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

  1. Oct 21, 2017 #1
    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.
     
  2. jcsd
  3. Oct 21, 2017 #2

    fresh_42

    Staff: Mentor

    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.
     
  4. Oct 21, 2017 #3
    Thanks so much.
     
  5. Oct 21, 2017 #4

    fresh_42

    Staff: Mentor

    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.
     
  6. Oct 22, 2017 #5

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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} ##).
     
  7. Oct 22, 2017 #6

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Try:

    http://detexify.kirelabs.org/symbols.html
     
  8. Oct 22, 2017 #7

    fresh_42

    Staff: Mentor

    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.
     
  9. Oct 25, 2017 #8
    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.
     
  10. Oct 25, 2017 #9

    fresh_42

    Staff: Mentor

    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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