A 3dim Poincare Algebra - isl(2,R)

[bob2]

The Poincare algebra is given by isl(2, R) ~ sl(2,R) + R^3. What exactly does the i stand for?
Thanks a lot in advance!

 

[fresh_42]

I have never seen a notation $isl(2,\mathbb{R})$. Can you give some more background? And what does $sl_2 + \mathbb{R}^3$ mean? And as far as I know, the Poincaré algebra is ten dimensional.

 

[bob2]

I have never seen a notation $isl(2,\mathbb{R})$. Can you give some more background? And what does $sl_2 + \mathbb{R}^3$ mean? And as far as I know, the Poincaré algebra is ten dimensional.

I'm sorry, my statement was incorrect. You are right about the Poincare algebra. I meant the Poincare algebra in 3dim (not 4dim) space and not the dimension of the Lie algebra. by sl(2,R) + R^3 the direct sum of the special linear algebra sl(2, R) and R^3 is denoted

 

[fresh_42]

I'm sorry, my statement was incorrect. You are right about the Poincare algebra. I meant the Poincare algebra in 3dim (not 4dim) space and not the dimension of the Lie algebra. by sl(2,R) + R^3 the direct sum of the special linear algebra sl(2, R) and R^3 is denoted

In this case my question is: $sl(2,\mathbb{R}) \oplus \mathbb{R}^3$ as a Lie algebra or simply the vector space? I assume $isl(2,\mathbb{R})$ is simply an abbreviation the author of your source uses for this (presumably Lie algebra) sum. And I further assume that it is not a direct product, but an indirect, i.e. I suppose $sl(2,\mathbb{R})$ to act non-trivially on $\mathbb{R}^3$.
If you can write down this action, you will get the multiplications in it.
And by the way, this one is six dimensional.

 

[fresh_42]

Let me make another guess. $sl(2,\mathbb{R})$ might formerly have been $o(2,1)$ which (I don't know without proving it) is isomorphic to $sl_2$. So if you consider Lorentz transformations plus translations on $\mathbb{R}^3 \cong \mathbb{R}^{2,1}$ and call this algebra $isl_2$, then it makes sense to write $isl_2 \cong sl_2 \ltimes \mathbb{R}^3$ or sloppy $isl_2 \text{ ~ } sl_2 + \mathbb{R}^3$

 

[bob2]

sl(2,R) as I remember is isomorphic to so(2,1). Thanks so much for your reply- this way of interpreting it makes sense. Sorry, that I am replying so late- I thought I had already replied