Chapter 19

Lie Algebras and Matrices

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Section 19.1Chapter Problems[37/100%]
Problem 19.1.1

Consider the n×nn \times n matrices EijE_{ij} having 1 in the (i,j)(i,j) position and 0 elsewhere, where i,j=1,…,ni, j = 1, \ldots , n (elementary matrices). Calculate the commutator. Discuss.

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Problem 19.1.2

Consider the vector space of the 2×22 \times 2 matrices over R\mathbb {R} and the three 2×22 \times 2 matrices with trace equal to 0

E=(0100),F=(0010),H=(100−1). 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}.

Show that we have a basis of a Lie algebra.

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Problem 19.1.3

Consider the matrices

h1=(100000001),h2=(000010000),h3=(001000100), h_{1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad h_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad h_{3} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, e=(010000010),f=(000101000). e = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \qquad f = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.

Show that the matrices form a basis of a Lie algebra.

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Problem 19.1.4

Let AA, BB be n×nn \times n matrices over C\mathbb {C}. Calculate tr⁡([A,B])\operatorname {tr}([A,B]). Discuss.

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Problem 19.1.5

An n×nn \times n matrix XX over C\mathbb {C} is skew-hermitian if X∗=−XX^{*} = -X. Show that the commutator of two skew-hermitian matrices is again skew-hermitian. Discuss.

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Problem 19.1.6

The Lie algebra su(m)\mathfrak {su}(m) consists of all m×mm \times m matrices XX over C\mathbb {C} with the conditions X∗=−XX^{*} = -X (i.e. XX is skew-hermitian) and tr⁡(X)=0\operatorname {tr}(X) = 0. Note that exp(X)\mathrm{exp}\left(X\right) is a unitary matrix. Find a basis for su(3)\mathfrak {su}(3).

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Problem 19.1.7

Any fixed element XX of a Lie algebra LL defines a linear transformation

ad⁡(X):Z→[X,Z]for anyZ∈L. \operatorname {ad}(X): Z \to [X,Z] \quad \text{for any} \quad Z \in L.

Show that for any K∈LK \in L we have

[ad⁡(Y),ad⁡(Z)]K=ad⁡([Y,Z])K. [\operatorname {ad}(Y), \operatorname {ad}(Z)]K = \operatorname {ad}([Y,Z])K.

The linear mapping ad⁡\operatorname {ad} gives a representation of the Lie algebra known as adjoint representation.

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Problem 19.1.8

There is only one non-commutative Lie algebra LL of dimension 2. If xx, yy are the generators (basis in LL), then

[x,y]=x. [x,y] = x.
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(i)

Find the adjoint representation of this Lie algebra. Let vv, ww be two elements of a Lie algebra. Then we define

ad⁡v(w):=[v,w] \operatorname {ad}v(w) := [v,w]

and wad⁡v:=[v,w]w\operatorname {ad}v := [v,w].

(ii)

The Killing form is defined by

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

for all x,y∈Lx, y \in L. Find the Killing form.

Problem 19.1.9

Consider the Lie algebra L=sl(2,F)L = \mathfrak {sl}(2,\mathbb {F}) with char⁡(F)≠2\operatorname {char}(\mathbb {F}) \neq 2. Take as the standard basis for LL the three matrices

E=(0100),F=(0010),H=(100−1). 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}.
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(i)

Find the commutators.

(ii)

Find the adjoint representation of LL with the ordered basis { E  H  F }\{ \, E \; H \; F \, \}.

(iii)

Show that LL is simple. If LL has no ideals except itself and 0, and if moreover [L,L]≠0[L,L] \neq 0, we call LL simple. A subspace II of a Lie algebra LL is called an ideal of LL if x∈Lx \in L, y∈Iy \in I together imply [x,y]∈I[x,y] \in I. In other words show that the Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) has no proper nontrivial ideals.

Problem 19.1.10

Consider the four-dimensional Lie algebra gl(2,R)\mathfrak {gl}(2,\mathbb {R}). The elementary matrices

e1=(1000),e2=(0100),e3=(0010),e4=(0001) 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 gl(2,R)\mathfrak {gl}(2,\mathbb {R}). Find the adjoint representation.

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Problem 19.1.11

Show that the matrices

J1=(00000−1010),J2=(001000−100),J3=(0−10100000) J_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad J_{2} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \qquad J_{3} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

form generators of a Lie algebra. Is the Lie algebra simple? A Lie algebra is simple if it contains no ideals other than LL and 00.

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Problem 19.1.12

The roots of a semisimple Lie algebra are the Lie algebra weights occurring in its adjoint representation. The set of roots forms the root system, and is completely determined by the semisimple Lie algebra. Consider the semisimple Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) with the generators

E=(0100),F=(0010),H=(100−1). 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}.

Find the roots.

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Problem 19.1.13

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

H=(100−1),E=(0100),F=(0010) 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 the commutators [H,E]=2E[H,E] = 2E, [H,F]=−2F[H,F] = -2F and [E,F]=H[E,F] = H.

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(i)

Define

C:=12H2+EF+FE. C := \frac{1}{2}H^{2} + EF + FE.

Find CC. Calculate the commutators [C,H][C,H], [C,E][C,E], [C,F][C,F]. Show that CC can be written in the form

C=12H2+H+2FE. C = \frac{1}{2}H^{2} + H + 2FE.
(ii)

Consider the vector in R2\mathbb {R}^{2}

v→=(10). \overrightarrow {v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}.

Calculate the vectors Hv→H\overrightarrow {v}, Ev→E\overrightarrow {v}, Fv→F\overrightarrow {v} and Cv→C\overrightarrow {v}. Give an interpretation.

(iii)

The universal enveloping algebra U(sl(2,R))U(\mathfrak {sl}(2,\mathbb {R})) of the Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) is the associative algebra with generators HH, EE, FF and the relations

HE−EH=2E,HF−FH=−2F,EF−FE=H. HE-EH = 2E, \qquad HF-FH = -2F, \qquad EF-FE = H.

Find a basis of the Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) so that all matrices are invertible. Find the inverse matrices of these matrices. Give the commutation relations.

(iv)

Which of the EE, FF, HH matrices are nonnormal? Use linear combinations to find a basis where all elements are normal matrices.

