Chapter 3

Lie Algebras

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Problem 3.1

Let $L$ be a Lie algebra and $a, b, h \in L$ . Assume that

[a,h] = 0, \qquad [b,h] = 0.

Calculate $[[a,b],h]$ .

Problem 3.2

Let $e_{1}, e_{2}, e_{3}$ be the generators of a three dimensional Lie algebra with the commutation relation

[e_{1},e_{3}] = e_{1}, \qquad [e_{2},e_{3}] = e_{1}+e_{2}, \qquad [e_{1},e_{2}] = 0 \, .

Find

[e_{1},[e_{2},e_{3}]] + [e_{3},[e_{1},e_{2}]] \, .

Problem 3.3

(i)(ii)

Let $L$ be a three dimensional Lie algebra with basis $a, b, c$ and commutation relations (solvable Lie algebra)

[a,b] = 0, \qquad [b,c] = a+kb, \qquad [c,a] = -ka+b, \qquad k > 0 \, .

(i)

Find the commutator $[[b,c],[c,a]]$ .

(ii)

Check the Jacobi identity

[[a,b],c] + [[c,a],b] + [[b,c],a] = 0 \, .

Problem 3.4

Let $L$ be a Lie algebra and let $I$ be an ideal of $L$ . We define the bracket $[\, , \, ] : L/I \to L/I$ by

[a+I, b+I] := [a,b] + I

where $a, b \in L$ . Show that the quotient vector space becomes a Lie algebra under this bracket.

Problem 3.5

Let $L$ be a Lie algebra. A linear map $D : L \to L$ is called a derivation if

D[a,b] = [Da,b] + [a,Db]

for all $a, b \in L$ . The collection of all derivatives of $L$ is denoted by $\operatorname {Der}(L)$ . Let $n \in \mathbb {N}$ . Show that (Leibniz rule)

D^{n}[a,b] = \sum _{j=0}^{n} {}^{n}C_{j} [D^{j}a, D^{n-j}b]

where the binomial coefficients ${}^{n}C_{j}$ are defined for an arbitrary field $\mathbb {F}$

{}^{n}C_{0} = {}^{n}C_{n} = 1, \qquad {}^{n}C_{j} = {}^{n-1}C_{j-1} + {}^{n-1}C_{j} \quad \text{for} \quad 0 < j < n \, .

Problem 3.6

A Lie algebra $L$ is called simple if $L$ is non-abelian and has no proper ideals. Show that for a simple Lie algebra the derived Lie algebra $L^{(1)} = [L,L]$ is equal to $L$ , i.e. $L^{(1)} = L$ .

Problem 3.7

A Lie algebra $L$ is called semisimple if $L \neq 0$ and $L$ has no abelian ideals $\neq 0$ . Show that a finite dimensional Lie algebra $L$ is semisimple if and only if $\operatorname {rad}L = 0$ .

Problem 3.8

Show that the adjoint representation of a semisimple Lie algebra is faithful. A faithful representation of a Lie algebra is a linear representation in which different elements of $L$ are represented by distinct linear mappings.

Problem 3.9

Consider the Lie algebra with basis $\{ e_{1}, e_{2}, e_{3}\}$ and the commutators

[e_{1},e_{2}] = e_{3}, \qquad [e_{2},e_{3}] = e_{1}, \qquad [e_{3},e_{1}] = e_{2} \, .

Find the adjoint representation.

Problem 3.10

Apply the Levi decomposition theorem to the finite dimensional Lie algebra $\mathfrak {gl}(2,\mathbb {R})$ . Consider as basis

X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad I_{2} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \, .

Problem 3.11

Consider a finite dimensional Lie algebra with basis elements $\{ X_{\alpha } : \alpha = 1, \ldots , r \}$ . The commutator may be expanded in terms of the basis

[X_{\alpha }, X_{\beta }] = \sum _{\gamma =1}^{r} C_{\alpha \beta }^{\gamma } X_{\gamma } \tag {1}

( $\alpha , \beta = 1, \ldots , r$ ) where $C_{\alpha \beta }^{\gamma }$ are called the structure constants. The structure constants $C_{\alpha \beta }^{\gamma }$ in (1) are constants. The structure constants, defined by the commutation relations (1) satisfy the relations

C_{\alpha \beta }^{\gamma } = -C_{\beta \alpha }^{\gamma }

\sum _{\rho =1}^{r} \left( C_{\alpha \beta }^{\rho } C_{\rho \gamma }^{\delta } + C_{\beta \gamma }^{\rho } C_{\rho \alpha }^{\delta } + C_{\gamma \alpha }^{\rho } C_{\rho \beta }^{\delta } \right) = 0

where $\alpha , \beta , \gamma = 1, \ldots , r$ . This is the Third Fundamental Theorem of Lie.

Problem 3.12

(i)(ii)

The Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ is spanned by the $2 \times 2$ matrices

h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}

with

[e,f] = h, \qquad [h,e] = 2e, \qquad [h,f] = -2f \, .

(i)

Let $x$ be a linear combination of the basis $h, e, f$

x = c_{1}h + c_{2}e + c_{3}f

where $c_{1}, c_{2}, c_{3} \in \mathbb {R}$ . Calculate the commutators $[h,x], [e,x], [f,x]$ .

(ii)

Show that the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ is simple.

Problem 3.13

The Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ is simple. Study the Lie algebra $\mathfrak {sl}(2,\mathbb {F})$ , where $\operatorname {char}\mathbb {F} = 2$ .

Problem 3.14

Show that for a simple Lie algebra $L$ the derived algebra $L^{(1)} = [L,L]$ is equal to $L$ .

Problem 3.15

The Lie algebra $\mathfrak {gl}(n,\mathbb {R})$ can be represented by $n \times n$ matrices over $\mathbb {R}$ . Consider the standard basis for the vector space of all $n \times n$ matrices over $\mathbb {R}$ consisting of the matrices $M_{ij}$ (having 1 in the $(i,j)$ -position and 0 elsewhere). Find the commutator

[M_{ij}, M_{k\ell }] \, .

Problem 3.16

The semisimple Lie algebra $\mathfrak {sl}(3,\mathbb {R})$ has dimension 8. The standard basis is given by

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

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

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

e_{13} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad f_{13} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \, .

Find the commutator table. Find the rank of the Lie algebra.

Problem 3.17

Give a Lie algebra which is semisimple, but not simple. Provide a matrix representation.

Problem 3.18

(i)(ii)(iii)

(i)

Classify all real two-dimensional Lie algebras, where $\mathbb {F} = \mathbb {R}$ . Denote the basis elements by $X$ and $Y$ .

(ii)

Find a representation of the non-abelian two-dimensional Lie algebra using differential operators.

(iii)

Find a representation of the non-abelian two-dimensional Lie algebra using $2 \times 2$ matrices.

Problem 3.19

Let $I_{n}$ be the $n \times n$ identity matrix. Can one find $n \times n$ matrices $A, B$ over $\mathbb {C}$ such that $[A,B] = I_{n}$ ?

Problem 3.20

(i)(ii)

(i)

Let $A$ be an associative algebra over a field $\mathbb {F}$ . We define the commutator

[a,b] := ab - ba

for all $a, b \in A$ . Show that with this definition of the commutator we have a Lie algebra.

(ii)

Give an example.

Problem 3.21

A finite-dimensional Lie algebra over a field $\mathbb {F}$ of characteristic 0 (for example $\mathbb {R}$ and $\mathbb {C}$ ) is called reductive if its adjoint representation is completely reducible. Give an example of a Lie algebra which is reductive but not semisimple.

Problem 3.22

(i)(ii)(iii)

Consider the vector space of analytic functions $f : \mathbb {R} \to \mathbb {R}$ . We define the bracket

[f,g] := f\frac{dg}{dx} - \frac{df}{dx}g \, .

(i)

Show that this defines an infinite dimensional Lie algebra.

(ii)

Consider the analytic functions

f(x) = \sin (x), \qquad g(x) = \cos (x), \qquad h(x) = 1 \, .

Show that these function for a basis of a finite dimensional Lie algebra. Is the Lie algebra semisimple?

(iii)

Let $m, n \in \mathbb {Z}$ . Consider

p(x) = x^{m}, \qquad q(x) = x^{n} \, .

Find $[p,q]$ .

Problem 3.23

Let $f, g, p, q : \mathbb {R} \to \mathbb {R}$ be analytic functions. Consider the differential operators

f(x)\frac{d}{dx} + p(x), \qquad g(x)\frac{d}{dx} + q(x) \, .

Calculate the commutator

\left[ f(x)\frac{d}{dx} + p(x), \, g(x)\frac{d}{dx} + q(x) \right] h(x)

where $h$ is an analytic function. Discuss.

