Math Problems Vault

Problem 3.7 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

A Lie algebra $L$ is called semisimple if $L \neq 0$ and $L$ has no abelian ideals $\neq 0$ . Show that a finite dimensional Lie algebra $L$ is semisimple if and only if $\operatorname {rad}L = 0$ .

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Problem 3.39 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

(i)(ii)

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(i)

Is the three dimensional Heisenberg algebra with the basis $\{ x,y,z\}$ and the commutators $[x,y] = z$ , $[x,z] = 0$ , $[y,z] = 0$ nilpotent?

(ii)

Find a matrix representation with $3 \times 3$ matrices.

Problem 3.40 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

If a Lie algebra is nilpotent, then the Killing form vanishes. Can we conclude that if the Killing form vanishes then the Lie algebra is nilpotent?

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Problem 3.41 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Show that the Lie algebra consisting of $3 \times 3$ matrices over $\mathbb {R}$ (or $\mathbb {C}$ ) with zeros on and below the main diagonal is nilpotent.

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Problem 3.42 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Consider the vector space of $3 \times 3$ matrices over $\mathbb {R}$ and the two linearly independent matrices

A = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix} \, .

Construct a Lie algebra by calculating the commutators. Thus $A$ and $B$ would be the generators of the Lie algebra.

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Problem 4.77 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

(i)(ii)

Consider the operators

H_1 = 1, \quad H_2 = x, \quad H_3 = \frac{\partial ^2}{\partial x^2}, \quad H_4 = i\frac{\partial }{\partial x} .

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(i)

Show that we have a nilpotent Lie algebra under the commutator.

(ii)

Let

\alpha _1(t) = cf(t), \quad \alpha _2(t) = c, \quad \alpha _3(t) = -\frac{1}{2}, \quad \alpha _4(t) = \frac{df}{dt} .

Consider the Hamilton operator

K = \sum _{j=1}^{4} \alpha _j(t)H_j

and the Schrodinger equation

i\frac{\partial \psi }{\partial t} = K\psi .

We write the solution of the Schrodinger equation in the form

\psi (x,t) = U(t,0)\psi (x,0)

where the unitary time evolution operator is given by

U(t,0) = \exp (\beta _1(t)H_1)\exp (\beta _2(t)H_2)\exp (\beta _3(t)H_3)\exp (\beta _4(t)H_4) .

Find the system of ordinary differential equations for $\beta_j(t)$ ( $j=1,2,3,4$ ) and solve them.

Problem 3.103 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Section 3.2 "Supplementary Problems"

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.

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Problem 3.104 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Section 3.2 "Supplementary Problems"

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.

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Solvable and nilpotent Lie algebras

References