Problem 19.1.14

Let { e,f }\{ \, e, f \, \} with

e=(0100),f=(−1000) e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad f = \begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix}

a basis for a Lie algebra. We have [e,f]=e[e,f] = e.

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(i)

Is { e⊗I2,f⊗I2 }\{ \, e \otimes I_{2}, f \otimes I_{2} \, \} a basis of a Lie algebra?

(ii)

Is { e⊗e,e⊗f,f⊗e,f⊗f }\{ \, e \otimes e, e \otimes f, f \otimes e, f \otimes f \, \} a basis of a Lie algebra?

Problem 19.1.15

We know that

E=(0100),H=(100−1),F=(0010) E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}

is an ordered basis of the simple Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) with

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

Consider

E~=(0001001000000000),H~=(1000010000−10000−1),F~=(0000000001001000). \widetilde{E} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad \widetilde{H} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}, \qquad \widetilde{F} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}.

Find the commutators [E~,H~][\widetilde{E},\widetilde{H}], [E~,F~][\widetilde{E},\widetilde{F}], [F~,H~][\widetilde{F},\widetilde{H}].

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Problem 19.1.16

The 2×22 \times 2 matrices

E=(0100),F=(0010),H=(100−1) 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 of the Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) with the commutation relations [E,H]=−2E[E,H] = -2E, [E,F]=H[E,F] = H, [F,H]=2F[F,H] = 2F.

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(i)

Let ⋆\star be the star product. Do the nine 4×44 \times 4 matrices

E⋆E,E⋆F,E⋆H,F⋆E,F⋆F,F⋆H,H⋆E,H⋆F,H⋆H E \star E, \quad E \star F, \quad E \star H, \quad F \star E, \quad F \star F, \quad F \star H, \quad H \star E, \quad H \star F, \quad H \star H

form a basis of a Lie algebra?

(ii)

Define the 4×44 \times 4 matrices

H~:=H⊗I2+I2⊗H,E~:=E⊗H−1+H⊗E,F~:=F⊗H−1+H⊗F. \widetilde{H} := H \otimes I_{2} + I_{2} \otimes H, \qquad \widetilde{E} := E \otimes H^{-1} + H \otimes E, \qquad \widetilde{F} := F \otimes H^{-1} + H \otimes F.

Find the commutators [H~,E~][\widetilde{H},\widetilde{E}], [H~,F~][\widetilde{H},\widetilde{F}], [E~,F~][\widetilde{E},\widetilde{F}].

(iii)

Consider

H~=H⊗I2+I2⊗H,E~=E⊗H+H⊗E,F~=F⊗H+H⊗F. \widetilde{H} = H \otimes I_{2} + I_{2} \otimes H, \qquad \widetilde{E} = E \otimes H + H \otimes E, \qquad \widetilde{F} = F \otimes H + H \otimes F.

Find the commutators [H~,E~][\widetilde{H},\widetilde{E}], [H~,F~][\widetilde{H},\widetilde{F}], [E~,F~][\widetilde{E},\widetilde{F}] and thus show that we have a basis of a Lie algebra. Is this Lie algebra isomorphic to sl(2,R)\mathfrak {sl}(2,\mathbb {R})?

Problem 19.1.17

A basis for the Lie algebra su(N)\mathfrak {su}(N), for odd NN, may be built from two unitary unimodular N×NN \times N matrices

g=(10000ω0000ω20⋮⋮⋮⋱⋮000…ωN−1),h=(010…0001…0⋮⋮⋮⋱⋮000…1100…0) g = \begin{pmatrix} 1 & 0 & 0 & & 0 \\ 0 & \omega & 0 & & 0 \\ 0 & 0 & \omega ^{2} & & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & \omega ^{N-1} \end{pmatrix}, \qquad h = \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{pmatrix}

where ω\omega is a primitive NN-th root of unity, i.e. with period not smaller than NN, here taken to be exp(4πi/N)\mathrm{exp}\left(4\pi i/N\right). We obviously have

hg=ωgh.(1) hg = \omega gh. \tag {1}
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(i)

Find gNg^{N} and hNh^{N}.

(ii)

Find tr⁡(g)\operatorname {tr}(g).

(iii)

Let m→=(m1,m2)\overrightarrow {m} = (m_{1},m_{2}), n→=(n1,n2)\overrightarrow {n} = (n_{1},n_{2}) and define

m→×n→:=m1n2−m2n1 \overrightarrow {m} \times \overrightarrow {n} := m_{1}n_{2} - m_{2}n_{1}

where m1=0,1,…,N−1m_{1} = 0,1,\ldots ,N-1 and m2=0,1,…,N−1m_{2} = 0,1,\ldots ,N-1. The complete set of unitary unimodular N×NN \times N matrices

Jm1,m2:=ωm1m2/2gm1hm2 J_{m_{1},m_{2}} := \omega ^{m_{1}m_{2}/2}g^{m_{1}}h^{m_{2}}

suffice to span the Lie algebra su(N)\mathfrak {su}(N), where J0,0=INJ_{0,0} = I_{N}. Find J∗J^{*}.

(iv)

Calculate Jm→Jn→J_{\overrightarrow {m}}J_{\overrightarrow {n}}.

(v)

Find the commutator [Jm→,Jn→][J_{\overrightarrow {m}}, J_{\overrightarrow {n}}].

Problem 19.1.18

Let LL be a finite dimensional Lie algebra and Z(L)Z(L) the center of LL. Show that ad⁡:L→gl(L)\operatorname {ad}: L \to \mathfrak {gl}(L) is a homomorphism of the Lie algebra LL with kernel Z(L)Z(L).

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Problem 19.1.19

Consider the 2×22 \times 2 matrices over R\mathbb {R}

A=(−1010),B=(010−1). A = \begin{pmatrix} -1 & 0 \\ 1 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix}.

Calculate the commutator C=[A,B]C = [A,B] and check whether CC can be written as a linear combination of AA and BB. If so we have a basis of a Lie algebra.