Problem 3.24

Let $f, g : W \to W$ be two maps, where $W$ is a topological vector space ( $u \in W$ ). Assume that the Gateaux derivative of $f$ and $g$ exists, i.e.

f'(u)[v] := \left. \frac{\partial f(u+\epsilon v)}{\partial \epsilon } \right|_{\epsilon =0}, \qquad g'(u)[v] := \left. \frac{\partial g(u+\epsilon v)}{\partial \epsilon } \right|_{\epsilon =0} \, . \tag {1}

The Lie product (or commutator) of $f$ and $g$ is defined by

[f,g](u) := f'(u)[g(u)] - g'(u)[f(u)] \, . \tag {2}

Let

f(u) := \frac{\partial u}{\partial t} - u\frac{\partial u}{\partial x} - \frac{\partial ^{3} u}{\partial x^{3}}, \qquad g(u) := \frac{\partial u}{\partial t} - \frac{\partial ^{2} u}{\partial x^{2}} \, . \tag {3}

Calculate the commutator $[f,g](u)$ .

Problem 3.25

(i)(ii)

Let $L$ be the real vector space $\mathbb {R}^{3}$ . Let $\overrightarrow {a}, \overrightarrow {b} \in L$ . Define

\overrightarrow {a} \times \overrightarrow {b} := \begin{pmatrix} a_{2}b_{3}-a_{3}b_{2} \\ a_{3}b_{1}-a_{1}b_{3} \\ a_{1}b_{2}-a_{2}b_{1} \end{pmatrix} \, . \tag {1}

Equation (1) is the cross or vector product.

(i)

Show that $L$ is a Lie algebra.

(ii)

Show that in general

\overrightarrow {a} \times (\overrightarrow {b} \times \overrightarrow {c}) \neq (\overrightarrow {a} \times \overrightarrow {b}) \times \overrightarrow {c} \, .

Problem 3.26

The simple Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ is spanned by the $2 \times 2$ matrices

h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}

with $[e,f] = h$ , $[h,e] = 2e$ , $[h,f] = -2f$ . Show that the bracket relations are also satisfied by the matrices

\rho _{2}(h) = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -2 \end{pmatrix},

\rho _{2}(e) = \begin{pmatrix} 0 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \qquad \rho _{2}(f) = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 2 & 0 \end{pmatrix}

and

\rho _{3}(h) = \begin{pmatrix} 3 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -3 \end{pmatrix},

\rho _{3}(e) = \begin{pmatrix} 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad \rho _{3}(f) = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \end{pmatrix} \, .

Problem 3.27

(i)(ii)

Consider as a basis for the simple Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ the invertible matrices with trace 0

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

(i)

Find the commutators.

(ii)

Show that the eigenvalues of these matrices form a group under multiplication.

Problem 3.28

The Lie algebra $\mathfrak {su}(n)$ are the $n \times n$ matrices $X$ with the conditions $X^{*} = -X$ , $\tr X = 0$ . Find a basis for $\mathfrak {su}(3)$ under the condition that the elements of the basis are orthogonal to each other with respect to the scalar product

\left\langle X, Y \right\rangle := \tr (XY^{*}) \, .

Problem 3.29

The Lie algebra $\mathfrak {su}(4)$ consists of all $4 \times 4$ matrices over $\mathbb {C}$ such that $A^{*} = -A$ , $\tr A = 0$ . The first condition is that the matrices are skew-hermitian. Give a basis for the Lie algebra under the condition that the elements of the basis are orthogonal to each other with respect to the scalar product

\left\langle A, B \right\rangle := \tr (AB^{*}) \, .

Problem 3.30

(i)(ii)

Let $X$ be a $2 \times 2$ matrix over $\mathbb {C}$ . Let

L = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \, .

(i)

Find the condition on $X$ such that $X^{*}L + LX = 0_{2}$ . Then $X$ is an element of the Lie algebra $\mathfrak {u}(1,1)$ .

(ii)

Give a basis of the Lie algebra. Is the Lie algebra semisimple?

Problem 3.31

There is only one simple complex Lie algebra of rank 1. If $\alpha = \lambda e_{1}$ is a nonzero root, $-\lambda e_{1}$ is the other. The canonical choice for $\lambda$ is $1/\sqrt{2}$ . The commutators for the basis

E_{-1/\sqrt{2}}, \quad H_{1}, \quad E_{+1/\sqrt{2}}

are

[H_{1}, E_{\pm 1/\sqrt{2}}] = \pm \frac{1}{\sqrt{2}} E_{\pm 1/\sqrt{2}}, \qquad [E_{+1/\sqrt{2}}, E_{-1/\sqrt{2}}] = \frac{1}{\sqrt{2}} H_{1} \, .

Find a $2 \times 2$ matrix representation for this semisimple Lie algebra.

Problem 3.32

(i)(ii)

Consider the Lie algebra $\mathfrak {gl}(2,\mathbb {R})$ . The $2 \times 2$ matrices

e_{1} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \qquad e_{2} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad e_{3} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad e_{4} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}

form a basis of $\mathfrak {gl}(2,\mathbb {R})$ .

(i)

Find the adjoint representation.

(ii)

Find the Killing form and the metric tensor for the Lie algebra. From these results decide whether the Lie algebra is semisimple.

Problem 3.33

Let $e_{1}, e_{2}, e_{3}$ be the generators of a three-dimensional Lie algebra with the commutation relation

[e_{1},e_{3}] = e_{1}, \qquad [e_{2},e_{3}] = e_{1}+e_{2}, \qquad [e_{1},e_{2}] = 0 \, .

Find the adjoint representation. Is the Lie algebra semisimple?

Problem 3.34

Let $I_{n}$ be the $n \times n$ unit matrix. Consider the $2n \times 2n$ matrices

E_{1} = \frac{1}{2} \begin{pmatrix} I_{n} & 0_{n} \\ 0_{n} & -I_{n} \end{pmatrix}, \qquad E_{2} = \frac{1}{2} \begin{pmatrix} 0_{n} & I_{n} \\ I_{n} & 0_{n} \end{pmatrix}, \qquad E_{3} = \frac{1}{2} \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix} \, .

Find the commutators $[E_{1},E_{2}], [E_{1},E_{3}], [E_{3},E_{2}]$ . Discuss.

Problem 3.35

Consider the Lie algebra $\mathfrak {u}(1,1)$ . A basis using $2 \times 2$ matrices is given by

\tau _{0} = \frac{1}{2} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \qquad \tau _{1} = \frac{1}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},

\tau _{2} = \frac{1}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \tau _{3} = \frac{1}{2} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \, .

Find the commutators. Find the $4 \times 4$ matrix

(A_{jk}) := \tr (\tau _{j}\tau _{k}) \qquad j,k = 0,1,2,3 \, .

Is the matrix invertible?

Problem 3.36

Let $H$ be a nonzero $n \times n$ hermitian matrix. Let $E$ be a nonzero $n \times n$ matrix. Assume that $[H,E] = aE$ , where $a \in \mathbb {R}$ and $a \neq 0$ . Show that the matrix $E$ cannot be hermitian.

Problem 3.37

Consider the vectors in $\mathbb {R}^{2}$

\alpha = \frac{1}{2\sqrt{3}}(1,\sqrt{3}), \qquad \beta = \frac{1}{2\sqrt{3}}(1,-\sqrt{3}) \, .

Then

\alpha + \beta = \frac{1}{\sqrt{3}}(1,0) \, .

Find eight $3 \times 3$ matrices

H_{1}, \; H_{2}, \; E_{\alpha }, \; E_{-\alpha }, \; E_{\beta }, \; E_{-\beta }, \; E_{\alpha +\beta }, \; E_{-(\alpha +\beta )}

such that the following commutation relations are satisfied

[H_{1}, E_{\pm \alpha }] = \pm \frac{1}{2\sqrt{3}}E_{\pm \alpha }, \qquad [H_{2}, E_{\pm \alpha }] = \pm \frac{1}{2}E_{\pm \alpha }

[H_{1}, E_{\pm \beta }] = \pm \frac{1}{2\sqrt{3}}E_{\pm \beta }, \qquad [H_{2}, E_{\pm \beta }] = \pm \frac{1}{2}E_{\pm \beta }

[H_{1}, E_{\pm (\alpha +\beta )}] = \frac{1}{\sqrt{3}}E_{\pm (\alpha +\beta )}, \qquad [H_{2}, E_{\pm (\alpha +\beta )}] = 0

[E_{\alpha }, E_{-\alpha }] = \frac{H_{1}}{2\sqrt{3}} + \frac{H_{2}}{2}, \qquad [E_{\beta }, E_{-\beta }] = \frac{H_{1}}{2\sqrt{3}} - \frac{H_{2}}{2},

[E_{\alpha +\beta }, E_{-(\alpha +\beta )}] = \frac{H_{1}}{\sqrt{3}}, \qquad [E_{\alpha }, E_{\beta }] = \frac{1}{\sqrt{6}}E_{\alpha +\beta },

[E_{\alpha }, E_{\alpha +\beta }] = 0, \qquad [E_{\beta }, E_{\alpha +\beta }] = 0

[E_{\alpha }, E_{-(\alpha +\beta )}] = -\frac{1}{\sqrt{6}}E_{-\beta }, \qquad [E_{\beta }, E_{-(\alpha +\beta )}] = \frac{1}{\sqrt{6}}E_{-\alpha }

and $[H_{1},H_{2}] = 0$ . We can assume that $H_{1}, H_{2}$ are diagonal matrices.

