1
$\begingroup$

Well, I'm studying (engineering-) quantum mechanics dealing with representation theory of Lie algebras. The books I read introduce irreducible representations of $su(2)$ which is heavily related with angular momenta of particles. However the problem is that they merely regard the representations of Lie algebra as matrix Lie algebra without any additional consideration. Though I found Ado's theorem, stating the existence of faithful representation of every Lie algebra (= existence of matrix Lie algebra isomorphic to original Lie algebra) it is also possible for us to find unfaithful representation of every Lie algebra and I cannot assure the faithfulness of (irreducible) representations of $su(2)$ the books suggest.

It would be better to start with an example.

Consider Lie group $SU(2)$ and its corresponding finite dimensional Lie algebra $su(2)$. I already know that with the basis elements $J_1 ,J_2 , J_3 \in\,su(2)$, $su(2)$ is closed under Lie bracket relation $[J_i,J_j]\,=i\epsilon_{ijk}J_k$.

Here I introduce an algebra representation $\rho$,

$\rho(J_1)\,=\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 &0 \\1 & 0 & 1\\0 & 1 & 0\end{bmatrix}$ $\rho(J_2)\,=\frac{1}{\sqrt{2}}\begin{bmatrix}0 & -i &0 \\i & 0 & -i\\0 & i & 0\end{bmatrix}$ $\rho(J_3)\,=\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 &0 \\0 & -1 & 0\\0 & 0 & 0\end{bmatrix}$

Clearly, $\rho(J_i)$'s satisfy same Lie bracket relation of $su(2)$ when we treat the bracket as commutator of matrices. Furthermore any element of $su(2)$ expressed by $aJ_1+bJ_2+cJ_3$ (linear combinations of 3 linearly independent vectors) can be mapped to $a\rho(J_1)+b\rho(J_2)+c\rho(J_3)$ (linear combinations of 3 linearly independent matrices).

Then my question is... does this representation $\rho$ represent matrix Lie algebra isomorphic to $su(2)$? At first I noticed that kernel of $\rho$ would be zero vector of $su(2)$ alone so $\rho$ may be faithful...

I cannot understand why the books I read just cope with the result of representations of basis elements of $su(2)$ and treat them as the basis elements of isomorphic matrix Lie algebra. Is it sufficient for us to confirm that certain representation of a Lie algebra mapping the basis elements, $\rho(J_i)$ is one-to-one and $\rho(J_i)$'s are again linearly independent?

$\endgroup$
  • 1
    $\begingroup$ $\mathfrak{su}(2)$ is a simple Lie algebra, so in this special case it turns out that every nontrivial representation of it must be faithful. In general, faithfulness is less important than it sounds. $\endgroup$ – Qiaochu Yuan Sep 26 '15 at 3:31
  • $\begingroup$ Constructing faithful representations of, say, nilpotent Lie algebras can be very interesting in geometry and topology, see John Milnor's article about affine crystallographic groups, left-invariant affine structures on Lie groups and faithful matrix representations. $\endgroup$ – Dietrich Burde Sep 26 '15 at 8:25
  • 1
    $\begingroup$ @QiaochuYuan simple Lie algebra... I get a new concept, and just find a paper saying that every non-trivial representation of finite dim. complex, simple Lie algebra is faithful, homepage.univie.ac.at/Dietrich.Burde/papers/… section 2.2. thanks to Dietrich... What a coincidence! $\endgroup$ – Discovery Sep 26 '15 at 11:36
  • $\begingroup$ Thank you for mentioning my paper. I hope it is helpful for you. $\endgroup$ – Dietrich Burde Sep 26 '15 at 17:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.