Complexifying su(2) to get sl(2,C)-group thread footnote

In summary, the conversation discusses the concept of complexification and how it relates to the matrix groups SL(2,C) and su(2). It is noted that su(2) is the skew hermitian ones and that su(2)C, the complexification of su(2), is isomorphic to sl(2,C). The group SL(2,C) is a representation of the group of boosts and turns, and there is a discussion about why it doesn't show up in descriptions instead of 4x4 Dirac spinors. The conversation ends with a quote from Archimedes about the inevitability of certain concepts.
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Complexifying su(2) to get sl(2,C)---group thread footnote

On the group thread midterm exam (which we never had to take!) it says what is the LA of the matrix group SL(2, C)
and the answer is the TRACE ZERO 2x2 matrices.
So that is what sl(2,C) is.
When you exponentiate one of the little critters, det = exp trace,
so the determinant is one which is what SL means.

Any X in sl(2,C) has a unique decomposition into skew hermitians that goes like this

X = (X - X*)/2 + i(X + X*)/2i

and these two skew hermitians
(X - X*)/2 and (X + X*)/2i
are trace zero, because trace is linear

check the skew hermitiandom of them:
(X - X*)* = (X* - X) = - (X - X*)

the other one checks because (1/2i)* = - (1/2i)
since conjugation does not change (X + X*)* = (X + X*)

so the upshot is that any X in sl(2,C) is composed
X = A + iB
of two matrices A and B in su(2)

Also on the midterm was the fact that su(2) is the skew hermitian ones: A* = - A.

There was this footnote on complexification of LAs and the above suffices to show, without much further ado, that su(2)C the complexification of su(2) is isomorphic to sl(2, C)
 
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SL(2,C) is a representation of the group of boosts and turns, so why doesn't it show up in our descriptions instead of the 4×4 Dirac spinors?
 
  • #3
Well, there you go: Topology/Non-Euclidian Geomerty, like poverty and ignorance: We will always have them with us.

Rudy

"Go Figure." - Archimedes
 
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FAQ: Complexifying su(2) to get sl(2,C)-group thread footnote

What is the purpose of complexifying su(2) to get sl(2,C)-group?

The purpose of complexifying su(2) to get sl(2,C)-group is to extend the special unitary group, su(2), to a larger complex group, sl(2,C). This allows for a more comprehensive understanding of the symmetries and properties of the group, and can aid in solving certain mathematical problems.

How is su(2) complexified to obtain sl(2,C)-group?

Su(2) is complexified by introducing complex coefficients in the generators of the group. This results in a larger, complexified Lie algebra, which is isomorphic to sl(2,C). This process is known as the complexification of Lie algebras.

What are the applications of complexifying su(2) to get sl(2,C)-group?

The applications of complexifying su(2) to get sl(2,C)-group include representation theory, quantum mechanics, and differential geometry. It can also be used in the study of Lie groups and their symmetries.

What are the differences between su(2) and sl(2,C)-group?

The main difference between su(2) and sl(2,C)-group is that su(2) is a real, compact group while sl(2,C) is a complex, non-compact group. Additionally, the dimension of su(2) is 3, while the dimension of sl(2,C) is 3+3=6.

Are there any real-world applications for complexifying su(2) to obtain sl(2,C)-group?

Yes, there are several real-world applications for complexifying su(2) to obtain sl(2,C)-group. For example, it is used in quantum mechanics to study the symmetries of particles and in differential geometry to understand the geometry of spaces with non-trivial curvature. It also has applications in signal processing and optimization problems.

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