Problem 3.38

The Lie algebra $\mathfrak {su}(5)$ consists of all $5 \times 5$ matrices with

X^{*} = -X, \qquad \tr X = 0

for $X \in \mathfrak {su}(5)$ . Give a basis of $\mathfrak {su}(5)$ . The basis elements should be orthogonal with respect to the scalar product $\tr (X_{j}X_{k}^{*})$ .

Problem 3.39

(i)(ii)

(i)

Is the three dimensional Heisenberg algebra with the basis $\{ x,y,z\}$ and the commutators $[x,y] = z$ , $[x,z] = 0$ , $[y,z] = 0$ nilpotent?

(ii)

Find a matrix representation with $3 \times 3$ matrices.

Problem 3.40

If a Lie algebra is nilpotent, then the Killing form vanishes. Can we conclude that if the Killing form vanishes then the Lie algebra is nilpotent?

Problem 3.41

Show that the Lie algebra consisting of $3 \times 3$ matrices over $\mathbb {R}$ (or $\mathbb {C}$ ) with zeros on and below the main diagonal is nilpotent.

Problem 3.42

Consider the vector space of $3 \times 3$ matrices over $\mathbb {R}$ and the two linearly independent matrices

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

Construct a Lie algebra by calculating the commutators. Thus $A$ and $B$ would be the generators of the Lie algebra.

Problem 3.43

Let $z \in \mathbb {C}$ . Consider the vector fields

V_{1} = -\frac{\partial }{\partial z} - \frac{\partial }{\partial \bar{z}}, \qquad V_{2} = i\frac{\partial }{\partial \bar{z}} - i\frac{\partial }{\partial z}, \qquad V_{3} = \frac{2}{i}\left( z\frac{\partial }{\partial \bar{z}} - \bar{z}\frac{\partial }{\partial z} \right) \, .

Find the commutators and show that we have a basis of a Lie algebra.

Problem 3.44

Consider the vector fields

V_{12} = x_{1}\frac{\partial }{\partial x_{2}} - x_{2}\frac{\partial }{\partial x_{1}}, \qquad V_{31} = x_{3}\frac{\partial }{\partial x_{1}} - x_{1}\frac{\partial }{\partial x_{3}}, \qquad V_{23} = x_{2}\frac{\partial }{\partial x_{3}} - x_{3}\frac{\partial }{\partial x_{2}} \, .

Find the commutators $[V_{12},V_{31}], [V_{31},V_{23}], [V_{23},V_{12}]$ and show that $V_{12}, V_{31}, V_{23}$ form a basis of a Lie algebra. Is the Lie algebra semisimple?

Problem 3.45

As an extension of the previous problem we can consider the vector fields ( $n \geq 3$ )

V_{jk} = -x_{j}\frac{\partial }{\partial x_{k}} + x_{k}\frac{\partial }{\partial x_{j}}, \qquad j = 1,2,\ldots ,n, \quad j < k \, .

There are $\frac{1}{2}n(n-1)$ such vector fields. Find the commutator $[V_{jk}, V_{k\ell }]$ .

Problem 3.46

The Lie algebra $\mathfrak {o}(2,1)$ admits the basis

D_{12} = -x_{1}\frac{\partial }{\partial x_{2}} + x_{2}\frac{\partial }{\partial x_{1}}, \qquad E_{23} = x_{2}\frac{\partial }{\partial x_{3}} + x_{3}\frac{\partial }{\partial x_{2}}, \qquad E_{31} = x_{3}\frac{\partial }{\partial x_{1}} + x_{1}\frac{\partial }{\partial x_{3}} \, .

Find the commutators.

Problem 3.47

Let $n \geq 2$ and $0 \leq p \leq n$ . The real semisimple Lie algebra $\mathfrak {o}(p,n-p)$ is of dimension $\frac{1}{2}n(n-1)$ and is spanned by the vector fields

D_{kj} = -D_{jk}, \qquad E_{kj} = E_{jk}

D_{jk} = -x_{j}\frac{\partial }{\partial x_{k}} + x_{k}\frac{\partial }{\partial x_{j}}, \qquad 1 \leq j < k \leq p, \quad p < j < k \leq n

E_{jk} = x_{j}\frac{\partial }{\partial x_{k}} + x_{k}\frac{\partial }{\partial x_{j}}, \qquad 1 \leq j \leq p, \quad p < k \leq n \, .

Find the commutators.

Problem 3.48

Let $z, w \in \mathbb {C}$ . Consider the vector fields

T_{1} = -\frac{w}{2i}\frac{\partial }{\partial z} - \frac{z}{2i}\frac{\partial }{\partial w}, \qquad T_{2} = -\frac{w}{2}\frac{\partial }{\partial z} + \frac{z}{2}\frac{\partial }{\partial w}, \qquad T_{3} = \frac{z}{2}\frac{\partial }{\partial z} - \frac{w}{2}\frac{\partial }{\partial w} \, .

Find the commutators $[T_{1},T_{2}], [T_{2},T_{3}], [T_{3},T_{1}]$ . Let

T_{+} := T_{1} + iT_{2}, \qquad T_{-} := T_{1} - iT_{2} \, .

Find $T_{+}$ and $T_{-}$ . Consider the function $z^{m}w^{n}$ , where $m$ is a nonnegative integer and $n$ is a negative integer with $n < -m$ . Show that $z^{m}w^{n}$ is an eigenfunction of $T_{3}$ . Find the eigenvalue. Find $T_{+}(z^{m}w^{n})$ and $T_{-}(z^{m}w^{n})$ .

Problem 3.49

Consider the operators

L_{x} = i\left( \cos (\phi )\cot (\alpha )\frac{\partial }{\partial \phi } + \sin (\phi )\frac{\partial }{\partial \alpha } \right)

L_{y} = i\left( \sin (\phi )\cot (\alpha )\frac{\partial }{\partial \phi } - \cos (\phi )\frac{\partial }{\partial \alpha } \right)

L_{z} = -i\frac{\partial }{\partial \phi } \, .

By calculating the commutators $[L_{x},L_{y}], [L_{y},L_{z}], [L_{z},L_{x}]$ show that we have a basis of a Lie algebra.

Problem 3.50

Consider the linear differential operators

S_{z} = -x\frac{d}{dx} + \ell , \qquad S_{-} = x, \qquad S_{+} = -x\frac{d^{2}}{dx^{2}} + 2\ell \frac{d}{dx}

acting on an analytic function $f$ . Find the commutators $[S_{+},S_{-}], [S_{+},S_{z}], [S_{-},S_{z}]$ . Discuss.

Problem 3.51

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices with the commutation relation

[\sigma _{1},\sigma _{2}] = 2i\sigma _{3}, \qquad [\sigma _{2},\sigma _{3}] = 2i\sigma _{1}, \qquad [\sigma _{3},\sigma _{1}] = 2i\sigma _{2} \, .

Thus we have

\mathfrak {su}(2) = \left\langle i\sigma _{1}, i\sigma _{2}, i\sigma _{3}\right\rangle \, .

Let $\otimes$ be the Kronecker product. Find the three commutators

[\sigma _{1} \otimes I_{2} + I_{2} \otimes \sigma _{2}, \; \sigma _{2} \otimes I_{2} + I_{2} \otimes \sigma _{3}]

[\sigma _{2} \otimes I_{2} + I_{2} \otimes \sigma _{3}, \; \sigma _{3} \otimes I_{2} + I_{2} \otimes \sigma _{1}]

[\sigma _{3} \otimes I_{2} + I_{2} \otimes \sigma _{1}, \; \sigma _{2} \otimes I_{2} + I_{2} \otimes \sigma _{3}] \, .

Discuss.

Problem 3.52

The Pauli spin matrices are given by

\sigma _{1} := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma _{2} := \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma _{3} := \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \, .

