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\begin{document}
\noindent {\bf Physics 251} \hfill {\bf Problem Set 4} \hfill {\bf
Spring 2019}
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\centerline{\sl DUE: \ FRIDAY, JUNE 7, 2019}
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\noindent
{\sl FINAL PROJECTS ALERT:}  The presentations of the final projects
will take place on Thursday June 6 from 3--6 pm and on Friday June 7 from 2:30--5:30 pm in ISB 231.   
You are strongly encouraged to attend both sessions. The slides from your
presentation will be posted to the class website.   

\myrule
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\noindent
1.~~~(a) Show that the Lie algebra of U($n$) can be written as a direct sum, $\mathfrak{u}(n)\iso\mathfrak{su}(n)\oplus\mathfrak{u}(1)$.
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(b) As for the corresponding Lie groups, show that U($n$)\,$\iso$\,SU($n$)$\otimes$U(1)/$\mathbb{Z}_n$.   
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\noindent {\sl HINT:}  Consider the homomorphism of $(A,e^{i\theta})\longmapsto e^{i\theta}A$, where $A\in{\rm SU}(n)$ and $e^{i\theta}\in{\rm U}(1)$.
What is the kernel of this homomorphism?


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\noindent
2. This problem concerns the Lie group SO(4) and its Lie algebra  $\mathfrak{so}(4)$.
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(a) Work out the Lie algebra $\mathfrak{so}(4)$ and verify that
$\mathfrak{so}(4)\iso \mathfrak{so}(3)\oplus \mathfrak{so}(3)$.
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\noindent {\sl HINT:} Show that there is a choice of basis for $\mathfrak{so}(4)$
consisting of $4\times 4$ anti\-symmetric matrices that contain
precisely two non-zero entries: 1 and $-1$.
Evaluate the commutation relations of these $\mathfrak{so}(4)$ generators.  Then,
by choosing a new basis consisting of sums and differences of pairs of the
old $\mathfrak{so}(4)$ generators, show that the resulting commutation relations
are isomorphic to the commutation relations of the Lie algebra
$\mathfrak{so}(3)\oplus \mathfrak{so}(3)$.
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(b) What is the universal covering group of SO(4)? What is the center
of SO(4)? Identify the adjoint group ${\rm Ad}\bigl({\rm SO}(4)\bigr)$.
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(c) Calculate the Killing form of $\mathfrak{so}(4)$ and verify that
this Lie algebra is semisimple and compact.

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\noindent
3. A Lie algebra $\mathfrak{g}$ is defined by the commutation
relations of the generators,
$$[e_a,e_b]=f_{ab}^c e_c\,.$$
Consider the finite-dimensional matrix representations of the
$e_a$.  We shall denote the corresponding generators in the
adjoint representation by $F_a$ and in an arbitrary irreducible
representation $R$ by $R_a$.  The dimension of the adjoint
representation, $d$, is equal to the dimension of the Lie
algebra $\mathfrak{g}$, while the dimension of the representation $R$ will be denoted by $d\ls{R}$.
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\clearpage

(a) Show that the Cartan-Killing metric $g_{ab}$ can be written
as $g_{ab}=\Tr(F_a F_b)$.
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(b) If $\mathfrak{g}$ is a simple real compact Lie algebra, prove that for
any irreducible representation~$R$,
$$\Tr(R_a R_b)= c\ls{R} g_{ab}\,,$$
where $c\ls{R}$ is called the {\it index} of the irreducible
representation $R$.
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\noindent {\sl HINT:} Choose a basis where $g_{ab}$ is
proportional to $\delta_{ab}$.  Then the $f_{ab}^c$ are
antisymmetric in all three indices.  Show that $\Tr\,[R_a,R_b]
R_c=\Tr\, R_a [R_b,R_c]$ and argue that this implies that $\Tr\, R_a
R_b$, viewed as the $ab$ element of a $d\times d$ matrix, commutes
with all Lie algebra
elements in the adjoint representation.  Finally, invoke Schur's
lemma.\footnote{Note that by complexifying the simple real compact Lie algebra,
one can easily show that the above result also holds for any simple complex
Lie algebra.}
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(c) The quadratic Casimir operator is defined as $C_2\equiv
g^{ab} e_a e_b$ where $g^{ab}$ is the inverse of $g_{ab}$.
Recall that $C_2$ commutes with all elements of the Lie
algebra.  Hence, by Schur's lemma, $C_2$ must be a multiple of
the identity operator.  Let us write $C_2=C_2(R)\id$ where $\id$ is
the $d_R\times d_R$ identity matrix and $C_2(R)$ is the
eigenvalue of the Casimir operator in the irreducible
representation $R$.  
Show that $C_2(R)$ is related to the index $c\ls{R}$ by
$$
C_2(R)={dc\ls{R}\over d_R}\,,
$$
where $d$ is the dimension of the Lie algebra $\mathfrak{g}$.
Check the above formula in
the case that $R$ is the adjoint representation.
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\noindent
{\sl HINT:} The matrix elements of the $R_a$ are $(R_a)_{ij}$, where
$i,j=1,\ldots,d_R$.  If you keep the matrix element indices explicit,
then the derivation of the above result is straightforward.

