Semi-Direct Product: Explained in Plain English

[nille40]

Hi!
Could someone please explain to me what semi-direct product is, in plain english? English isn't my native language, so I would really appreciate if mathematical lingo could be avoided.

I would also appreciate some online resources. I've tried www.mathworld.com,[/URL] but that wasn't very easy to understand...

Thanks in advance,
Nille

 

[suyver]

I can only think of the dot-product for vectors. If you have two vectors:
$$A= \left(\begin{array}{c}A_x\\ A_y\\ A_z\end{array}\right)$$
and
$$B= \left(\begin{array}{c}B_x\\ B_y\\ B_z\end{array}\right)$$

Then the dot-product will be:
$$A\cdot B= \left(\begin{array}{c}A_x B_x\\ A_y B_y\\ A_z B_z\end{array}\right)$$

Hope this helps!

 

[nille40]

Thanks for responding!
However, it is not the dot-product I am looking for.

A semi-direct product is an operation between two groups. And that's pretty much all I know :)

Nille

 

[Lonewolf]

Are you familiar with normal and quotient groups?

 

[suyver]

http://dominia.org/djao/semidirect.pdf

 

[nille40]

Thanks! Actually, I think I've read that document, and thus failed to understand it... :)

Do you have any suggestion on where I should start reading? I've read about homo- and automorphisms, and I understand these pretty well. It's the final step, on which I need a clear definition, preferably in natural language, as opposed to mathematical language.

As for normal and quoutien groups - I have no idea. I'm learning (unfortunatly) the swedish lingo, so it's hard to say.

Thanks!
Nille

 

[lethe]

perhaps it would be useful to look at an example.

the most important one that i know of is the Poincaré group, which is the semidirect product of the abelian translation group generated by $P^\mu$ and the rotation group generated by $J^{\mu\nu}$. composition is defined in the obvious way.

or an even simpler example, consider the following actions of the real line.

translations:
$$\begin{align*}T(a):&\mathbb{R}\longrightarrow\mathbb{R}\\ &x\longmapsto x+a \end{align*}$$

and scales:
$$\begin{align*} S(m):&\mathbb{R}\longrightarrow\mathbb{R},\quad m\neq 0\\ &x\longmapsto mx \end{align*}$$

the first one makes a group that is isomorphic to the reals under addition, and the second one is a group that is isomorphic to the reals without zero, under multiplication.

you can compose the two in the obvious way:

$$(b,n)(a,m)=(na+b,nm)$$
which we find from the equation $n(mx+a)+b=nmx+na+b$

notice that if instead we had an honest to goodness direct product on our hands, the rule would be
$$(b,n)(a,m)=(a+b,nm)$$

 

[nille40]

Thanks! That's probably the best description of semi-direct product I've read! And I've scanned the larger part of the internet...:)

Thank you very much!
Nille

 

[lethe]

Originally posted by nille40
Thanks! That's probably the best description of semi-direct product I've read! And I've scanned the larger part of the internet...:)

Thank you very much!
Nille

i hope that helped. the point here is that instead of two copies of a group that act independently as they would in a direct product, you get ordered pairs where one of the groups acts normally, and the other group acts through some homomorphism on the first group.

the example i gave above is very natural and easy to understand, so read the paper linked above with this example in mind, and try to associate all the objects described in that paper with the translations and scales i described above.

 

[matt grime]

Originally posted by lethe
perhaps it would be useful to look at an example.

the most important one that i know of is the Poincaré group, which is the semidirect product of the abelian translation group generated by $P^\mu$ and the rotation group generated by $J^{\mu\nu}$.

if that's the most obvious/important one you know you ought to get out more! the simplest one is of course the semi direct product of C_2 and C_3 that is not C_6 ie it is S_3

note tongue firmly planted in cheek, but in general A_n semi direct prod with C_2 is fairly important!

 

[lethe]

Originally posted by matt grime
if that's the most obvious/important one you know you ought to get out more! the simplest one is of course the semi direct product of C_2 and C_3 that is not C_6 ie it is S_3

note tongue firmly planted in cheek, but in general A_n semi direct prod with C_2 is fairly important!

well, i think the Poincaré group is the most important group in all of physics, and is therefore more important than S₆, but i suppose it is a matter of opinion, so choose whatever group you want.

 

[matt grime]

Originally posted by lethe
well, i think the Poincaré group is the most important group in all of physics, and is therefore more important than S₆, but i suppose it is a matter of opinion, so choose whatever group you want.

What about the Lorentz group? Or the thingy bob group associated to a manifold (pontrjagin? picard?) How's the affine transformations most important? Serious question, unlike my previous answer, which was genuinely facetious. I mean what does it control? This was posted in linear algebra not physics, and here the Poincare group might be less important than all the permutation groups (which are finite), and therefore Hecke algebras. And it's a small step from there via duality theroems to objects of great importance in theoretical physics, but that's a whole new topic (in Lie theory etc)
In fact we get all D_n as semidirect products too. Indeed as the Poincare group deals in affine transformations and not linear ones... ok tongue still in cheek for that one. Any takers to explain PSL_3(F_2)?

 

[lethe]

Originally posted by matt grime
What about the Lorentz group? Or the thingy bob group associated to a manifold (pontrjagin? picard?) How's the affine transformations most important?

yes, like i said, it is a matter of opinion, so take whichever you want and call it the most important.

Serious question, unlike my previous answer, which was genuinely facetious. I mean what does it control? This was posted in linear algebra not physics, and here the Poincare group might be less important than all the permutation groups (which are finite), and therefore Hecke algebras.

it is a good point. i might argue that the Poincaré group is the most important group in physics, but it is probably not such an important group to a mathematician.

And it's a small step from there via duality theroems to objects of great importance in theoretical physics, but that's a whole new topic (in Lie theory etc)
In fact we get all D_n as semidirect products too. Indeed as the Poincare group deals in affine transformations and not linear ones... ok tongue still in cheek for that one. Any takers to explain PSL_3(F_2)?

what is F₂?

 

[matt grime]

F_2 is (here) the field with two elements, PSL_3(F_2) is the symmetries of the fano plane which I seem to have some memory as being Octonionic and therefore interesting in the John Baez view point on Mathematical physics.

 

[lethe]

Originally posted by matt grime
F_2 is (here) the field with two elements

thanks. i am too used to seeing Z₂, but i guess i should have known from context.