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Problem 19.1.20

A Chevalley basis for the semisimple Lie algebra sl(3,R)\mathfrak {sl}(3,\mathbb {R}) is given by

X1=(000001000),X2=(010000000),X3=(001000000), X_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \qquad X_{2} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad X_{3} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, Y1=(000000010),Y2=(000100000),Y3=(000000100), Y_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \qquad Y_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad Y_{3} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, H1=(00001000−1),H2=(1000−10000) H_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \qquad H_{2} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}

where Yj=XjTY_{j} = X_{j}^{T} for j=1,2,3j = 1,2,3. The Lie algebra has rank 2 owing to H1H_{1}, H2H_{2} and [H1,H2]=03[H_{1},H_{2}] = 0_{3}. Another basis could be formed by looking at the linear combinations

Uj=Xj+Yj,Vj=Xj−Vj. U_{j} = X_{j}+Y_{j}, \qquad V_{j} = X_{j}-V_{j}.
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(i)

Find the table of the commutator.

(ii)

Calculate the vectors of commutators

\begin{pmatrix} {[H_{1},X_{1}]} \\[H_{2},X_{1}] \end{pmatrix}, \qquad \begin{pmatrix} {[H_{1},X_{2}]} \\[H_{2},X_{2}] \end{pmatrix}, \qquad \begin{pmatrix} {[H_{1},X_{3}]} \\[H_{2},X_{3}] \end{pmatrix}

and thus find the roots.

Problem 19.1.21

Consider the 3×33 \times 3 matrices

h1=(100000001),h2=(000010000),h3=(001000100), h_{1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad h_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad h_{3} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, e=(010000010),f=(000101000). e = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \qquad f = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.

Show that the matrices form a basis of a Lie algebra.

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Problem 19.1.22

Consider the two 2×22 \times 2 matrices

A=(100−1),B=(0110) A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

where AA is the Pauli spin matrix σ3\sigma_{3} and BB the Pauli spin matrix σ1\sigma_{1}. The two matrices AA, BB are linearly independent. Let AA, BB be the generators of a Lie algebra. Classify the Lie algebra generated.

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Problem 19.1.23

The gl(1∣1)\mathfrak {gl}(1|1) superalgebra involves two even (denoted by HH and ZZ) and two odd (denoted by EE, FF) generators. The following commutation and anti-commutation relations hold

[Z,E]=[Z,F]=[Z,H]=0,[H,E]=E,[H,F]=−F,[E,F]+=Z [Z,E] = [Z,F] = [Z,H] = 0, \qquad [H,E] = E, \qquad [H,F] = -F, \qquad [E,F]_{+} = Z

and E2=F2=0E^{2} = F^{2} = 0. Find a 2×22 \times 2 matrix representation.

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Problem 19.1.24

A basis of the simple Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) is given by the traceless 2×22 \times 2 matrices

X1=12(0110),X2=12(−1001),X3=12(01−10). X_{1} = \frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad X_{2} = \frac{1}{2}\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad X_{3} = \frac{1}{2}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.
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(i)

Find the commutators [X1,X2][X_{1},X_{2}], [X2,X3][X_{2},X_{3}], [X3,X1][X_{3},X_{1}].

(ii)

Let z∈Cz \in \mathbb {C}. Find exp(zX1)\mathrm{exp}\left(zX_{1}\right), exp(zX2)\mathrm{exp}\left(zX_{2}\right), exp(zX3)\mathrm{exp}\left(zX_{3}\right).

(iii)

Let u,v,r∈Ru, v, r \in \mathbb {R}. Show that

g(u,v,r)=euX3erX2evX3 g(u,v,r) = e^{uX_{3}}e^{rX_{2}}e^{vX_{3}} =er/2(−sin⁡(u/2)sin⁡(v/2)cos⁡(v/2)sin⁡(u/2)−cos⁡(u/2)sin⁡(v/2)cos⁡(u/2)cos⁡(v/2)) = e^{r/2}\begin{pmatrix} -\sin (u/2)\sin (v/2) & \cos (v/2)\sin (u/2) \\ -\cos (u/2)\sin (v/2) & \cos (u/2)\cos (v/2) \end{pmatrix} +e−r/2(cos⁡(u/2)cos⁡(v/2)cos⁡(u/2)sin⁡(v/2)−cos⁡(v/2)sin⁡(u/2)−sin⁡(u/2)sin⁡(v/2)). + e^{-r/2}\begin{pmatrix} \cos (u/2)\cos (v/2) & \cos (u/2)\sin (v/2) \\ -\cos (v/2)\sin (u/2) & -\sin (u/2)\sin (v/2) \end{pmatrix}.
(iv)

Find g(u,v,r)−1g(u,v,r)^{-1}.

Problem 19.1.25

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

Hj:=Ej,j−Ej+1,j+1,j∈{1,2,…,n}. H_{j} := E_{j,j} - E_{j+1,j+1}, \qquad j \in \{ 1,2,\ldots ,n\} .

Show that Ei,jE_{i,j} are root vectors for i≠ji \neq j, i.e. there exists λH,i,j∈F\lambda_{H,i,j} \in \mathbb {F} such that

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

for all H∈span⁡{ H1,…,Hn }H \in \operatorname {span}\{ \, H_{1}, \ldots , H_{n} \, \}.

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Problem 19.1.26

Consider the vector space R3\mathbb {R}^{3} and the vector product ×\times. The vector product is not associative. The associator of three vectors u→\overrightarrow {u}, v→\overrightarrow {v}, w→\overrightarrow {w} is defined by

ass⁡(u→×(v→×w→)):=(u→×v→)×w→−u→×(v→×w→). \operatorname {ass}(\overrightarrow {u} \times (\overrightarrow {v} \times \overrightarrow {w})) := (\overrightarrow {u} \times \overrightarrow {v}) \times \overrightarrow {w} - \overrightarrow {u} \times (\overrightarrow {v} \times \overrightarrow {w}).

The associator measures the failure of associativity.

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(i)

Consider the unit vectors

u→=(100),v→=(010),w→=(001). \overrightarrow {u} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad \overrightarrow {v} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad \overrightarrow {w} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.

Find the associator.

(ii)

Consider the normalized vectors

u→=12(101),v→=(010),w→=12(10−1). \overrightarrow {u} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \qquad \overrightarrow {v} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad \overrightarrow {w} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}.