Their multiplication table is

\sigma _{1}\sigma _{2} = -\sigma _{2}\sigma _{1} = i\sigma _{3}, \qquad \sigma _{2}\sigma _{3} = -\sigma _{3}\sigma _{2} = i\sigma _{1}, \qquad \sigma _{3}\sigma _{1} = -\sigma _{1}\sigma _{3} = i\sigma _{2} \, .

Let

A := \sigma _{1} \otimes I_{2}, \qquad B := \sigma _{2} \otimes I_{2}, \qquad C := \sigma _{3} \otimes I_{2} \, .

Find the commutators $[A,B], [B,C], [C,A]$ .

Problem 3.53

Consider the Pauli spin matrices $\sigma_{1}, \sigma_{2}, \sigma_{3}$ and the $4 \times 4$ matrices

\sigma _{1} \otimes \sigma _{2}, \qquad \sigma _{2} \otimes \sigma _{3}, \qquad \sigma _{3} \otimes \sigma _{2} \, .

Find the commutators

[\sigma _{1} \otimes \sigma _{2}, \sigma _{2} \otimes \sigma _{3}], \qquad [\sigma _{2} \otimes \sigma _{3}, \sigma _{3} \otimes \sigma _{1}], \qquad [\sigma _{3} \otimes \sigma _{1}, \sigma _{1} \otimes \sigma _{2}] \, .

Discuss.

Problem 3.54

Consider the $4 \times 4$ matrices

S_{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes I_{2}, \qquad S_{2} = I_{2} \otimes \begin{pmatrix} 1 & 0 \\ 0 & \omega \end{pmatrix}

where $\omega = \mathrm{exp}\left(i\pi \right) = -1$ . Find $S_{1}^{2}, S_{2}^{2}, S_{1}S_{2}S_{1}^{-1}S_{2}^{-1}$ and the commutator $[S_{1},S_{2}]$ .

Problem 3.55

(i)(ii)(iii)

Let $\omega = \mathrm{exp}\left(2\pi i/4\right) \equiv \mathrm{exp}\left(\pi i/2\right) = i$ . Consider the four $64 \times 64$ invertible matrices

S_{1} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \omega & 0 & 0 \\ 0 & 0 & \omega ^{2} & 0 \\ 0 & 0 & 0 & \omega ^{3} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \omega ^{3} & 0 & 0 \\ 0 & 0 & \omega ^{2} & 0 \\ 0 & 0 & 0 & \omega \end{pmatrix} \otimes I_{4}

S_{2} = I_{4} \otimes \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix} \otimes I_{4}

S_{3} = I_{4} \otimes \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \omega & 0 & 0 \\ 0 & 0 & \omega ^{2} & 0 \\ 0 & 0 & 0 & \omega ^{3} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \omega ^{3} & 0 & 0 \\ 0 & 0 & \omega ^{2} & 0 \\ 0 & 0 & 0 & \omega \end{pmatrix}

S_{4} = I_{4} \otimes I_{4} \otimes \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix} \, .

(i)

Find $S_{j}^{4}$ for $j = 1,2,3,4$ .

(ii)

Find

S_{1}S_{2}S_{1}^{-1}S_{2}^{-1}, \quad S_{2}S_{3}S_{2}^{-1}S_{3}^{-1}, \quad S_{3}S_{4}S_{3}^{-1}S_{4}^{-1}, \quad S_{4}S_{1}S_{4}^{-1}S_{1}^{-1} \, .

(iii)

Find the commutators $[S_{1},S_{2}], [S_{2},S_{3}], [S_{3},S_{4}], [S_{4},S_{1}]$ .

Problem 3.56

(i)(ii)

(i)

Consider the non-commutative two dimensional Lie algebra with $[A,B] = A$ and

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

Show that the matrices

\{ A \otimes I_{2} + I_{2} \otimes A, \; B \otimes I_{2} + I_{2} \otimes B \}

also form a non-commutative Lie algebra under the commutator, where $\otimes$ denotes the Kronecker product. Discuss.

(ii)

We denote by $\mathfrak {sl}(2,\mathbb {R})$ the semisimple Lie algebra spanned by a basis $\{ E,F,H\}$ with the Lie bracket defined by

[H,E] = 2E, \qquad [H,F] = -2F, \qquad [E,F] = H \, .

Using $2 \times 2$ matrices we have the representation

H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \, .

Let $\otimes$ be the Kronecker product. Do the three $4 \times 4$ matrices

H \otimes I_{2} + I_{2} \otimes H, \qquad F \otimes I_{2} + I_{2} \otimes F, \qquad E \otimes I_{2} + I_{2} \otimes E

form a basis of a Lie algebra? Calculate the commutators.

Problem 3.57

Consider the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with

x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

and the commutators $[x,h] = -2x$ , $[x,y] = h$ , $[y,h] = 2y$ . Consider

h \mapsto h \otimes I_{2} + I_{2} \otimes h, \qquad x \mapsto x \otimes h + h \otimes x, \qquad y \mapsto y \otimes h + h \otimes y \, .

Find the three commutators

[h\otimes I_{2}+I_{2}\otimes h, \; x\otimes h+h\otimes x], \quad [h\otimes I_{2}+I_{2}\otimes h, \; y\otimes h+h\otimes y], \quad [x\otimes h+h\otimes x, \; y\otimes h+h\otimes y]

and thus show that we have a Lie algebra. Is this Lie algebra isomorphic to $\mathfrak {sl}(2,\mathbb {R})$ ?

Problem 3.58

The four matrices

X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad I_{2} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

form a basis of the Lie algebra $\mathfrak {gl}(2,\mathbb {R})$ . We have

[X,Y] = H, \qquad [H,X] = 2X, \qquad [H,Y] = -2Y

and $[I_{2},X] = [I_{2},Y] = [I_{2},H] = 0_{2}$ . Can one find an invertible matrix $S$ over $\mathbb {R}$ such that

SXS^{-1} = Y, \qquad SYS^{-1} = X, \qquad SHS^{-1} = -H \; ?

Problem 3.59

Let $A$ be an arbitrary $n \times n$ matrix over $\mathbb {C}$ . Let $T_{1}, T_{2}$ be $n \times n$ matrices over $\mathbb {C}$ satisfying

T_{1}^{*}A + AT_{1} = 0, \qquad T_{2}^{*}A + AT_{2} = 0 \, .

Show that

[T_{1},T_{2}]^{*}A + A[T_{1},T_{2}] = 0 \, .

Problem 3.60

Show that the dimension of a Cartan sub-Lie algebra is not in general the maximal dimension of an abelian sub-Lie algebra. Consider the Lie algebra $\mathfrak {sl}(2n,\mathbb {C})$ of $2n \times 2n$ matrices of trace 0.

Problem 3.61

Show that the Lie algebras $\mathfrak {sl}(2,\mathbb {C})$ and $\mathfrak {so}(3,\mathbb {C})$ are isomorphic.

Problem 3.62

(i)(ii)

Let $L$ be any Lie algebra. If $x, y \in L$ , define

\kappa (x,y) := \tr (\operatorname {ad}x \operatorname {ad}y)

where $\kappa$ is called the Killing form and $\tr (\cdot )$ denotes the trace.

(i)

Show that

\kappa ([x,y],z) = \kappa (x,[y,z]) \, .

(ii)

If $B_{1}$ and $B_{2}$ are subspaces of a Lie algebra $L$ , we define $[B_{1},B_{2}]$ to be the subspace spanned by all products $[b_{1},b_{2}]$ with $b_{1} \in B_{1}$ and $b_{2} \in B_{2}$ . This is the set of sums

\sum _{j} [b_{1j}, b_{2j}]

where $b_{1j} \in B_{1}$ and $b_{2j} \in B_{2}$ . Let

L_{(1)} := [L,L], \qquad L_{(2)} := [L,L_{(1)}], \qquad \ldots , \qquad L_{(k)} := [L,L_{(k-1)}] \, .

A Lie algebra $L$ is called nilpotent if

L_{(n)} = \{ 0\}

for some positive integer $n$ . Prove that if $L$ is nilpotent, the Killing form of $L$ is identically zero.

Problem 3.63

Let $\mathfrak {gl}(n,\mathbb {R})$ be the Lie algebra of all $n \times n$ matrices over $\mathbb {R}$ . Let

x \in \mathfrak {gl}(n,\mathbb {R}) \tag {1}

have $n$ distinct eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ in $\mathbb {R}$ . Prove that the eigenvalues of the $n^{2} \times n^{2}$ matrix

\operatorname {ad}x \tag {2}

are the $n^{2}$ scalars $\lambda_{i} - \lambda_{j}$ , where $i,j = 1,2,\ldots ,n$ .

