Intereting Posts

Recursive formula for variance
Is the statement $A \in A$ true or false?
A inverse Trigonometric multiple Integrals
Difference between “space” and “algebraic structure”
Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$
Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$
Ellipse with non-orthogonal minor and major axes?
Permutation and Combination- Rowing a Boat
Let $H$ be a subgroup of $G$ with $=n$. Let $N$ be the kernel of the left-multiplication action on the cosets of $H$. Show $$ divides $n!$
Sum of discrete and continuous random variables with uniform distribution
Monotone class theorem vs Dynkin $\pi-\lambda$ theorem
Prove that $SL(n,R)$ is connected.
Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.
A question about a proof in one of Sierpiński's papers
Space of bounded functions is reflexive if the domain is finite

I’m (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps.

(1) Prove that the vector space of $2\times 2$ Hermitian matrices $H$ is isomorphic to Minkowski space.

(2) Demonstrate that the action of $SU(2)$ on $H$ by $X\mapsto AXA^{\dagger}$ induces a group homomorphism $SL(2,\mathbb{C})\to L$.

- Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?
- How to understand Weyl chambers?
- How to differentiate a homomorphism between two Lie groups
- How to deduce the Weyl group of type D?
- Is the Lie Algebra of a connected abelian group abelian?
- Tangent vectors to coadjoint orbits

(3) Prove this is 2:1 by observing that every $2\times 2$ complex matrix can be written as $X+iY$ with $X$ and $Y$ Hermitian.

I still need to prove that this map is **surjective** though. Here I am completely stuck. All the books and internet resources I have found either gloss over it, or state that it’s true but don’t prove it.

Could someone possibly give me a proper proof, with mathematician’s rigour?!

P.S. My own attempts at a proof have fallen down. I tried the following

(a) Derive a formula for the inverse map locally. I can’t see any good way to attack this though.

(b) Prove that the associated Lie algebra homomorphism is an isomorphism. I know theoretically this should be possible, but practically it seems a nightmare!

- orthogonal group of a quadratic vector space
- Representation theory of associative algebras applied to Lie Algebras
- What's a good place to learn Lie groups?
- Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?
- Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?
- Three-dimensional simple Lie algebras over the rationals
- How to prove that $B^\vee$ is a base for coroots?
- Homology and Euler characteristics of the classical Lie groups
- Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups
- Why Lie algebras of type $B_2$ and $C_2$ are isomorphic?

You have shown that $SL(2,\mathbb C)/K \to L$ is an injective homomrphism of Lie groups, where $K$ is the kernel of the representation, which is a discrete subgroup of $SL(2,\mathbb C)$. In other words, $SL(2)/K$ is (isomorphic to) a Lie subgroup of $L$. But by dimensionality reasons you are actually done: since $L$ is connected, there is a 1-to-1 correspondence between connected Lie subgroups and subalgebras of $L$’s Lie algebra. This means that any connected Lie subgroup of the same dimension must be $L$ itself.

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