Find the associator.

Problem 19.1.27

Consider vectors in the vector space R3\mathbb {R}^{3} and the vector product ×\times. Consider the mapping of the vectors in R3\mathbb {R}^{3} into 3×33 \times 3 skew-symmetric matrices

(abc)↔(0c−b−c0ab−a0). \begin{pmatrix} a \\ b \\ c \end{pmatrix} \leftrightarrow \begin{pmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{pmatrix}.

Calculate the vector product

(a2b2c2)×(a1b1c1) \begin{pmatrix} a_{2} \\ b_{2} \\ c_{2} \end{pmatrix} \times \begin{pmatrix} a_{1} \\ b_{1} \\ c_{1} \end{pmatrix}

and the commutator [M1,M2][M_{1},M_{2}], where

M1=(0c1−b1−c10a1b1−a10),M2=(0c2−b2−c20a2b2−a20). M_{1} = \begin{pmatrix} 0 & c_{1} & -b_{1} \\ -c_{1} & 0 & a_{1} \\ b_{1} & -a_{1} & 0 \end{pmatrix}, \qquad M_{2} = \begin{pmatrix} 0 & c_{2} & -b_{2} \\ -c_{2} & 0 & a_{2} \\ b_{2} & -a_{2} & 0 \end{pmatrix}.

Discuss.

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Problem 19.1.28

A basis for the Lie algebra su(3)\mathfrak {su}(3) is given by the eight 3×33 \times 3 skew-hermitian matrices

X1=(0i0i00000),X2=(010−100000), X_{1} = \begin{pmatrix} 0 & i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad X_{2} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, X3=(i000−i0000),X4=(00i000i00), X_{3} = \begin{pmatrix} i & 0 & 0 \\ 0 & -i & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad X_{4} = \begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}, X5=(001000−100),X6=(00000i0i0), X_{5} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \qquad X_{6} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & i \\ 0 & i & 0 \end{pmatrix}, X7=(0000010−10),X8=13(i000i000−2i). X_{7} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}, \qquad X_{8} = \frac{1}{\sqrt{3}}\begin{pmatrix} i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & -2i \end{pmatrix}.

Find

∑j=18(Xj⊗Xj),∑j=18((XjXk)⊗Xj). \sum _{j=1}^{8}(X_{j} \otimes X_{j}), \qquad \sum _{j=1}^{8}((X_{j}X_{k}) \otimes X_{j}).

Express the results as Kronecker products of two 3×33 \times 3 matrices.

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Problem 19.1.29

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

A=(0100),B=(0001). A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.

Show that the matrices

{ A⊗I2+I2⊗A,  B⊗I2+I2⊗B } \{ \, 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. Discuss.

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Problem 19.1.30

Consider the three 3×33 \times 3 matrices

A=(010−100000),B=(00−1000100),C=(0000010−10) A = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \qquad C = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}

which satisfy the commutation relations [A,B]=C[A,B] = C, [B,C]=A[B,C] = A, [C,A]=B[C,A] = B. Thus we have a basis of the simple Lie algebra so(3)\mathfrak {so}(3). Calculate the commutators

[A⊗I3,B⊗I3],[B⊗I3,C⊗I3],[C⊗I3,A⊗I3]. [A \otimes I_{3}, B \otimes I_{3}], \qquad [B \otimes I_{3}, C \otimes I_{3}], \qquad [C \otimes I_{3}, A \otimes I_{3}].
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Problem 19.1.31
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(i)

The Heisenberg algebra h(3)\mathfrak {h}(3) is a three-dimensional non-commutative Lie algebra with basis E1E_{1}, E2E_{2}, E3E_{3} and

[E1,E2]=E3,[E1,E3]=[E2,E3]=0. [E_{1},E_{2}] = E_{3}, \qquad [E_{1},E_{3}] = [E_{2},E_{3}] = 0.

From the matrix representation

T=(0t1t300t2000),t1,t2,t3∈R T = \begin{pmatrix} 0 & t_{1} & t_{3} \\ 0 & 0 & t_{2} \\ 0 & 0 & 0 \end{pmatrix}, \qquad t_{1}, t_{2}, t_{3} \in \mathbb {R}

of an element T=t1E1+t2E2+t3E3T = t_{1}E_{1}+t_{2}E_{2}+t_{3}E_{3} in h(3)\mathfrak {h}(3). Find exp(T)\mathrm{exp}\left(T\right).

(ii)

The corresponding Lie group H(3)H(3) is the group of 3×33 \times 3 upper triangular matrices with 1's on the main diagonal

H(3)={ g=(1α1α301α2001) }. H(3) = \left\{ \, g = \begin{pmatrix} 1 & \alpha _{1} & \alpha _{3} \\ 0 & 1 & \alpha _{2} \\ 0 & 0 & 1 \end{pmatrix} \, \right\} .

Find g−1g^{-1} and gTg−1gTg^{-1}.

Problem 19.1.32

Let A1,A2,…,ArA_{1}, A_{2}, \ldots , A_{r} be n×nn \times n matrices which form a basis of a non-commutative Lie algebra. Let B1,B2,…,BsB_{1}, B_{2}, \ldots , B_{s} be n×nn \times n matrices which form a basis of a commutative Lie algebra. Let ⊗\otimes be the Kronecker product and form all elements

Aj⊗Bk,j=1,…,r,k=1,…,s. A_{j} \otimes B_{k}, \qquad j = 1, \ldots , r, \quad k = 1, \ldots , s.

Calculate the commutators

[Aj⊗Bk,Aℓ⊗Bm],j,ℓ=1,…,r,k,m=1,…,s. [A_{j} \otimes B_{k}, A_{\ell } \otimes B_{m}], \qquad j,\ell = 1,\ldots ,r, \quad k,m = 1,\ldots ,s.

Discuss.

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Problem 19.1.33

Consider the Heisenberg algebra h(1)\mathfrak {h}(1) with the generators given by n×nn \times n matrices HH, X1X_{1}, X2X_{2} and the commutation relations

[H,X1]=0n,[H,X2]=0n,[X1,X2]=H. [H,X_{1}] = 0_{n}, \qquad [H,X_{2}] = 0_{n}, \qquad [X_{1},X_{2}] = H.

