Verifying bases and commutator tables of small concrete Lie algebras

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

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

Show that we have a basis of a Lie algebra.

Consider the matrices

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

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

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

Any fixed element $X$ of a Lie algebra $L$ defines a linear transformation

Show that for any $K \in L$ we have

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

There is only one non-commutative Lie algebra $L$ of dimension 2. If $x$, $y$ are the generators (basis in $L$), then

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

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

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

Show that the matrices

form generators of a Lie algebra. Is the Lie algebra simple? A Lie algebra is simple if it contains no ideals other than $L$ and $0$.

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

with the commutators $[H,E] = 2E$, $[H,F] = -2F$ and $[E,F] = H$.

We know that

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

Consider

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

The $2 \times 2$ matrices

form a basis of the Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with the commutation relations $[E,H] = -2E$, $[E,F] = H$, $[F,H] = 2F$.

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

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

Consider the $2 \times 2$ matrices over $\mathbb {R}$

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

Consider the $3 \times 3$ matrices

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

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

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

Calculate the vector product

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

Discuss.

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

Find

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

Consider the three $3 \times 3$ matrices

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

Let $L$ be a finite dimensional Lie algebra. Let $C^{\infty }(S^{1})$ be the set of all infinitely differentiable functions, where $S^{1}$ is the unit circle manifold. In the product space $L \otimes C^{\infty }(S^{1})$ we define the Lie bracket ($g_{1}, g_{2} \in L$ and $f_{1}, f_{2} \in C^{\infty }(S^{1})$)

Calculate

The elements (generators) $Z_{1}, Z_{2}, \ldots , Z_{r}$ of an $r$-dimensional Lie algebra satisfy the conditions

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

Suppose that $X$ is some other linear combination such that

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

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

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 $\mathfrak {sl}(4,\mathbb {C})$.

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.

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