Computer implementation and computer algebra

Give an efficient implementation of a group in C++ and LISP. For C++ use map<pair<G,G>,G> or map<G,map<G,G> > from the Standard Template Library. The map class in C++ implements a many to one relationship between keys and values. The map class is a template class map<keytype,valuetype> which can be used in similar way to arrays, i.e. m[key]=value. Consider for example the simplest non-trivial group $\left\{ a,b\right\}$ with group operation $\cdot$ given by $a \cdot a = b \cdot b = a$ and $a \cdot b = b \cdot a = b$.

Consider the permutation group $S_{3}$. Write a C++ program for the composition of the group elements using the map class of the Standard Template Library. Then implement the inverse of each group element. Finally we determine the conjugacy classes. The group consists of six elements which we denote by $a[0], a[1], \ldots , a[5]$. The neutral (identity) element is denoted by $a[0]$. Thus we have

for $j=0,1,\ldots ,5$.

Consider the $2 \times 2$ matrix

where $-\pi \leq \alpha_{0} < \pi$, $-\pi \leq \alpha_{1} < \pi$, $0 \leq \alpha_{2} \leq \pi$. Write a Computer algebra program in SymbolicC++ that calculates

and thus finds the generators

Thus

Let $\alpha \in \mathbb {R}$. Given the matrix

Write a SymbolicC++ program that calculates the vector

and shows that $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-x_{3}^{2}$ is invariant.

Let $L$ be a Lie algebra. One analyzes the structure of a Lie algebra $L$ by studying its ideals. A subspace $I$ of a Lie algebra $L$ is called an ideal of $L$ if $x \in L$ and $y \in I$ together imply $[x,y] \in I$. If $L$ has no ideals except itself and 0, and if moreover $[L,L] \neq 0$ we call the Lie algebra $L$ simple.

If $L$ is a Lie algebra and $x \in L$, the operator $\operatorname {ad}X$ that maps $y$ to $[x,y]$ is a linear transformation of $L$ into itself

Then $x \to \operatorname {ad}x$ is a representation of the Lie algebra $L$ with $L$ itself considered as the vector space of the representation. The Killing form of a Lie algebra is the symmetric bilinear form

where $\tr$ denotes the trace. If $x_{j}$ ($j = 0,1,\ldots ,n-1$) form a basis of $L$ then

is called the metric tensor for $L$. Note that a Lie algebra is semisimple if and only if the matrix $(g_{jk})$ is nonsingular, i.e., $\det ((g_{jk})) \neq 0$.

We consider the simple Lie algebra $\mathfrak {so}(3)$ with the basis elements $x(0), x(1), x(2)$ and the commutation relations

We find

Write a SymbolicC++ program that calculates the metric tensor for this Lie algebra using a two-dimensional array comm(i,j) to store the commutators, and determines whether the Lie algebra is semisimple.

Let $P = \left\{ P_0, P_2, \ldots , P_{k-1} \right\}$ be a semi-group of permutations on the set $\Omega = \left\{ 0,1,\ldots ,n-1 \right\}$. A semi-group is closed under the group operation, but the inverse and identity need not be a member of the set. The transitivity set (or orbit) containing $i \in \Omega$ is the set of images under the action of products of elements of $P$. Write a C++ program which finds the orbits given $P$ and $\Omega$.

Let $\mathcal{B}_n$ denote the braid group on $n-1$ strands. $\mathcal{B}_n$ is generated by the elementary braids (generators) $\left\{ b_1, b_2, \ldots , b_{n-1} \right\}$ with the braid relations

Actually one should better write $b_{12}, b_{23}, \ldots , b_{n-1n}$ instead of $b_1, b_2, \ldots , b_{n-1}$. 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

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

and the corresponding inverse operation

Use computer algebra to show that $B_1, B_2, \ldots , B_{n-1}$ satisfy the braid condition

if

Write a C++ program that generates all $n \times n$ permutation matrices. In main we use these matrices to find the permutation matrix that satisfies

where $\oplus$ is the direct sum.

Let $n$ be a positive integer. There are exactly as many irreducible representations of the permutation group $S_n$ as there are partitions $\lambda_j$ of $n$

For example for $n=4$ we have the 5 partitions 4000 3100 2200 2110 1111 Let $k, n$ be positive integers. The number of partitions $p(1,n)$ can be found from the recursion

A number of Hamilton systems can be written in the form

where $L$ and $A$ are time-dependent $n \times n$ matrices. This is called the Lax representation of the Hamilton system. We find that

and the that $\tr [L^k]$ ($k=1,2,\ldots$) are first integrals. Consider the Hamilton function (Toda lattice)

Introducing the quantities

and cyclic boundary conditions (i.e., $q_4 \equiv q_1$) we find that the Hamilton equations of motion take the form (with $b_3 = 0$)

where $j=1,2$. Introducing the matrices (Lax pair)

the equations of motion can be written as Lax representation. From $L$ we find the first integral as $\tr [L^n]$, where $n = 1,2,\ldots$, where $\tr$ denotes the trace. We obtain

Write a SymbolicC++ program lax.cpp that finds $[L,A]$ and shows that $\tr (L)$, $\tr (L^2)$ and the determinant of $L$ are first integrals.

The $n$-qubit Pauli group is defined by

where $\sigma_x, \sigma_y, \sigma_z$ are the $2\times 2$ Pauli matrices and $I_2$ is the $2\times 2$ identity matrix. The dimension of the Hilbert space under consideration is $\dim \mathcal{H} = 2^n$. Thus each element of the Pauli group $\mathcal{P}_n$ is (up to an overall phase $\pm 1,\pm i$) a Kronecker product of Pauli matrices and $2\times 2$ identity matrices acting on $n$ qubits. The order of the Pauli group is $2^{2n+2}$. Thus for $n=1$ we have the order 16. For $n=2$ we have order 64. Write a SymbolicC++ program that implements the Pauli group for $n=2$.

Let $b_1^\dagger , b_2^\dagger$ be Bose creation operator and let $I$ be the identity operator. The semi-simple Lie algebra $\mathfrak {su}(1,1)$ is generated by

with the commutation relations

We use the ordering $K_+, K_-, K_0$ for the basis. Write a computer algebra program that finds the ajoint representation for the Lie algebra.

The exceptional Lie algebra $\mathfrak {g}_2$ has rank 2 and dimension 14. A basis is given by

with the commutation relations $[H_1,H_2] = 0$ and

Thus

Thus $H_1, X_1, Y_1$ and $H_2, X_2, Y_2$ each span the sub Lie algebra $\mathfrak {sl}(2,\mathbb {C})$ of $\mathfrak {g}_2$. The commutator table for $\mathfrak {g}_2$ (Table 4.1) is, where entry $(i,j)$ gives $[\text{row}_i, \text{column}_j]$ for $i < j$ and is left blank (not part of the table) otherwise:

Note that $\mathfrak {sl}(3,\mathbb {C})$ is a sub Lie algebra of $\mathfrak {g}_2$. Write a SymbolicC++ program that finds the adjoint representation.

Let $b_1, b_2, b_3$ be Bose annihilation operators. Show that

are a representation of the Lie algebra $\mathfrak {su}(3)$. Write a SymbolicC++ program that implements this representation.