Find the commutators for the matrices H⊗InH \otimes I_{n}, X1⊗InX_{1} \otimes I_{n}, X2⊗InX_{2} \otimes I_{n}.

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Problem 19.1.34
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(i)

Let AA, BB be n×nn \times n matrices. Calculate the commutator

[A⊗In+In⊗A,B⊗B] [A \otimes I_{n} + I_{n} \otimes A, B \otimes B]

and express the result using the commutator [A,B][A,B].

(ii)

Assume that [A,B]=B[A,B] = B, i.e. AA and BB form a basis of a two-dimensional Lie algebra. Express the result from (i) using [A,B]=B[A,B] = B. Discuss.

Problem 19.1.35

Let LL be a finite dimensional Lie algebra. Let C∞(S1)C^{\infty }(S^{1}) be the set of all infinitely differentiable functions, where S1S^{1} is the unit circle manifold. In the product space L⊗C∞(S1)L \otimes C^{\infty }(S^{1}) we define the Lie bracket (g1,g2∈Lg_{1}, g_{2} \in L and f1,f2∈C∞(S1)f_{1}, f_{2} \in C^{\infty }(S^{1}))

[g1⊗f1,g2⊗f2]:=[g1,g2]⊗(f1f2). [g_{1} \otimes f_{1}, g_{2} \otimes f_{2}] := [g_{1},g_{2}] \otimes (f_{1}f_{2}).

Calculate

[g1⊗f1,[g2⊗f2,g3⊗f3]]+[g3⊗f3,[g1⊗f1,g2⊗f2]]+[g2⊗f2,[g3⊗f3,g1⊗f1]]. [g_{1} \otimes f_{1}, [g_{2} \otimes f_{2}, g_{3} \otimes f_{3}]] + [g_{3} \otimes f_{3}, [g_{1} \otimes f_{1}, g_{2} \otimes f_{2}]] + [g_{2} \otimes f_{2}, [g_{3} \otimes f_{3}, g_{1} \otimes f_{1}]].
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Problem 19.1.36

The elements (generators) Z1,Z2,…,ZrZ_{1}, Z_{2}, \ldots , Z_{r} of an rr-dimensional Lie algebra satisfy the conditions

[Zμ,Zν]=∑τ=1rcμντZτ [Z_{\mu }, Z_{\nu }] = \sum _{\tau =1}^{r} c^{\tau }_{\mu \nu }Z_{\tau }

with cμντ=−cνμτc^{\tau }_{\mu \nu } = -c^{\tau }_{\nu \mu }, where the cμντc^{\tau }_{\mu \nu }'s are called the structure constants. Let AA be an arbitrary linear combination of the elements

A=∑μ=1raμZμ. A = \sum _{\mu =1}^{r} a^{\mu }Z_{\mu }.

Suppose that XX is some other linear combination such that

X=∑ν=1rbνZνand[A,X]=ρX. X = \sum _{\nu =1}^{r} b^{\nu }Z_{\nu } \qquad \text{and} \qquad [A,X] = \rho X.

This equation has the form of an eigenvalue equation, where ρ\rho is the corresponding eigenvalue and XX the corresponding eigenvector. Assume that the Lie algebra is represented by matrices. Find the secular equation for the eigenvalues ρ\rho.

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Problem 19.1.37

Let cσλτc^{\tau }_{\sigma \lambda } be the structure constants of a Lie algebra. We define

gσλ=gλσ=∑ρ=1r∑τ=1rcσρτcλτρandgσλgσλ=δσλ. g_{\sigma \lambda } = g_{\lambda \sigma } = \sum _{\rho =1}^{r}\sum _{\tau =1}^{r} c^{\tau }_{\sigma \rho }c^{\rho }_{\lambda \tau } \qquad \text{and} \qquad g^{\sigma \lambda }g_{\sigma \lambda } = \delta ^{\lambda }_{\sigma }.

A Lie algebra LL is called semisimple if and only if det⁡(∣gσλ∣)≠0\operatorname {det}\left(\left|g_{\sigma \lambda }\right|\right) \neq 0. We assume in the following that the Lie algebra is semisimple. We define

C:=∑ρ=1r∑σ=1rgρσXρXσ. C := \sum _{\rho =1}^{r}\sum _{\sigma =1}^{r} g^{\rho \sigma }X_{\rho }X_{\sigma }.

The operator CC is called Casimir operator. Let XτX_{\tau } be an element of the Lie algebra LL. Calculate the commutator [C,Xτ][C,X_{\tau }].

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Section 19.2Supplementary Problems[21/0%]
Problem 19.2.1

Given the basis of the Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R})

E=(0100),F=(0010),H=(100−1) 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}

with the commutation relations [E,F]=H[E,F] = H, [E,H]=−2E[E,H] = -2E, [F,H]=−2F[F,H] = -2F.

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(i)

Let z∈Cz \in \mathbb {C}. We define

E~=ezH/2⊗E+E⊗e−z‾H/2,F~=ez‾H/2⊗F+F⊗e−zH/2,H~=H⊗I2+I2⊗H. \widetilde{E} = e^{zH/2} \otimes E + E \otimes e^{-\overline{z}H/2}, \qquad \widetilde{F} = e^{\overline{z}H/2} \otimes F + F \otimes e^{-zH/2}, \qquad \widetilde{H} = H \otimes I_{2} + I_{2} \otimes H.

Find the commutators [E~,F~][\widetilde{E},\widetilde{F}], [E~,H~][\widetilde{E},\widetilde{H}], [F~,H~][\widetilde{F},\widetilde{H}].

(ii)

Let U(sl(2,R))U(\mathfrak {sl}(2,\mathbb {R})) be the universal enveloping algebra. Then any element of U(sl(2,R))U(\mathfrak {sl}(2,\mathbb {R})) can be expressed as a sum of product of the form FjHkEℓF^{j}H^{k}E^{\ell } where j,k,ℓ≥0j,k,\ell \geq 0. Show that

EF2=−2F+2FH+F2E. EF^{2} = -2F+2FH+F^{2}E.
Remark.
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Utilize that EF2≡[E,F2]+F2EEF^{2} \equiv [E,F^{2}]+F^{2}E.

