- #1

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R^4 |X SL(2,C)

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

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- #1

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R^4 |X SL(2,C)

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

- #2

fresh_42

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- #3

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Thanks so much.

- #4

fresh_42

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$$

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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R^4 |X SL(2,C)

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

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.

$$

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.

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

WWGD

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Try:

R^4 |X SL(2,C)

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

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

- #7

fresh_42

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

- #9

fresh_42

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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.

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

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