Problem 3.64

Consider the Lie algebra $\mathfrak {so}(4)$ . Show that the Lie algebra $\mathfrak {so}(4)$ is semisimple, but not simple. Consider as a basis of $\mathfrak {so}(4)$ the six vector fields

V_{12} = x_{2}\frac{\partial }{\partial x_{1}} - x_{1}\frac{\partial }{\partial x_{2}}, \qquad V_{23} = x_{3}\frac{\partial }{\partial x_{2}} - x_{2}\frac{\partial }{\partial x_{3}}, \qquad V_{31} = x_{1}\frac{\partial }{\partial x_{3}} - x_{3}\frac{\partial }{\partial x_{1}}

W_{41} = x_{1}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{1}}, \qquad W_{42} = x_{2}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{2}}, \qquad W_{43} = x_{3}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{3}} \, .

Problem 3.65

(i)(ii)

Consider the semisimple Lie algebra $\mathfrak {so}(2,1)$ whose elements $X_{1}, X_{2}, X_{3}$ satisfy the commutation relations

[X_{1},X_{2}] = X_{3}, \qquad [X_{2},X_{3}] = -X_{1}, \qquad [X_{3},X_{1}] = X_{2} \, .

(i)

Find the adjoint representation.

(ii)

Let $Y_{j}$ ( $j = 1,\ldots ,n$ ) be a basis for a Lie algebra. Then $g_{jk} = \kappa (Y_{j},Y_{k})$ is called the metric tensor. From the adjoint representation find the metric tensor.

Problem 3.66

Consider the two matrices

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

Show that

[\mathrm{exp}\left(tA\right), \mathrm{exp}\left(tB\right)] = t^{2}[A,B] \, .

Problem 3.67

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Show that the $n \times n$ matrices $R$ over $\mathbb {C}$ satisfying $R^{*}A + AR = 0$ form a Lie algebra.

Problem 3.68

Show that the Lie algebras $\mathfrak {so}(3)$ and $\mathfrak {su}(2)$ of the Lie groups $SO(3)$ and $SU(2)$ are isomorphic.

Problem 3.69

Consider the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with the basis $H, X, Y$ and the commutators

[H,X] = 2X, \qquad [H,Y] = -2Y, \qquad [X,Y] = H

with

H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \, .

Assume that $\overrightarrow {v} \in \mathbb {R}$ is an eigenvector of $H$ , i.e. $H\overrightarrow {v} = \alpha \overrightarrow {v}$ , where $\alpha$ is the eigenvalue. Calculate $H(X\overrightarrow {v})$ . Discuss.

Problem 3.70

Let $S^{1} := \left\{ (x_{1},x_{2}) : x_{1}^{2}+x_{2}^{2} = 1\right\}$ . Let $\operatorname {Diff}^{+}(S^{1})$ be the space of orientation-preserving diffeomorphism of $S^{1} \to S^{1}$ . With the product given by the composition of diffeomorphism it is an infinite dimensional Lie group. The Lie algebra is the space of vector fields on $S^{1}$ . A Lie subalgebra of the complexification of the Lie algebra of $\operatorname {Diff}^{+}(S^{1})$ is given by

L := \left\{ f(z)\frac{d}{dz} \, : \, f(z) \in \mathbb {C}{}[z,z^{-1}] \right\} \, .

This is a subspace of the space of vector fields on $S^{1}$ , regarding $S^{1}$ as the unit circle in $\mathbb {C}$ . We choose a basis $\{ L_{n}\}_{n\in \mathbb {Z}}$ of $L$ defining

L_{n} := -z^{n+1}\frac{d}{dz} \, .

Find the commutator $[L_{n},L_{m}]$ .

Problem 3.71

(i)(ii)

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. Let $I_{2}$ be the $2 \times 2$ identity matrix and $0_{2}$ be the $2 \times 2$ zero matrix. Consider the four $4 \times 4$ matrices

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

J_{3} = \frac{i}{2}\begin{pmatrix} 0_{2} & \sigma _{3} \\ -\sigma _{3} & 0_{2} \end{pmatrix}, \qquad J_{0} = \frac{i}{2}\begin{pmatrix} 0_{2} & I_{2} \\ -I_{2} & 0_{2} \end{pmatrix} \, .

(i)

Find the commutators.

(ii)

Consider the six $4 \times 4$ matrices

K_{1} = \frac{i}{2}\begin{pmatrix} 0_{2} & \sigma _{3} \\ \sigma _{3} & 0_{2} \end{pmatrix}, \qquad K_{2} = \frac{i}{2}\begin{pmatrix} I_{2} & 0_{2} \\ 0_{2} & -I_{2} \end{pmatrix}, \qquad K_{3} = -\frac{i}{2}\begin{pmatrix} 0_{2} & \sigma _{1} \\ \sigma _{1} & 0_{2} \end{pmatrix},

Q_{1} = -\frac{i}{2}\begin{pmatrix} \sigma _{3} & 0_{2} \\ 0_{2} & -\sigma _{3} \end{pmatrix}, \qquad Q_{2} = \frac{i}{2}\begin{pmatrix} 0_{2} & I_{2} \\ I_{2} & 0_{2} \end{pmatrix}, \qquad Q_{3} = \frac{i}{2}\begin{pmatrix} \sigma _{1} & 0_{2} \\ 0_{2} & -\sigma _{1} \end{pmatrix} \, .

Find the commutators. Find also the commutators $[K_{j},J_{k}], [Q_{j},J_{k}]$ , where $j,k = 1,2,3$ and $[K_{j},J_{0}], [Q_{j},J_{0}]$ . Thus show that we have a basis of a 10 dimensional Lie algebra.

Problem 3.72

(i)(ii)

(i)

Consider the matrices

N_{1} = \begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad N_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & i \\ 0 & 0 & 0 \end{pmatrix}, \qquad L = \begin{pmatrix} 0 & -i/2 & 0 \\ i/2 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \, .

Show that they form a basis of a Lie algebra.

(ii)

Is the Lie algebra semisimple?

Problem 3.73

(i)(ii)

(i)

Consider the differential operators

J_{3} = \frac{i}{2}\left( p\frac{\partial }{\partial x} - x\frac{\partial }{\partial p} \right)

and

K_{1} = \frac{i}{2}\left( p\frac{\partial }{\partial p} - x\frac{\partial }{\partial x} \right), \qquad K_{2} = -\frac{i}{2}\left( x\frac{\partial }{\partial p} + p\frac{\partial }{\partial x} \right) \, .

Find the commutators. Show that we have a basis for a semisimple Lie algebra.

(ii)

Consider the differential operators

\tilde{J}_{3} = \frac{1}{4}\left( x^{2} - \frac{d^{2}}{dx^{2}} \right)

and

\tilde{K}_{1} = \frac{i}{4}\left( x\frac{d}{dx} + \frac{d}{dx}x \right), \qquad \tilde{K}_{2} = \frac{1}{4}\left( x^{2} + \frac{d^{2}}{dx^{2}} \right) \, .

Find the commutators. Show that we have a basis for a semisimple Lie algebra. Are the two Lie algebras isomorphic?

Problem 3.74

Show that the vector fields

J_{1} = -\frac{i}{2}\left( \left( p_{2}\frac{\partial }{\partial x_{1}} + p_{1}\frac{\partial }{\partial x_{2}} \right) - \left( x_{2}\frac{\partial }{\partial p_{1}} + x_{1}\frac{\partial }{\partial p_{2}} \right) \right)

J_{2} = -\frac{i}{2}\left( \left( x_{1}\frac{\partial }{\partial x_{2}} - x_{2}\frac{\partial }{\partial x_{1}} \right) + \left( p_{1}\frac{\partial }{\partial p_{2}} - p_{2}\frac{\partial }{\partial p_{1}} \right) \right)

J_{3} = -\frac{i}{2}\left( \left( p_{1}\frac{\partial }{\partial x_{1}} - p_{2}\frac{\partial }{\partial x_{2}} \right) - \left( x_{1}\frac{\partial }{\partial p_{1}} - x_{2}\frac{\partial }{\partial p_{2}} \right) \right)

J_{0} = -\frac{i}{2}\left( \left( p_{1}\frac{\partial }{\partial x_{1}} + p_{2}\frac{\partial }{\partial x_{2}} \right) - \left( x_{1}\frac{\partial }{\partial p_{1}} + x_{2}\frac{\partial }{\partial p_{2}} \right) \right)

K_{1} = -\frac{i}{2}\left( \left( p_{1}\frac{\partial }{\partial x_{1}} - p_{2}\frac{\partial }{\partial x_{2}} \right) + \left( x_{1}\frac{\partial }{\partial p_{1}} - x_{2}\frac{\partial }{\partial p_{2}} \right) \right)