Problem 19.2.2

Study the Lie algebra sl(2,F)\mathfrak {sl}(2,\mathbb {F}), where char⁡(F)=2\operatorname {char}(\mathbb {F}) = 2.

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Problem 19.2.3

Do the matrices

E+=(010001000),E−=(000100010),H=(10000000−1) E_{+} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \qquad E_{-} = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \qquad H = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}

form a basis for the Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R})?

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Problem 19.2.4

Show that a basis of the Lie algebra sl(2,C)\mathfrak {sl}(2,\mathbb {C}) is given by

12(0ii0),12(01−10),12(−i00i), \frac{1}{2}\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \qquad \frac{1}{2}\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad \frac{1}{2}\begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}, 12(0110),12(0−ii0),12(−1001). \frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \frac{1}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \frac{1}{2}\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}.
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Problem 19.2.5

Given the spin matrices

S+=(0100),S−=(0010). S_{+} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad S_{-} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.

Consider the 4×44 \times 4 matrices

(02S−0202),(0202S+02),(S+0202S+),(S−0202S−). \begin{pmatrix} 0_{2} & S_{-} \\ 0_{2} & 0_{2} \end{pmatrix}, \qquad \begin{pmatrix} 0_{2} & 0_{2} \\ S_{+} & 0_{2} \end{pmatrix}, \qquad \begin{pmatrix} S_{+} & 0_{2} \\ 0_{2} & S_{+} \end{pmatrix}, \qquad \begin{pmatrix} S_{-} & 0_{2} \\ 0_{2} & S_{-} \end{pmatrix}.

Calculate the commutators of these matrices and extend the set so that one finds a basis of a Lie algebra.

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Problem 19.2.6

Let L1L_{1} and L2L_{2} be two Lie algebras. Let φ:L1→L2\varphi : L_{1} \to L_{2} be a Lie algebra homomorphism. Show that im⁡(φ)\operatorname {im}(\varphi ) is a Lie subalgebra of L2L_{2} and ker⁡(φ)\ker (\varphi ) is an ideal in L1L_{1}.

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Problem 19.2.7

Let AA, BB, CC nonzero 3×33 \times 3 matrices with the commutation relations

[A,B]=C,[B,C]=A,[C,A]=B [A,B] = C, \qquad [B,C] = A, \qquad [C,A] = B

i.e. AA, BB, CC form a basis of the Lie algebra so(3)\mathfrak {so}(3). Find the Lie algebra generated by the 9×99 \times 9 matrices

H=A⊗I3+I3⊗B+A⊗B,K=A⊗I3+I3⊗B+B⊗A. H = A \otimes I_{3} + I_{3} \otimes B + A \otimes B, \qquad K = A \otimes I_{3} + I_{3} \otimes B + B \otimes A.
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Problem 19.2.8

Consider the Pauli spin matrices σ0=I2\sigma_{0} = I_{2}, σ1\sigma_{1}, σ2\sigma_{2}, σ3\sigma_{3} and the sixteen 4×44 \times 4 matrices (j=0,1,2,3j = 0,1,2,3)

(σj0202σj),(σj0202−σj),(02σjσj02),(02σj−σj02). \begin{pmatrix} \sigma _{j} & 0_{2} \\ 0_{2} & \sigma _{j} \end{pmatrix}, \qquad \begin{pmatrix} \sigma _{j} & 0_{2} \\ 0_{2} & -\sigma _{j} \end{pmatrix}, \qquad \begin{pmatrix} 0_{2} & \sigma _{j} \\ \sigma _{j} & 0_{2} \end{pmatrix}, \qquad \begin{pmatrix} 0_{2} & \sigma _{j} \\ -\sigma _{j} & 0_{2} \end{pmatrix}.

Do the sixteen 4×44 \times 4 matrices form a basis of a Lie algebra under the commutator?

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Problem 19.2.9

Consider the Lie algebra of real-skew symmetric 3×33 \times 3 matrices

A=(0a3−a2−a30a1a2−a10). A = \begin{pmatrix} 0 & a_{3} & -a_{2} \\ -a_{3} & 0 & a_{1} \\ a_{2} & -a_{1} & 0 \end{pmatrix}.

Let RR be a real orthogonal 3×33 \times 3 matrix, i.e. RRT=I3RR^{T} = I_{3}. Show that RARTRAR^{T} is a real-skew symmetric matrix.

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Problem 19.2.10

The Lie group SU(2,2)SU(2,2) is defined as the group of transformation on the four dimensional complex space C4\mathbb {C}^{4} leaving invariant the indefinite quadratic form

∣z1∣2+∣z2∣2−∣z3∣2−∣z4∣2. \left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}-\left|z_{3}\right|^{2}-\left|z_{4}\right|^{2}.

The Lie algebra su(2,2)\mathfrak {su}(2,2) is defined as the 4×44 \times 4 matrices XX with trace 0 and X∗L+LX=04X^{*}L+LX = 0_{4}, where LL is the 4×44 \times 4 matrix

L=(−I20202I2). L = \begin{pmatrix} -I_{2} & 0_{2} \\ 0_{2} & I_{2} \end{pmatrix}.

Is

X=(0001001001001000) X = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}

an element of the Lie algebra su(2,2)\mathfrak {su}(2,2)? Find exp(zX)\mathrm{exp}\left(zX\right). Discuss.

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Problem 19.2.11

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

h1=(1000−10000),h2=(00001000−1), 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}, e1=(010000000),e2=(000001000), 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}, f1=(000100000),f2=(000000010), 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}, e13=(001000000),f13=(000000100). 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.

