Math Problems Vault

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