Intereting Posts

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …
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Proof that Epicycloids are Algebraic Curves?
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$f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
$A \oplus B = A \oplus C$ imply $B = C$?
$X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
Capelli Lemma for polynomials
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Does this really converge to 1/e? (Massaging a sum)
Does there exist a $(m,n)\in\mathbb N$ such that $m^3-2^n=3$?

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 $[J_i,J_j]\,=i\epsilon_{ijk}J_k$.

- An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.
- Let $\rho$ f.d. rep of a nilpotent Lie algebra such that $\rm{det} \rho(X) = 0$, $\forall X$. Then $\exists v \neq 0$: $\rho(X)v = 0, \forall X$.
- Expression for the Maurer-Cartan form of a matrix group
- Exponential of a polynomial of the differential operator
- A covering map from a differentiable manifold
- Under what conditions is the exponential map on a Lie algebra injective?

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?

- Showing the Lie Algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are not isomorphic.
- Examples about that $\exp(X+Y)=\exp(X) \exp(Y)$ does not imply $=0$ where $X,Y$ are $n \times n $ matrix
- Show that the Lie algebra generated by x, y with relations $ad(x)^2(y) = ad(y)^5(x) = 0$ is infinite dimensional and construct a basis
- Lie algebra-like structure corresponding to noncrystallographic root systems
- Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)
- Cover and extension of a Lie group
- Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
- Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra #2
- Product of exponential of matrices
- Automorphism group of a lie algebra as a lie subgroup of $GL(\frak g)$

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- Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$