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Problem 19.2.12

Let EjkE_{jk} (j,k=1,2,3,4j,k = 1,2,3,4) be the elementary matrices in the vector space of 4×44 \times 4 matrices. Show that the 15 matrices

X1=E12,X2=E23,X3=E13,X4=E34,X5=E24,X6=E14 X_{1} = E_{12}, \qquad X_{2} = E_{23}, \qquad X_{3} = E_{13}, \qquad X_{4} = E_{34}, \qquad X_{5} = E_{24}, \qquad X_{6} = E_{14} Y1=E21,Y2=E32,Y3=E31,Y4=E43,Y5=E42,Y6=E41 Y_{1} = E_{21}, \qquad Y_{2} = E_{32}, \qquad Y_{3} = E_{31}, \qquad Y_{4} = E_{43}, \qquad Y_{5} = E_{42}, \qquad Y_{6} = E_{41} H1=14diag⁡(3,−1,−1,−1,−1),H2=12diag⁡(1,1,−1,−1), H_{1} = \frac{1}{4}\operatorname {diag}(3,-1,-1,-1,-1), \qquad H_{2} = \frac{1}{2}\operatorname {diag}(1,1,-1,-1), H3=14diag⁡(1,1,1,−3) H_{3} = \frac{1}{4}\operatorname {diag}(1,1,1,-3)

form a basis (Cartan-Weyl basis) of the Lie algebra sl(4,C)\mathfrak {sl}(4,\mathbb {C}).

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Problem 19.2.13

Let AA, BB, CC be 3×33 \times 3 nonzero matrices such that (Lie algebra so(3)\mathfrak {so}(3))

[A,B]=C,[B,C]=A,[C,A]=B. [A,B] = C, \qquad [B,C] = A, \qquad [C,A] = B.

Find the commutators of the 9×99 \times 9 matrices

A⊗I3+I3⊗A,B⊗I3+I3⊗B,C⊗I3+I3⊗C. A \otimes I_{3} + I_{3} \otimes A, \qquad B \otimes I_{3} + I_{3} \otimes B, \qquad C \otimes I_{3} + I_{3} \otimes C.
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Problem 19.2.14

The isomorphism of the Lie algebras sl(2,C)\mathfrak {sl}(2,\mathbb {C}) and so(3,C)\mathfrak {so}(3,\mathbb {C}) has the form

(abc−a)↔(0b−c−i(b+c)c−b02iai(b+c)−2ia0). \begin{pmatrix} a & b \\ c & -a \end{pmatrix} \leftrightarrow \begin{pmatrix} 0 & b-c & -i(b+c) \\ c-b & 0 & 2ia \\ i(b+c) & -2ia & 0 \end{pmatrix}.

Let z∈Cz \in \mathbb {C}. Find

exp(z(abc−a)),exp(z(0b−c−i(b+c)c−b02iai(b+c)−2ia0)). \mathrm{exp}\left(z\begin{pmatrix} a & b \\ c & -a \end{pmatrix}\right), \qquad \mathrm{exp}\left(z\begin{pmatrix} 0 & b-c & -i(b+c) \\ c-b & 0 & 2ia \\ i(b+c) & -2ia & 0 \end{pmatrix}\right).
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Problem 19.2.15

Consider a four-dimensional vector space with basis e1e_{1}, e2e_{2}, e3e_{3}, e4e_{4}. Assume that the non-zero commutators are given

[e2,e3]=e1,[e1,e4]=2e1,[e2,e4]=e2,[e3,e4]=e2+e3. [e_{2},e_{3}] = e_{1}, \qquad [e_{1},e_{4}] = 2e_{1}, \qquad [e_{2},e_{4}] = e_{2}, \qquad [e_{3},e_{4}] = e_{2}+e_{3}.

Do these relations define a Lie algebra? For example we have

[e1,[e2,e3]]+[e3,[e1,e2]]+[e2,[e3,e1]]=[e1,e1]+[e3,0]+[e2,0]=0. [e_{1},[e_{2},e_{3}]]+[e_{3},[e_{1},e_{2}]]+[e_{2},[e_{3},e_{1}]] = [e_{1},e_{1}]+[e_{3},0]+[e_{2},0] = 0.

If so find the adjoint representation.

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Problem 19.2.16

Let AA, BB, CC be nonzero n×nn \times n matrices. Assume that [A,B]=0n[A,B] = 0_{n} and [C,A]=0n[C,A] = 0_{n}. Can we conclude that [B,C]=0n[B,C] = 0_{n}? Is the Jacobi identity

[A,[B,C]]+[C,[A,B]]+[B,[C,A]]=0n [A,[B,C]]+[C,[A,B]]+[B,[C,A]] = 0_{n}

of any use?

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Problem 19.2.17
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(i)

Find the Lie algebra generated by the 2×22 \times 2 matrices

A=(0001),B=(00−10). A = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 \\ -1 & 0 \end{pmatrix}.

Note that [A,B]=A+B[A,B] = A+B.

(ii)

Find the Lie algebra generated by the 3×33 \times 3 matrices

A=(00001000−1),B=(000−100000),C=(000000100). A = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad C = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}.

Note that [A,B]=B[A,B] = B, [B,C]=03[B,C] = 0_{3}, [C,A]=C[C,A] = C.

Problem 19.2.18

In the decomposition of the simple Lie algebra sl(3,R)\mathfrak {sl}(3,\mathbb {R}) one finds the 3×33 \times 3 matrices

A=(a11a120a21a22000−a11−a22),B=(00b1300b23b31b320) A = \begin{pmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & -a_{11}-a_{22} \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & b_{13} \\ 0 & 0 & b_{23} \\ b_{31} & b_{32} & 0 \end{pmatrix}

where ajk,bjk∈Ra_{jk}, b_{jk} \in \mathbb {R}. Find the commutators [A,A′][A,A'], [B,B′][B,B'] and [A,B][A,B]. Discuss. Note that

[A,B]=(00a12b23+a22b1300a21b13+a11b23−a22b31−a21b32−a11b32−a12b310). [A,B] = \begin{pmatrix} 0 & 0 & a_{12}b_{23}+a_{22}b_{13} \\ 0 & 0 & a_{21}b_{13}+a_{11}b_{23} \\ -a_{22}b_{31}-a_{21}b_{32} & -a_{11}b_{32}-a_{12}b_{31} & 0 \end{pmatrix}.

Thus the matrix of the commutator [A,B][A,B] is of the form of matrix BB.

