Dimension of SL(2,H), SL(2,R), SL(2,C), SL(2,O) - Proving 15 & 45

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In summary, the dimension of the special linear algebra sl(2,R) is 3, which is isomorphic to the Lorentz algebra so(2,1) with dimensions 2+1=3. Similarly, the dimension of sl(2,C) is 6, isomorphic to so(3,1) with dimensions 3+2+1=6. For sl(2,H) and sl(2,O), the dimensions are 15 and 45 respectively, isomorphic to so(5,1) and so(9,1) with dimensions 5+4+3+2+1=15 and 9+8+7+6+5+4+3+2+1
  • #1
BruceG
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dimension of sl(2,R) = 1*(2*2-1) = 3, is isomorphic to so(2,1) : 2+1 = 3
dimension of sl(2,C) = 2*(2*2-1) = 6, is isomorphic to so(3,1) : 3+2+1 = 6
dimension of sl(2,H) = 15, is isomorphic to so(5,1) : 5+4+3+2+1 = 15
dimension of sl(2,O) = 45, is isomorphic to so(9,1) : 9+8+7+6+5+4+3+2+1 = 45

How do you prove the 15 and 45?
Naively I would expect 4*(2*2-1)=12 and 8*(2*2-1)=24.
 
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  • #2
The tricky part is the definition of sl(2,H) and sl(2,O). John Baez discusses it (but not in all the details) http://math.ucr.edu/home/baez/octonions/node11.html" .
 
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  • #3
OK, ta, I got it.

So with R and C the set of traceless matrices are closed (if a,b are traceless then so is [a,b]). So once you've counted the traceless matrices you've got the whole algebra.

H and O are not commutative, so if you start with the set of traceless matrices, then to close off the algebra you have to add in some matrices of the form [a,b] which are not traceless.

Hence 15 > 12
and 45 > 24

This then makes this result all the more fascinating: take any normed division algebra K and generate sl(2,K) according to the above procedure and you magically end up with something isomorphic to the Lorentz algebra so(dim(K)+1,1).

There must be a deeper reasoning behind all this: I'll read more of Baez to see what he has to say, but any other references on this would be appreciated.
 

Related to Dimension of SL(2,H), SL(2,R), SL(2,C), SL(2,O) - Proving 15 & 45

Question 1: What is SL(2,H)?

SL(2,H) is the special linear group of 2x2 matrices with entries in the division algebra of quaternions, H. This group is also known as the quaternionic special linear group.

Question 2: What is SL(2,R)?

SL(2,R) is the special linear group of 2x2 real matrices with determinant 1. This group is also known as the real special linear group.

Question 3: What is SL(2,C)?

SL(2,C) is the special linear group of 2x2 complex matrices with determinant 1. This group is also known as the complex special linear group.

Question 4: What is SL(2,O)?

SL(2,O) is the special linear group of 2x2 octonionic matrices with determinant 1. This group is also known as the octonionic special linear group.

Question 5: How can you prove the dimension of 15 and 45 for these special linear groups?

The dimension of a special linear group can be proven by using the fact that the determinant of a matrix in the group must equal 1. This condition creates a system of equations that can be solved to find the number of independent parameters in the matrices, which is equal to the dimension of the group. For these particular special linear groups, the dimension can be shown to be 15 for SL(2,H) and SL(2,R), and 45 for SL(2,C) and SL(2,O).

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