Representations of Lie algebras

Define on $V_{n}$ the structure of an $\mathfrak {su}(2)$-module, using formulas (2), returning to the basis elements

and associating with them the linear operators $K_{1}, K_{2}, K_{3}$.

Let $L= \left\langle e_{-1}, e_{0}, e_{1}\right\rangle$ be a simple Lie algebra with multiplication table (1) over an algebraically closed field $F$ of characteristic $p>2$. Consider the square matrices of order $p$

Verify that the correspondence $e_{i} \mapsto E_{i}, \quad i=-1,0,1$, establishes an irreducible matrix representation of the Lie algebra $L$, depending on three parameters $\left(\lambda , \beta , \gamma_{0}\right)$. Here

(This entirely new situation is typical for representations over a field of finite characteristic.)

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

Find the adjoint representation.

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

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

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

and

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

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

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

Find the adjoint representation. Is the Lie algebra semisimple?

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

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

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

Then

Find eight $3 \times 3$ matrices

such that the following commutation relations are satisfied

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

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

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

Consider the vector fields

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?

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

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

Find the three commutators

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

The four matrices

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

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

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

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

with

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.

Consider the spin-matrices (spin-1)

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

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

The $2 \times 2$ matrices

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 ($m_{j} = j(k-j+1)$)

satisfy the same commutation relations?

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

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.

Let $z_1, z_2$ be complex variables. Show that differential operators (vector fields)

form a basis of a Lie algebra under the commutator.

Let $j = 0, 1/2, 1, 3/2, \ldots$. Show that the differential operators

form a basis of a Lie algebra under the commutator.

Let $j = 0, 1/2, 1, 3/2, \ldots$. Show that the differential operators

form a basis of a Lie algebra under the commutator.

Consider the differential operators

where $n$ is an integer.

Consider the vector fields

They satisfy the commutation relations

Thus we have a realization of the Lie algebra $\mathfrak {sl}(2,\mathbb {C})$. Extend the vector fields to the operators

Find the condition on the analytic functions $f_1, f_2, f_3$ such that $W_1, W_2, W_3$ also satisfy the commutation relations

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

Give faithful representations of $\mathfrak {sl}(2,\mathbb {R})$ using (i) $2\times 2$ matrices, (ii) $3\times 3$ matrices, (iii) vector fields, (iv) second order differential operators, (v) Bose operators, (vi) Fermi operators.

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

with the commutation relations

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

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

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

Thus

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