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Problem 19.2.19

Let σ1\sigma_{1}, σ2\sigma_{2}, σ3\sigma_{3} be the Pauli spin matrices with the commutation relations

[σ1,σ2]=2iσ3,[σ2,σ3]=2iσ1,[σ3,σ1]=2iσ2. [\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 su(2)=span⁡{ iσ1,iσ2,iσ3 }\mathfrak {su}(2) = \operatorname {span}\{ \, i\sigma_{1}, i\sigma_{2}, i\sigma_{3} \, \}. Then

[σ1⊗I2+I2⊗σ2,σ2⊗I2+I2⊗σ3]=2i(σ3⊗I2+I2⊗σ1). [\sigma _{1} \otimes I_{2} + I_{2} \otimes \sigma _{2}, \sigma _{2} \otimes I_{2} + I_{2} \otimes \sigma _{3}] = 2i(\sigma _{3} \otimes I_{2} + I_{2} \otimes \sigma _{1}).
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(i)

Find the commutators

[σ2⊗I2+I2⊗σ3,σ3⊗I2+I2⊗σ1],[σ3⊗I2+I2⊗σ1,σ2⊗I2+I2⊗σ3]. [\sigma _{2} \otimes I_{2} + I_{2} \otimes \sigma _{3}, \sigma _{3} \otimes I_{2} + I_{2} \otimes \sigma _{1}], \qquad [\sigma _{3} \otimes I_{2} + I_{2} \otimes \sigma _{1}, \sigma _{2} \otimes I_{2} + I_{2} \otimes \sigma _{3}].

Discuss.

(ii)

We know that the Pauli spin matrices σ1\sigma_{1}, σ2\sigma_{2}, σ3\sigma_{3} are elements of the Lie algebra u(2)\mathfrak {u}(2), but not su(2)\mathfrak {su}(2) since det⁡(σj)=−1\operatorname {det}\left(\sigma_{j}\right) = -1 for j=1,2,3j = 1,2,3. Are the nine 4×44 \times 4 matrices

σj⊗σk,j,k=1,2,3 \sigma _{j} \otimes \sigma _{k}, \qquad j,k = 1,2,3

elements of the Lie algebra su(4)\mathfrak {su}(4)?

(iii)

Let ⋆\star be the star operation. Find the commutators

[σ1,σ2],[σ2,σ3],[σ3,σ1],[σ1⋆σ1,σ2⋆σ2],[σ2⋆σ2,σ3⋆σ3],[σ3⋆σ3,σ1⋆σ1]. [\sigma _{1},\sigma _{2}], \qquad [\sigma _{2},\sigma _{3}], \qquad [\sigma _{3},\sigma _{1}], \qquad [\sigma _{1}\star \sigma _{1}, \sigma _{2}\star \sigma _{2}], \qquad [\sigma _{2}\star \sigma _{2}, \sigma _{3}\star \sigma _{3}], \qquad [\sigma _{3}\star \sigma _{3}, \sigma _{1}\star \sigma _{1}].

Find the anti-commutators [σ1,σ2]+[\sigma_{1},\sigma_{2}]_{+}, [σ2,σ3]+[\sigma_{2},\sigma_{3}]_{+}, [σ3,σ1]+[\sigma_{3},\sigma_{1}]_{+},

[σ1⋆σ1,σ2⋆σ2]+,[σ2⋆σ2,σ3⋆σ3]+,[σ3⋆σ3,σ1⋆σ1]+. [\sigma _{1}\star \sigma _{1}, \sigma _{2}\star \sigma _{2}]_{+}, \qquad [\sigma _{2}\star \sigma _{2}, \sigma _{3}\star \sigma _{3}]_{+}, \qquad [\sigma _{3}\star \sigma _{3}, \sigma _{1}\star \sigma _{1}]_{+}.
Problem 19.2.20

Let α∈R\alpha \in \mathbb {R}. Show that the matrices

L1=(α+i00α),L2=(0100),L3=(0i00) L_{1} = \begin{pmatrix} \alpha +i & 0 \\ 0 & \alpha \end{pmatrix}, \qquad L_{2} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad L_{3} = \begin{pmatrix} 0 & i \\ 0 & 0 \end{pmatrix}

satisfy the commutation relations

[L1,L2]=L3,[L2,L3]=02,[L3,L1]=L2. [L_{1},L_{2}] = L_{3}, \qquad [L_{2},L_{3}] = 0_{2}, \qquad [L_{3},L_{1}] = L_{2}.
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Problem 19.2.21

Consider the four-dimensional real vector space with a basis e0e_{0}, e1e_{1}, e2e_{2}, e3e_{3}. The Godel quaternion algebra G\mathbb {G} is defined by the non-commutative multiplication

e0ek=ek=eke0,(e0)2=e0 e_{0}e_{k} = e_{k} = e_{k}e_{0}, \qquad (e_{0})^{2} = e_{0} ejek=(−1)ℓϵjkℓeℓ−(−1)kδjke0,j,k,ℓ=1,2,3. e_{j}e_{k} = (-1)^{\ell }\epsilon _{jk\ell }e_{\ell } - (-1)^{k}\delta _{jk}e_{0}, \qquad j,k,\ell = 1,2,3.

Let q0q_{0}, q1q_{1}, q2q_{2}, q3q_{3} be arbitrary real numbers. We call the vector

q=q0e0+q1e1+q2e2+q3e3 q = q_{0}e_{0}+q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{3}

a Godel quaternion. We define the basis

e11:=12(e0+e3),e22:=12(e0−e3), e_{11} := \frac{1}{2}(e_{0}+e_{3}), \qquad e_{22} := \frac{1}{2}(e_{0}-e_{3}), e12:=12(e1+e2),e21:=12(e1−e2). e_{12} := \frac{1}{2}(e_{1}+e_{2}), \qquad e_{21} := \frac{1}{2}(e_{1}-e_{2}).

Show that the ejke_{jk} satisfy the multiplication law

ejkers=δkrejs,j,k,r,s=1,2. e_{jk}e_{rs} = \delta _{kr}e_{js}, \qquad j,k,r,s = 1,2.

Show that every Godel quaternion qq can be written as

q=∑j,k=12qjkejk. q = \sum _{j,k=1}^{2} q_{jk}e_{jk}.

Show that

e0=(1001),e1=(0110),e2=(01−10),e3=(100−1) e_{0} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad e_{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad e_{2} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad e_{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

is a matrix representation.

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