What does this math symbol mean? R^4 |X SL(2,C)

[pellman]

It is an X like in a cross product but with a vertical line connecting upper-left and lower-left endpoints. I will write it as |X . Example context:

R^4 |X SL(2,C)

where the R and C are real numbers and complex numbers.

 

[fresh_42]

It means the semidirect product of the groups $\mathbb{R}^4$ and $SL(2,\mathbb{C})$ where the latter is a normal subgroup of it, i.e. the former operates on the latter.

 

[pellman]

It means the semidirect product of the groups $\mathbb{R}^4$ and $SL(2,\mathbb{C})$ where the latter is a normal subgroup of it, i.e. the former operates on the latter.

Thanks so much.

 

[fresh_42]

But I don't know how the operation goes. Often it is a conjugation, but $\mathbb{R}^4$ is additive and $SL(2,\mathbb{C})$ multiplicative. So maybe in this case the Euclidean space is the normal subgroup, because not all authors follow the rule:
$$
K \leq G = H \ltimes K \trianglerighteq H
$$
Some use it the other way around. In any case is it important how the group multiplication is defined to see which one is the normal one.

 

[dextercioby]

It is an X like in a cross product but with a vertical line connecting upper-left and lower-left endpoints. I will write it as |X . Example context:

R^4 |X SL(2,C)

where the R and C are real numbers and complex numbers.

But I don't know how the operation goes. Often it is a conjugation, but $\mathbb{R}^4$ is additive and $SL(2,\mathbb{C})$ multiplicative. So maybe in this case the Euclidean space is the normal subgroup, because not all authors follow the rule:
$$
K \leq G = H \ltimes K \trianglerighteq H
$$
Some use it the other way around. In any case is it important how the group multiplication is defined to see which one is the normal one.

Well, the inhomogenous $\text{SL}(2,\mathbb{C})$ is the semidirect product of $(\mathbb{R}^4, +)$ and $(\text{SL}(2,\mathbb{C}),\cdot)$, where + stands for the addition of 4-tuples of real numbers and the dot stands for matrix multiplication. The group $(\mathbb{R}^4, +)$ is an abelian normal subgroup of the semidirect product group.

More precisely:

$$\text{ISL}(2,\mathbb{C}) := \mathbb{R}^4 \ltimes \text{SL}(2,\mathbb{C}) =\left\{ (x,A), x\in\mathbb{R}^4, A \in\text{SL}(2,\mathbb{C}) | (x.A) \ltimes (y,B) = (x+A \star b, A \cdot B), \forall x,y\in\mathbb{R}^4, \forall A,B\in\text{SL}(2,\mathbb{C})\right\} $$

$$ (\mathbb{R}^4, +) \trianglerighteq \text{ISL}(2,\mathbb{C}) $$

The $A \star b$ is the action of $\text{SL}(2,\mathbb{C})$ on the 4D-Minkowski spacetime defined by a matrix representation of the restricted Lorentz group ($A\in \text{SL}(2,\mathbb{C}) \mapsto \bf{\Lambda}_{A} \in \mathcal{L}_{+}^{\uparrow}$).

 

[WWGD]

It is an X like in a cross product but with a vertical line connecting upper-left and lower-left endpoints. I will write it as |X . Example context:

R^4 |X SL(2,C)

where the R and C are real numbers and complex numbers.

Try:

http://detexify.kirelabs.org/symbols.html

 

[fresh_42]

This is what I meant by my second post: If $\mathbb{R}^4 \trianglelefteq ISL(2,\mathbb{C})$, then it would be better to write $ISL(2,\mathbb{C}) = \mathbb{R}^4 \rtimes SL(2,\mathbb{C})$ but unfortunately not all authors follow this convention, which means the easy mnemonic that $\rtimes \Leftrightarrow \geq + \trianglelefteq$ doesn't work in those cases. Therefore the automorphism has to be mentioned, because there are a lot of unspoken conventions involved here. Theoretically, i.e. I haven't checked whether it is impossible, there could be a multiplication which makes $SL(2,\mathbb{C})$ the normal and $\mathbb{R}^4$ the ordinary subgroup. If I had to bet, I'd say it is possible, because I know of a proper semidirect product of the two-dimensional non-Abelian Lie algebra with $\mathfrak{sl}(2,\mathbb{C})$ where $\mathfrak{sl}(2,\mathbb{C})$ is the ideal. It should be integrable to the groups.

 

[pellman]

I have been looking at some definitions of semidirect product online and many of them start with suppose that G is a group with subgroups H and N where N is a normal subgroup and G = HN.

What does HN mean? I've never come across that notation in the physics literature.

 

[fresh_42]

I have been looking at some definitions of semidirect product online and many of them start with suppose that G is a group with subgroups H and N where N is a normal subgroup and G = HN.

What does HN mean? I've never come across that notation in the physics literature.

It means, you can write the elements $g\in G$ as $g=h \cdot n$ with $h\in H , n\in N$. This doesn't need to be unique, any product will do. It is the group generated by all elements of $H$ and $N$. Since $N$ is normal, you can always sort $N$ to the right.