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

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  • #1
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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.
 

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  • #2
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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.
 
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  • #3
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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.
 
  • #4
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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.
 
  • #5
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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} ##).
 
  • #6
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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
 
  • #7
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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.
 
  • #8
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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.
 
  • #9
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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.
 

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