K_{2} = \frac{i}{2}\left( -\left( x_{1}\frac{\partial }{\partial x_{1}} + x_{2}\frac{\partial }{\partial x_{2}} \right) + \left( p_{1}\frac{\partial }{\partial p_{1}} + p_{2}\frac{\partial }{\partial p_{2}} \right) \right)

K_{3} = \frac{i}{2}\left( \left( p_{2}\frac{\partial }{\partial x_{1}} + p_{1}\frac{\partial }{\partial x_{2}} \right) + \left( x_{2}\frac{\partial }{\partial p_{1}} + x_{1}\frac{\partial }{\partial p_{2}} \right) \right)

Q_{1} = \frac{i}{2}\left( \left( x_{1}\frac{\partial }{\partial x_{1}} - x_{2}\frac{\partial }{\partial x_{2}} \right) - \left( p_{1}\frac{\partial }{\partial p_{1}} - p_{2}\frac{\partial }{\partial p_{2}} \right) \right)

Q_{2} = -\frac{i}{2}\left( \left( p_{1}\frac{\partial }{\partial x_{1}} + p_{2}\frac{\partial }{\partial x_{2}} \right) + \left( x_{1}\frac{\partial }{\partial p_{1}} + x_{2}\frac{\partial }{\partial p_{2}} \right) \right)

Q_{3} = -\frac{i}{2}\left( \left( x_{2}\frac{\partial }{\partial x_{1}} + x_{1}\frac{\partial }{\partial x_{2}} \right) - \left( p_{2}\frac{\partial }{\partial p_{1}} + p_{1}\frac{\partial }{\partial p_{2}} \right) \right)

form a basis of a Lie algebra under the commutator.

Problem 3.75

Consider the spin-matrices (spin-1)

S_{+} = \begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{pmatrix}, \qquad S_{-} = \begin{pmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{pmatrix}, \qquad S_{z} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \, .

Find the commutators and thus show that we have a basis of a Lie algebra. Show that we have a faithful representation using differential operators by

S_{+} \to z\frac{\partial }{\partial \zeta }, \qquad S_{-} \to \zeta \frac{\partial }{\partial z}, \qquad S_{z} \to \frac{1}{2}\left( z\frac{\partial }{\partial z} - \zeta \frac{\partial }{\partial \zeta } \right) \, .

Problem 3.76

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . Consider the expression

(A \otimes B - B \otimes A) \oplus (-[A,B])

where $\otimes$ denotes the Kronecker product, $\oplus$ denotes the direct sum and $[\, , \, ]$ denotes the commutator. Find the trace of this expression. This expression plays a role for universal enveloping algebras.

Problem 3.77

(i)(ii)

Consider the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with the basis $e_{1}, e_{2}, e_{3}$ and the commutators

[e_{2},e_{1}] = 2e_{1}, \qquad [e_{2},e_{3}] = -2e_{3}, \qquad [e_{1},e_{3}] = e_{2} \, .

(i)

Find the adjoint representation.

(ii)

Calculate $\mathrm{exp}\left(\operatorname {ad}e_{1}\right), \mathrm{exp}\left(\operatorname {ad}e_{2}\right), \mathrm{exp}\left(\operatorname {ad}e_{3}\right)$ .

Problem 3.78

The semisimple Lie algebra $\mathfrak {sl}(3,\mathbb {C})$ has dimension 8 and rank 2. The standard basis is given by

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

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

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

e_{13} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad f_{13} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \, .

Note that $e_{1} = f_{1}^{T}$ , $e_{2} = f_{2}^{T}$ , $e_{13} = f_{13}^{T}$ . Find the commutation relation table.

Problem 3.79

(i)(ii)

Consider the $4 \times 4$ matrices

J_{+} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad J_{-} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},

J_{3} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad E = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \, .

(i)

Find the commutators and thus show that we have a basis of a Lie algebra. Is the Lie algebra semisimple?

(ii)

Let $\alpha , \beta , \gamma , \delta \in \mathbb {R}$ . Calculate

\mathrm{exp}\left(\alpha J_{+}\right), \quad \mathrm{exp}\left(\beta J_{-}\right), \quad \mathrm{exp}\left(\gamma J_{3}\right), \quad \mathrm{exp}\left(\delta E\right) \, .

Show that $\mathrm{exp}\left(\alpha J_{+}\right)\mathrm{exp}\left(\delta E\right) = \mathrm{exp}\left(\delta E\right)\mathrm{exp}\left(\alpha J_{+}\right)$ .

Problem 3.80

(i)(ii)

(i)

Let $U, V$ be $n \times n$ unitary matrices such that the commutator $[U,V] = W$ is again a unitary matrix, i.e. $W^{*} = W^{-1}$ . Show that

R + R^{-1} = I_{n}

where $R = U^{-1}V^{-1}UV$ is the group commutator of $U^{-1}$ and $V^{-1}$ .

(ii)

Give an example for such matrices $U, V$ for the case $n=2$ .

Problem 3.81

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $[A,B] \equiv AB-BA = 0_{n}$ . Let $\otimes$ be the Kronecker product. Can we conclude that

A \otimes B - B \otimes A = 0_{n^{2}} \; ?

Problem 3.82

Consider the Lie algebra $\mathfrak {sl}(3,\mathbb {C})$ . Given the roots find the structure constants. The semisimple Lie algebra $\mathfrak {sl}(3,\mathbb {C})$ has dimension 8 and rank 2. The root system of $\mathfrak {sl}(3,\mathbb {C})$ consists of 6 vectors which go from the origin to vertices of regular hexagon. Check the Jacobi identities for the resulting structure constants.

Problem 3.83

Consider the semisimple Lie algebra $\mathfrak {sl}(n+1,\mathbb {F})$ . Let $E_{i,j}$ ( $i,j \in \{ 1,2,\ldots ,n+1\}$ ) denote the standard basis, i.e. $(n+1) \times (n+1)$ matrices with all entries zero except for the entry in the $i$ th row and $j$ th column which is one. We can form a Cartan-Weyl basis with

H_{j} := E_{j,j} - E_{j+1,j+1}, \qquad j \in \{ 1,2,\ldots ,n\} \, .

Show that $E_{i,j}$ are root vectors for $i \neq j$ , i.e. there exists $\lambda_{H,i,j} \in \mathbb {F}$ such that

[H,E_{i,j}] = \lambda _{H,i,j}E_{i,j}

for all $H \in \left\langle H_{1},\ldots ,H_{n}\right\rangle$ .

Problem 3.84

(i)(ii)

(i)

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . Let $z \in \mathbb {C}$ . Let $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {C}^{n}$ . Calculate

\mathrm{exp}\left(z(A \otimes B)\right)(\overrightarrow {u} \otimes \overrightarrow {v})

where $\otimes$ denotes the Kronecker product.

(ii)

Simplify the result from (i) if $A^{2} = I_{n}$ and $B^{2} = I_{n}$ .

Problem 3.85

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . What is the condition on $A, B$ such that

[A \otimes B, B \otimes A] = 0 \; ?

Problem 3.86

Let $A, B, H$ be $n \times n$ matrices such that

[H,A] = 0, \qquad [H,B] = 0 \, .

Show that

[H \otimes I_{n} + I_{n} \otimes H, A \otimes B] = 0 \, .

Problem 3.87

(i)(ii)

Let $A_{1}, A_{2}, A_{3}$ be $n \times n$ matrices over $\mathbb {C}$ . The ternary commutator $[A_{1},A_{2},A_{3}]$ (also called the ternutator) is defined as

\begin{align} [A_{1},A_{2},A_{3}] & := \sum _{\pi \in S_{3}} \operatorname {sgn}(\pi ) A_{\pi (1)}A_{\pi (2)}A_{\pi (3)} \\ & \equiv A_{1}A_{2}A_{3} + A_{2}A_{3}A_{1} + A_{3}A_{1}A_{2} - A_{1}A_{3}A_{2} - A_{2}A_{1}A_{3} - A_{3}A_{2}A_{1} \, . \end{align}

(i)

Let $n=2$ and consider the Pauli spin matrices $\sigma_{x}, \sigma_{y}, \sigma_{z}$ . Calculate the ternutator

[\sigma _{x}, \sigma _{y}, \sigma _{z}] \, .

(ii)

Calculate

A_{1}\otimes A_{2}\otimes A_{3} + A_{2}\otimes A_{3}\otimes A_{1} + A_{3}\otimes A_{1}\otimes A_{2} - A_{1}\otimes A_{3}\otimes A_{2} - A_{2}\otimes A_{1}\otimes A_{3} - A_{3}\otimes A_{2}\otimes A_{1} \, .

Problem 3.88

Let $c, x \in \mathbb {R}$ . Consider the operators

K_{1} = -ic\cos (x), \qquad K_{2} = -ic\sin (x), \qquad K_{3} = -\frac{d}{dx} \, .

