- #1
Anthony
- 83
- 0
Hi guys - long time reader first time poster!
I'm currently getting to grips with the topic of Lie Algebras, and I've come across something that's baffled me somewhat. I've been asked to show:
[tex]so(4) = su(2) \oplus su(2)[/tex]
Where the lower so(n) denotes the Lie Algebra of SO(n) etc. Now, in a previous question, I was asked to show:
[tex]u(2) = su(2) \oplus \mathbb{R} [/tex]
Where [tex]\mathbb{R}[/tex] denotes the set of constant (real) multiples of the matrices of the form [tex]i\mathbb{I}_2[/tex]. This was easy enough; I showed that for each [tex]v\in u(2)[/tex] there exists a [tex]x \in su(2)[/tex] and a [tex]y \in \mathbb{R}[/tex] such that v=x+y, and also that if [tex]A \in su(2) \cap \mathbb{R} [/tex] then A=0.
However, in this new case, I assume the [tex]\oplus[/tex] means the the matrix direct sum, but surely if this is the case, then it's false - since the matrices in so(4) take the form:
[tex]\left(\begin{array}{cccc}0&a&b&c\\-a&0&d&e\\-b&-d&0&f\\-c&-e&-f&0\end{array}\right)[/tex]
Which is not of the form of a matrix direct sum. If anyone could give me any hints as to where my confusion lies, I'd be very greatful - although if you could keep the hints sufficiently vague, as to not to do all the work for me!
I'm currently getting to grips with the topic of Lie Algebras, and I've come across something that's baffled me somewhat. I've been asked to show:
[tex]so(4) = su(2) \oplus su(2)[/tex]
Where the lower so(n) denotes the Lie Algebra of SO(n) etc. Now, in a previous question, I was asked to show:
[tex]u(2) = su(2) \oplus \mathbb{R} [/tex]
Where [tex]\mathbb{R}[/tex] denotes the set of constant (real) multiples of the matrices of the form [tex]i\mathbb{I}_2[/tex]. This was easy enough; I showed that for each [tex]v\in u(2)[/tex] there exists a [tex]x \in su(2)[/tex] and a [tex]y \in \mathbb{R}[/tex] such that v=x+y, and also that if [tex]A \in su(2) \cap \mathbb{R} [/tex] then A=0.
However, in this new case, I assume the [tex]\oplus[/tex] means the the matrix direct sum, but surely if this is the case, then it's false - since the matrices in so(4) take the form:
[tex]\left(\begin{array}{cccc}0&a&b&c\\-a&0&d&e\\-b&-d&0&f\\-c&-e&-f&0\end{array}\right)[/tex]
Which is not of the form of a matrix direct sum. If anyone could give me any hints as to where my confusion lies, I'd be very greatful - although if you could keep the hints sufficiently vague, as to not to do all the work for me!