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(d) Compute the index of an arbitrary irreducible
representation of $\mathfrak{su}(2)$.
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(e) Compute the index of the defining representation of $\mathfrak{su}(3)$.  
Generalize this result to $\mathfrak{su}(n)$.

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\noindent
4. Various subalgebras of $\mathfrak{su}(3)$ may be identified with specific
subsets of the $\mathfrak{su}(3)$ generators.
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(a) Show that the Gell-Mann matrices $\lambda_1$, $\lambda_2$, and
$\lambda_3$ generate an $\mathfrak{su}(2)$ subalgebra.
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(b) Show that the Gell-Mann matrices $\lambda_2$, $\lambda_5$, and
$\lambda_7$ generate an $\mathfrak{so}(3)$ subalgebra.  (Why do you think I called this
an $\mathfrak{so}(3)$ subalgebra rather than an $\mathfrak{su}(2)$ subalgebra?)
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(c) Decompose (if necessary) the
three-dimensional irreducible representation of $\mathfrak{su}(3)$ into representations that
are irreducible under the subalgebras of parts (a) and (b).  
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\clearpage

\noindent
5. Consider the simple Lie algebra $\mathfrak{g}$ generated by the ten $4\times 4$ matrices:
$\sigma_a\otimes \id$, $\sigma_a\otimes\tau_1$, $\sigma_a\otimes\tau_3$ and
$\id\otimes\tau_2$, where $(\id,\sigma_a)$ and
$(\id,\tau_a)$ are the $2\times 2$ identity and
Pauli matrices in orthogonal spaces.  For example, since $\tau_3=
\bigl({1\atop 0}{\phm 0\atop -1}\bigr)$, we obtain in block matrix form:
$$
\sigma_a\otimes\tau_3=\left(\begin{array}{c|c} \sigma_a \phm\!\! & \boldsymbol{0} \\ \hline
\boldsymbol{0} \phm\!\! & -\sigma_a\end{array}\right)\,,\qquad (a=1,2,3)\,,
$$
%$\sigma_a\otimes\tau_3=\bigl({\sigma_a\atop 0}{0\atop -\sigma_a}\bigr)$.
where $\boldsymbol{0}$ is the $2\times 2$ zero matrix.
The remaining seven matrices can be likewise obtained.
Take $H_1=\sigma_3\otimes \id$ and $H_2=\sigma_3\otimes\tau_3$ as the generators of the
Cartan subalgebra.  Note that if $A$, $B$, $C$, and $D$ are $2\times 2$ matrices,
then $(A\otimes B)(C\otimes D)=AC\otimes BD$.
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(a) Find the roots of $\mathfrak{g}$.  Normalize the roots such that the shortest root vector has length~1.  What is the rank of $\mathfrak{g}$?
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(b) Determine the simple roots and evaluate the corresponding Cartan matrix.  
Deduce the Dynkin diagram for this Lie algebra and identify it by name.  
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(c) The fundamental weights $\boldsymbol{m}_i$ are defined in terms of the simple roots $\boldsymbol{\alpha}_j\in\Pi$  such that
$$
\frac{2(\boldsymbol{m}_i,\boldsymbol{\alpha}_j)}{(\boldsymbol{\alpha}_j,\boldsymbol{\alpha}_j)}=\delta_{ij}\,,
\qquad\quad \text{for $i,j=1,2,\ldots,r$}\,,
$$
where $r\equiv{\rm rank}~\mathfrak{g}$.
Using the results of part (b), determine all the
fundamental weights of $\mathfrak{g}$.
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\noindent {\sl HINT:} 
Expand the $\boldsymbol{m}_i$ as a linear combination of the simple roots and solve for the coefficients.
%that if $A$, $B$, $C$, and $D$ are $2\times 2$ matrices, then
%the
%following is an identity for $4\times 4$ matrices:
%$(A\otimes B)(C\otimes D)=AC\otimes BD\,.$

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(d) [{\textsl{EXTRA CREDIT}] Each of the $r$ fundamental weights is the highest
weight for an irreducible representation of
$\mathfrak{g}$. Collectively, these are called the fundamental (or basic)
representations of $\mathfrak{g}$.  For each fundamental
representation of $\mathfrak{g}$, compute the complete set of weights
and draw the corresponding weight diagrams.\footnote{The weight
  diagrams should be plotted on a two dimensional plane, where the
  axes correspond to the diagonalized
  generators normalized such that the shortest root vector has length 1.}
What are the
corresponding dimensions of the fundamental representations of
$\mathfrak{g}$.
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\noindent {\sl HINT:}
In this example, all weights of the fundamental representations of $\mathfrak{g}$ appear with multiplicity equal to one.
The complete set of weights for the irreducible representations of $\mathfrak{sp}(2,\mathbb{C})\iso\mathfrak{so}(5,\mathbb{C})$ corresponding to 
the highest weights $\boldsymbol{m}_1$ and $\boldsymbol{m}_2$, respectively,
can be obtained by the method of block weight diagrams described in Robert N.~Cahn, \textit{Semi-Simple
Lie Algebras and Their Representations} (Dover Publications, Inc., Mineola, NY, 2006).\footnote{However, note that Cahn
defines the Cartan matrix that is the transpose of our definition.}

\end{document}