Find the commutators and thus show that we have a basis of a Lie algebra. Is the Lie algebra semisimple?

Problem 3.89

(i)(ii)(iii)

$SU(2)$ is the Lie group of all $2 \times 2$ unitary unimodular matrices, i.e., the group of all matrices of the form

A = \begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix}, \qquad \left|a\right|^{2} + \left|b\right|^{2} = 1

where $a, b \in \mathbb {C}$ and $\bar{a}$ is the complex conjugate of $a$ .

(i)

Find the inverse of $A$ .

(ii)

The Lie algebra $\mathfrak {su}(2)$ of the Lie group $SU(2)$ can be identified with the vector space of all $2 \times 2$ complex skew-hermitian matrices of trace 0. A basis of $\mathfrak {su}(2)$ is given by

G_{1} = \begin{pmatrix} 0 & i/2 \\ i/2 & 0 \end{pmatrix} = \frac{i}{2}\sigma _{x},

G_{2} = \begin{pmatrix} 0 & 1/2 \\ -1/2 & 0 \end{pmatrix} = \frac{i}{2}\sigma _{y},

G_{3} = \begin{pmatrix} -i/2 & 0 \\ 0 & i/2 \end{pmatrix} = -\frac{i}{2}\sigma _{z} \, .

Find the commutation relations $[G_{1},G_{2}], [G_{3},G_{1}], [G_{2},G_{3}]$ .

(iii)

Find the matrix

M = \mathrm{exp}\left(\phi _{1}G_{3}\right)\mathrm{exp}\left(\theta G_{1}\right)\mathrm{exp}\left(\phi _{2}G_{3}\right) \, .

Problem 3.90

(i)(ii)

Consider the real Lie algebra $L$ constructed from a linear vector space with basis

\left\{ A_{ij} : i,j = 1,2,\ldots ,n \right\} \, .

We assume that the structure of the Lie algebra is given by the commutation relations

[A_{ij}, A_{k\ell }] = \delta _{kj}A_{i\ell } - \delta _{\ell i}A_{kj}

where $\delta_{ij}$ is the Kronecker symbol. Hence we consider the Lie algebra $\mathfrak {gl}(n,\mathbb {R})$ . The dimension of $L$ is $n^{2}$ .

(i)

Show that the Lie algebra is not simple.

(ii)

Show that the mapping

A_{ij} \mapsto x_{i}\frac{\partial }{\partial x_{j}}

is a faithful representation of the Lie algebra $\mathfrak {gl}(n,\mathbb {R})$ .

Problem 3.91

Consider the Lie group $O(2,\mathbb {R})$ and its Lie algebra $\mathfrak {o}(2,\mathbb {R})$ . Show that $O(2,\mathbb {R})$ is not commutative. Discuss the properties of $\mathfrak {o}(2,\mathbb {R})$ .

Problem 3.92

The Euclidean group $E(2)$ in the plane is associated with the Lie algebra $\mathfrak {e}(2)$ defined by the commutation relations

[X_{1},X_{2}] = X_{3}, \qquad [X_{1},X_{3}] = -X_{2}, \qquad [X_{2},X_{3}] = 0 \, .

Show that the Lie algebra is not semisimple.

Problem 3.93

(i)(ii)

The vector space of $n \times n$ matrices over $\mathbb {R}$ form a Hilbert space with the scalar product defined

\left\langle A, B \right\rangle := \tr (AB^{*}) \, .

This implies a norm

\left\| A\right\| ^{2} = \tr (AA^{*})

and a distance $\left\| A-B\right\|^{2} = \tr ((A-B)(A-B)^{*})$ . Consider the matrices

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

which are elements of the simple Lie algebra $\mathfrak {so}(3)$ .

(i)

Are the matrices orthogonal to each other?

(ii)

Find the distance between $A$ and $B$ .

Problem 3.94

Let $A, B$ be $n \times n$ matrices over $\mathbb {C}$ . We define the quasi-multiplication

A \bullet B := \frac{1}{2}(AB+BA) \, .

Obviously $A \bullet B = B \bullet A$ . Show that $(A^{2} \bullet B) \bullet A = A^{2} \bullet (B \bullet A)$ . This is called the Jordan identity. This is an example of a Jordan algebra.

Problem 3.95

The $2 \times 2$ matrices

E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad H = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

form a basis for the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with the commutators $[H,E] = 2E$ , $[H,F] = -2F$ , $[E,F] = H$ . Let $k \geq 1$ . Do the matrices ( $m_{j} = j(k-j+1)$ )

\rho _{k}(H) = \begin{pmatrix} k & 0 & 0 & \ldots & 0 \\ 0 & k-2 & 0 & \ldots & 0 \\ 0 & 0 & k-4 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & -k \end{pmatrix}

\rho _{k}(F) = \begin{pmatrix} 0 & 0 & 0 & \ldots & \cdot & 0 \\ 1 & 0 & 0 & \ldots & \cdot & 0 \\ 0 & 1 & 0 & \ldots & \cdot & 0 \\ \vdots & \vdots & \vdots & \ldots & \cdot & \vdots \\ 0 & 0 & 0 & \ldots & 1 & 0 \end{pmatrix}, \qquad \rho _{k}(E) = \begin{pmatrix} 0 & m_{1} & 0 & \ldots & 0 \\ 0 & 0 & m_{2} & \ldots & 0 \\ 0 & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ldots & m_{k} \\ 0 & 0 & 0 & \ldots & 0 \end{pmatrix}

satisfy the same commutation relations?

Problem 3.96

The exceptional Lie algebra $\mathfrak {g}_{2}$ has the simple roots

(0,1,-1) \qquad (1,-2,1) \, .

Find the Cartan matrix. Verify the properties of the Cartan matrix.

Problem 3.97

If $\{ X_{j} : j = 1,2,\ldots ,m \}$ form a basis for a finite dimensional Lie algebra $L$ then

g_{ij} = K(X_{i},X_{j})

is called the metric tensor for $L$ . In terms of the structure constants,

g_{ij} = \sum _{r,s=1}^{m} C_{is}^{r}C_{jr}^{s} \, .

For a given semisimple Lie algebra $L$ with basis $\{ X_{j} : j = 1,\ldots ,m\}$ , the Casimir operator is a quantity (which is not an element of the Lie algebra) that commutes with each element of the Lie algebra. The quadratic Casimir operator $C$ of a given semisimple Lie algebra $L$ is defined by

C := \sum _{i=1}^{m}\sum _{j=1}^{m} (g^{ij})X_{i}X_{j}

where $(g^{ij})$ is the inverse matrix of $(g_{ij})$ . Note that the Casimir operator is independent of the choice of the basis. Consider the Lie algebra $\mathfrak {so}(3)$

L_{1} := -i\left( x_{2}\frac{\partial }{\partial x_{3}} - x_{3}\frac{\partial }{\partial x_{2}} \right)

L_{2} := -i\left( x_{3}\frac{\partial }{\partial x_{1}} - x_{1}\frac{\partial }{\partial x_{3}} \right)

L_{3} := -i\left( x_{1}\frac{\partial }{\partial x_{2}} - x_{2}\frac{\partial }{\partial x_{1}} \right)

with the commutation relations $[L_{1},L_{2}] = iL_{3}$ , $[L_{2},L_{3}] = iL_{1}$ , $[L_{3},L_{1}] = iL_{2}$ . Find the quadratic Casimir operator.

Problem 3.98

Let $\mathbb {Z}_{2} = \{ 0,1\}$ be the finite group of integers under addition modulo 2. The super-Lie algebra $\mathfrak {g} = \mathfrak {g}_{0}+\mathfrak {g}_{1}$ is a $\mathbb {Z}_{2}$ -graded Lie algebra with a super-Lie bracket $[\, , \, ]$ which satisfies

[A,B] = AB - (-1)^{d(A)d(B)}BA

and (super Jacobi identity)

[A,[B,C]] = [[A,B],C] + (-1)^{d(A)d(B)}[B,[A,C]]

where $d(X) = 0$ (1) if $A \in \mathfrak {g}_{0}$ ( $\mathfrak {g}_{1}$ ). $d(A)$ is called the degree of $A$ . We denote by $[\, , \, ]_{-}$ the commutator and by $[\, , \, ]_{+}$ the anticommutator. The super-Lie algebra $\mathfrak {sl}(m/n)$ is defined by

\mathfrak {sl}(m/n) := \left\{ X = \begin{pmatrix} R & S \\ T & U \end{pmatrix} \, : \, \operatorname {Str}(X) := \tr (R) - \tr (U) = 0 \right\}

where $R$ is an $m \times m$ matrix, $S$ a $m \times n$ matrix, $T$ an $n \times m$ matrix and $D$ an $n \times n$ matrix. We assume that $m \neq n$ . Consider the super-Lie algebra $\mathfrak {sl}(2/1)$ . A basis is given by

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

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

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

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

i.e. we have four even elements $E_{1}, F_{1}, H_{1}, H_{2}$ and four odd elements $E_{2}, F_{2}, G_{1}, G_{2}$ . Find the non-vanishing commutators and anticommutators.

