Supplementary Problems

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

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

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

Do the matrices

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

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

Given the spin matrices

Consider the $4 \times 4$ matrices

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

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

Let $A$, $B$, $C$ nonzero $3 \times 3$ matrices with the commutation relations

i.e. $A$, $B$, $C$ form a basis of the Lie algebra $\mathfrak {so}(3)$. Find the Lie algebra generated by the $9 \times 9$ matrices

Consider the Pauli spin matrices $\sigma_{0} = I_{2}$, $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ and the sixteen $4 \times 4$ matrices ($j = 0,1,2,3$)

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

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

Let $R$ be a real orthogonal $3 \times 3$ matrix, i.e. $RR^{T} = I_{3}$. Show that $RAR^{T}$ is a real-skew symmetric matrix.

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

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

Is

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

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

Find the commutator table.

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

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

Let $A$, $B$, $C$ be $3 \times 3$ nonzero matrices such that (Lie algebra $\mathfrak {so}(3)$)

Find the commutators of the $9 \times 9$ matrices

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

Let $z \in \mathbb {C}$. Find

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

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

If so find the adjoint representation.

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

of any use?

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

where $a_{jk}, b_{jk} \in \mathbb {R}$. Find the commutators $[A,A']$, $[B,B']$ and $[A,B]$. Discuss. Note that

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

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

Thus we have $\mathfrak {su}(2) = \operatorname {span}\{ \, i\sigma_{1}, i\sigma_{2}, i\sigma_{3} \, \}$. Then

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

satisfy the commutation relations

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

Let $q_{0}$, $q_{1}$, $q_{2}$, $q_{3}$ be arbitrary real numbers. We call the vector

a Godel quaternion. We define the basis

Show that the $e_{jk}$ satisfy the multiplication law

Show that every Godel quaternion $q$ can be written as

Show that

is a matrix representation.