetilde{R})$ ?

Problem 20.2.10

Let $R$ be a $4 \times 4$ matrix and

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

Let $\widetilde{R} = PR$ . Suppose that $\widetilde{R}$ satisfies

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

Show the following ten matrices satisfy this equation

R_{0} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \qquad R_{1} = \begin{pmatrix} 1 & 0 & 0 & i \\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ i & 0 & 0 & 1 \end{pmatrix},

R_{2} = \begin{pmatrix} 1+t & 0 & 0 & 1 \\ 0 & (1+t^{2})^{1/2} & 1 & 0 \\ 0 & 1 & (1+t^{2})^{1/2} & 0 \\ 1 & 0 & 0 & 1-t \end{pmatrix}, \qquad R_{3} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & s & 0 & 0 \\ 0 & 1-t & st & 0 \\ 1 & 0 & 0 & -t \end{pmatrix},

R_{4} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & st & 1-t & 0 \\ 0 & 0 & s & 0 \\ 0 & 0 & 0 & -t \end{pmatrix}, \qquad R_{5} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & r & 0 & 0 \\ 0 & 1-rt & t & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix},

R_{6} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & r & 0 & 0 \\ 0 & 1-rt & t & 0 \\ 0 & 0 & 0 & -rt \end{pmatrix}, \qquad R_{7} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & s & 0 & 0 \\ 0 & 0 & s & 0 \\ 1 & 0 & 0 & s' \end{pmatrix},

R_{8} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & r & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & t \end{pmatrix}, \qquad R_{9} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & t & 0 \\ 0 & t & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}

with $\widetilde{R}_{j} = PR_{j}$ ( $j=0,1,\ldots ,9$ ), $s^{2}=1$ , $(s')^{2}=1$ and $s,p,t,r$ are arbitrary.

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Problem 20.2.11

(i)(ii)(iii)(iv)

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(i)

Let $A$ , $B$ be invertible $n \times n$ matrices with $AB \neq BA$ . Assume that

ABA = BAB \qquad \text{and} \qquad ABBA = I_{n} .

Show that $A^{4} = B^{4} = I_{n}$ .

(ii)

Find all $2 \times 2$ matrices $A$ and $B$ which satisfy the conditions given in (i).

(iii)

Find all $3 \times 3$ matrices $A$ and $B$ which satisfy the conditions given in (i).

(iv)

Find all $4 \times 4$ matrices $A$ and $B$ which satisfy the conditions given in (i).

Problem 20.2.12

Let $S$ be the unitary $4 \times 4$ matrix

S = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & i & 0 \\ 1 & 0 & 0 & 1 \\ -i & 0 & 0 & i \\ 0 & 1 & -i & 0 \end{pmatrix} .

Consider the $16 \times 16$ matrices $S \otimes I_{2} \otimes I_{2}$ , $I_{2} \otimes S \otimes I_{2}$ , $I_{2} \otimes I_{2} \otimes S$ . Note that $I_{4} = I_{2} \otimes I_{2}$ . Show that the braid-like relations

(S \otimes I_{4})(I_{2} \otimes S \otimes I_{2})(S \otimes I_{4}) = (I_{2} \otimes S \otimes I_{2})(S \otimes I_{4})(I_{2} \otimes S \otimes I_{2})

(I_{2} \otimes S \otimes I_{2})(I_{4} \otimes S)(I_{2} \otimes S \otimes I_{2}) = (I_{4} \otimes S)(I_{2} \otimes S \otimes I_{2})(I_{4} \otimes S)

are satisfied.

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Problem 20.2.13

(i)(ii)

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(i)

Find all nonzero $4 \times 4$ matrices $R$ such that

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

We obtain $8 \times 8 = 64$ conditions.

(ii)

Can one find $R$ 's with the additional condition $R^{2} = I_{4}$ ?

Problem 20.2.14

(i)(ii)(iii)

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(i)

Let $L_{1}$ , $L_{2}$ be $2 \times 2$ matrices and $R$ a $4 \times 4$ matrix. Find the conditions on $L_{1}$ , $L_{2}$ , $R$ such that

(L_{1} \otimes L_{2})R = R(L_{2} \otimes L_{1}) .

Simplify the system of equations using Grobner bases.

(ii)

Let

T = \sigma _{2} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}

and $T_{1} = T \otimes I_{2}$ , $T_{2} = I_{2} \otimes T$ . Find all invertible $4 \times 4$ matrices $R$ over $\mathbb {C}$ such that

RT_{1}T_{2} = T_{2}T_{1}R .

(iii)

Let $A$ , $B$ be given nonzero $2 \times 2$ matrices. Find all $4 \times 4$ matrices $R$ such that

R(A \otimes B) = (B \otimes A)R .

Find all $4 \times 4$ matrices $R$ such that $R(A \otimes B) = (A \otimes B)R$ .

Problem 20.2.15

Do the matrices

R_{1} = \begin{pmatrix} a & 0 & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & c & 0 & 0 \\ 0 & 0 & 0 & d \end{pmatrix}, \qquad R_{2} = \begin{pmatrix} 0 & 0 & 0 & a \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ d & 0 & 0 & 0 \end{pmatrix}

satisfy the Yang-Baxter relation

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

Or what are the conditions on $a,b,c,d$ so that the condition is satisfied?

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Problem 20.2.16

The braid linking matrix $B$ is a square symmetric $k \times k$ matrix defined by $B = (b_{ij})$ with $b_{ii}$ the sum of half-twists in the $i$ -th branch, $b_{ij}$ the sum of the crossings between the $i$ -th and the $j$ -th branches of the ribbon graph with standard insertion. Thus the $i$ -th diagonal element of $B$ is the local torsion of the $i$ -th branch. The off-diagonal elements of $B$ are twice the linking numbers of the ribbon graph for the $i$ -th and $j$ -th branches. Consider the braid linking matrix

B = \begin{pmatrix} -1 & 0 & -1 \\ 0 & 2 & -1 \\ -1 & -1 & 0 \end{pmatrix} .

Discuss. Draw a graph.

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