Problem 3.99

Let $q > 0$ . The algebra $\mathfrak {su}_{q}(2)$ is generated by $X_{+}, X_{-} = (X_{+})^{*}$ and $H = H^{*}$ with deformed commutation relations

[H,X_{+}] = 2X_{+}, \qquad [H,X_{-}] = -2X_{-}, \qquad [X_{+},X_{-}] = \frac{q^{H}-q^{-H}}{q-q^{-1}} \, .

Show by direct computation of the commutators that this algebra has the same matrix representation as the undeformed Lie algebra $\mathfrak {su}(2)$ , i.e.

X_{+} \mapsto \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad X_{-} \mapsto \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad H \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \, .

Problem 3.100

Let $V, W$ be smooth vector fields defined in $\mathbb {R}^{n}$ . Let $f,g : \mathbb {R}^{n} \to \mathbb {R}$ be smooth functions. Consider now the pairs $(V,f), (W,g)$ . One defines a commutator of such pairs as

[(V,f),(W,g)] := ([V,W], L_{V}g - L_{W}f)

where $L_{V}(\cdot )$ denotes the Lie derivative. Now $[V,W]$ is a vector field again and $L_{V}g - L_{V}f$ is a smooth function again. The Lie derivative is linear and obeys the product rule. Consider $n=2$ . Let

V = u_{2}\frac{\partial }{\partial u_{1}} - u_{1}\frac{\partial }{\partial u_{2}}, \qquad W = u_{1}\frac{\partial }{\partial u_{1}} + u_{2}\frac{\partial }{\partial u_{2}}

and $f(u_{1},u_{2}) = g(u_{1},u_{2}) = u_{1}^{2}+u_{2}^{2}$ . Calculate the commutator.

Section 3.1Programming Problems[2/100%]

Problem 3.101

(i)(ii)

(i)

Let $L$ be a three dimensional Lie algebra with basis $a, b, c$ and the commutation relations (simple Lie algebra $\mathfrak {so}(3)$ )

[a,b] = c, \qquad [b,c] = a, \qquad [c,a] = b \, .

Write a SymbolicC++ program to test whether the Jacobi identity is satisfied.

(ii)

Let $L$ be a three dimensional Lie algebra with basis $a, b, c$ and the commutation relations (solvable Lie algebra)

[a,b] = 0, \qquad [b,c] = a+kb, \qquad [c,a] = -ka+b, \qquad k > 0 \, .

Write a SymbolicC++ program to test whether the Jacobi identity is satisfied.

Problem 3.102

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

(\operatorname {ad}x)y := [x,y] \, .

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

K(x,y) := \tr (\operatorname {ad}x\operatorname {ad}y)

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

g_{jk} = K(x_{j},x_{k})

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

[x(0),x(1)] = x(2), \qquad [x(1),x(2)] = x(0), \qquad [x(2),x(0)] = x(1) \, .

We find

\operatorname {ad}x(0) = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad \operatorname {ad}x(1) = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix},

\operatorname {ad}x(2) = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \, .

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.

Section 3.2Supplementary Problems[8/0%]

Problem 3.103

Let $L$ be a Lie algebra. Given $H, K \subset L$ . We define their bracket by

[H,K] := \left\langle [h,k] \, : \, h \in H, \, k \in K \right\rangle \, .

The derived Lie algebra of $L$ is the ideal $[L,L]$ . Show that the quotient Lie algebra $L/[L,L]$ is an abelian Lie algebra.

Problem 3.104

Show that the Lie algebra consisting of $n \times n$ matrices over $\mathbb {R}$ (or $\mathbb {C}$ ) with zeros below the main diagonal is solvable but not nilpotent.

Problem 3.105

(i)(ii)

(i)

Let $L$ be a finite-dimensional Lie algebra over a field $\mathbb {F}$ . The linear map $\kappa : L \times L \to \mathbb {F}$ defined by

\kappa (a,b) := \tr (\operatorname {ad}(a)\operatorname {ad}(b))

is called the Cartan-Killing form or Killing form of $L$ , where $\tr$ denotes the trace. Show that

\kappa ([a,b],c) = \kappa (a,[b,c]) \, .

(ii)

Consider the Cartan-Killing form of the finite-dimensional linear Lie algebra $\mathfrak {gl}(n,\mathbb {R})$ . Show that

\kappa (X,Y) = \tr ((\operatorname {ad}X)(\operatorname {ad}Y)) = 2n\tr (XY) - 2(\tr X)(\tr Y) \, .

Problem 3.106

Let $A, B$ be $2 \times 2$ matrices over $\mathbb {C}$ . We define

A \star B := \begin{pmatrix} a_{11} & 0 & 0 & a_{12} \\ 0 & b_{11} & b_{12} & 0 \\ 0 & b_{21} & b_{22} & 0 \\ a_{21} & 0 & 0 & a_{22} \end{pmatrix} \, .

Consider the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with the basis

h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \, .

Find $h \star h$ , $e \star e$ , $f \star f$ . Calculate the commutators

[e\star e, f\star f], \qquad [h\star h, e\star e], \qquad [h\star h, f\star f] \, .

Discuss.

Problem 3.107

Do the vector fields

\frac{\partial }{\partial x_{j}}, \qquad x_{k}\frac{\partial }{\partial x_{j}}, \qquad x_{\ell }x_{k}\frac{\partial }{\partial x_{j}}, \qquad j,k,\ell = 1,\ldots ,n

form a Lie algebra under the commutator? If so classify the Lie algebra. Consider first the case $n=1$ .

Problem 3.108

Let $s = 0, 1/2, 1, 3/2, \ldots$ . The spin $s$ generators of the Lie algebra $\mathfrak {sl}(2,\mathbb {C})$ at site $k$ are given by

S_{+,k} = z_{k}^{2}\frac{\partial }{\partial z_{k}} - 2sz_{k}, \qquad S_{-,k} = -\frac{\partial }{\partial z_{k}}, \qquad S_{3,k} = z_{k}\frac{\partial }{\partial z_{k}} - s \, .

Find the commutators.

Problem 3.109

Let $k \geq 1$ . Do the vector fields

\frac{\partial }{\partial x}, \quad \frac{\partial }{\partial u}, \quad x\frac{\partial }{\partial u}, \quad \ldots , \quad x^{k}\frac{\partial }{\partial u}, \quad x\frac{\partial }{\partial x} + (k+1)u\frac{\partial }{\partial u}

form a basis of a Lie algebra under the commutator? If so classify the Lie algebra.

Problem 3.110

(i)(ii)(iii)

The infinitesimal generators (vector fields) of the six dimensional Lie algebra $\mathfrak {so}(4)$ can be written in terms of the variables $(x_{1},x_{2},x_{3},x_{4})$ as

X_{1} = x_{3}\frac{\partial }{\partial x_{2}} - x_{2}\frac{\partial }{\partial x_{3}}, \qquad X_{2} = x_{1}\frac{\partial }{\partial x_{3}} - x_{3}\frac{\partial }{\partial x_{1}}, \qquad X_{3} = x_{2}\frac{\partial }{\partial x_{1}} - x_{1}\frac{\partial }{\partial x_{2}},

Y_{1} = x_{1}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{1}}, \qquad Y_{2} = x_{2}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{2}}, \qquad Y_{3} = x_{3}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{3}} \, .

(i)

Find the commutators.

(ii)

Show that the Lie algebra is semisimple, but not simple.

(iii)

Find elements of the compact Lie group $SO(4)$ by calculating

\mathrm{exp}\left(\alpha _{1}X_{1}+\alpha _{2}X_{2}+\alpha _{3}X_{3}+\beta _{1}Y_{1}+\beta _{2}Y_{2}+\beta _{3}Y_{3}\right)

and

\mathrm{exp}\left(\alpha _{1}X_{1}\right)\mathrm{exp}\left(\alpha _{2}X_{2}\right)\mathrm{exp}\left(\alpha _{3}X_{3}\right)\mathrm{exp}\left(\beta _{1}Y_{1}\right)\mathrm{exp}\left(\beta _{2}Y_{2}\right)\mathrm{exp}\left(\beta _{3}Y_{3}\right)

where $\alpha_{j}, \beta_{j} \in \mathbb {R}$ .