The colorful world of ##2\times 2## complex matrices

[mma]

TL;DR

They are related to important Lie groups and Lie algebras, spinors, quaternions and biquaternions, hyperbolic geometry, Special Relativity, and so on. Looking for a monography about them.

The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the double cover of SO^+(3,1). The Iwasawa decomposition of SL(2,\mathbb C) is a sphere bundle over the 3-dimensional hyperbolic space. And many things I haven't mentioned or don't know about. Is there a monography on them?

 

[WWGD]

This is a strong area for @fresh_42 .

 

[fresh_42]

TL;DR Summary: They are related to important Lie groups and Lie algebras, spinors, quaternions and biquaternions, hyperbolic geometry, Special Relativity, and so on. Looking for a monography about them.

Is there a monography on them?

Not that I knew of. And thinking about it, I doubt it.

Imagine someone had written such a book. Whom would it be supposed to sell to? It would be a collection of low-dimensional examples in about a dozen fields without ever elaborating on those fields that each would require at least one book in its own right. People are interested in the various areas of mathematics, not in low-dimensional examples that aren't even necessarily typical. $\mathbb{M}(2,\mathbb{C})$ is even in physics only responsible for the weak force. Such a book would be at odds with all possible, professional buyer groups. Where on your bookshelf would such a book sit? It would fit in a dozen of sections and none.

Maybe except for entertainment. I would like to read such a book under this aspect. However, I'm afraid that $\mathbb{M}(2,\mathbb{C})$ or even $\operatorname{SU}(2,\mathbb{C})$ are simply not prominent enough. I possess a book titled "100 Years Set Theory" that is written with rigor and for entertainment. It covers subjects as Hilbert's hotel, Sierpinski and Peano curves and so on. It was published at a time when set theory was in everybody's mind because some didacticians here thought it would be a good idea to introduce Venn diagrams to elementary schools and call it set theory. The publisher prudently conceals the print run, but I very much doubt that it was a bestseller. Set theory quickly became a slogan and a synonym for modern mathematics that nobody understood - and we are talking about Venn diagrams! Now imagine a book about the $2$-sphere or quaternions.

I have written some insight articles about $\operatorname{SU}(2,\mathbb{C})$
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/
on a website dedicated to physics, where particularly this group, $\operatorname{SU}(2,\mathbb{C})$, plays a decisive role in the field. I have no idea about the click rates of these, but I doubt that they are very high. So even students who are directly affected presumably don't read an article that was written for them. The corresponding threads report 2,000+ clicks. For comparison: "Why is this line $[0,2\pi)$ not homeomorphic to the unit circle?" - a more or less trivial topic - has 5,000+ clicks and "Continuity of the Determinant" - trivial, too - has 6,000+ clicks!

Btw., you have forgotten to list the cross-product!

 

[mma]

Not that I knew of. And thinking about it, I doubt it.

Imagine someone had written such a book. Whom would it be supposed to sell to? It would be a collection of low-dimensional examples in about a dozen fields without ever elaborating on those fields that each would require at least one book in its own right. People are interested in the various areas of mathematics, not in low-dimensional examples that aren't even necessarily typical. $\mathbb{M}(2,\mathbb{C})$ is even in physics only responsible for the weak force. Such a book would be at odds with all possible, professional buyer groups. Where on your bookshelf would such a book sit? It would fit in a dozen of sections and none.

Maybe except for entertainment. I would like to read such a book under this aspect. However, I'm afraid that $\mathbb{M}(2,\mathbb{C})$ or even $\operatorname{SU}(2,\mathbb{C})$ are simply not prominent enough. I possess a book titled "100 Years Set Theory" that is written with rigor and for entertainment. It covers subjects as Hilbert's hotel, Sierpinski and Peano curves and so on. It was published at a time when set theory was in everybody's mind because some didacticians here thought it would be a good idea to introduce Venn diagrams to elementary schools and call it set theory. The publisher prudently conceals the print run, but I very much doubt that it was a bestseller. Set theory quickly became a slogan and a synonym for modern mathematics that nobody understood - and we are talking about Venn diagrams! Now imagine a book about the $2$-sphere or quaternions.

I have written some insight articles about $\operatorname{SU}(2,\mathbb{C})$
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/
on a website dedicated to physics, where particularly this group, $\operatorname{SU}(2,\mathbb{C})$, plays a decisive role in the field. I have no idea about the click rates of these, but I doubt that they are very high. So even students who are directly affected presumably don't read an article that was written for them. The corresponding threads report 2,000+ clicks. For comparison: "Why is this line $[0,2\pi)$ not homeomorphic to the unit circle?" - a more or less trivial topic - has 5,000+ clicks and "Continuity of the Determinant" - trivial, too - has 6,000+ clicks!

Btw., you have forgotten to list the cross-product!

Wise thoughts. Indeed, I did not take into account the commercial aspects at all. And indeed, I have omitted vector multiplication and even the relevant Grassmann and Clifford algebras from head to toe. Special thanks for the references to your Insights. They alone made it worth writing here.

 

[martinbn]

Not what you are looking for but Lang has a book titled $SL_2(\mathbb R)$.

 

[mma]

Nice hit! Next will be $SL_2,(\mathbb{C})$, then $GL_2(\mathbb{C})$ then we arrive to $\mathbb{M}(2,\mathbb{C})$! :) Looking forward it :)

 

[martinbn]

Nice hit! Next will be $SL_2,(\mathbb{C})$, then $GL_2(\mathbb{C})$ then we arrive to $\mathbb{M}(2,\mathbb{C})$! :) Looking forward it :)

Not for the things he does in the book. There $SL_2(\mathbb R)$ is the most complecated compare to the complex ones.

 

[mma]

There $SL_2(\mathbb R)$ is the most complecated compare to the complex ones.

Why?

 

[mma]

One more question. This is the structure of the set of 2 by 2 complex matrices:

Here, the only non-standard symbol is $\mathfrak E_3=isu(2)$. It is the 3-dimensional real Euclidean space of traceless, Hermitian matrices, with the half-anticommutator as dot product. What should I write instead of the red "????"— that is, what is the meaning of the elements of $sl(2,\mathbb C)$ whose determinant is 1?

 

[WWGD]

@fresh_42

 

[martinbn]

What should I write instead of the red "????"— that is, what is the meaning of the elements of $sl(2,\mathbb C)$ whose determinant is 1?

Are you asking about the name and notation for that set? Why do you need a name and notation?

 

[fresh_42]

What should I write instead of the red "????"— that is, what is the meaning of the elements of $sl(2,\mathbb C)$ whose determinant is 1?

If I read the table correctly, and I'm not sure it even makes sense, then $su(2)\oplus \mathfrak{E}_3=su(2)\oplus isu(2)=su(2)\oplus su(2).$

Here are some standard isomorphisms:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/#Spheres

 

[mma]

Are you asking about the name and notation for that set? Why do you need a name and notation?

All the other objects in the picture have their own names, designations, and significance somewhere. I wonder if this one does too, or is it an exception?

 

[fresh_42]

All the other objects in the picture have their own names, designations, and significance somewhere. I wonder if this one does too, or is it an exception?

What do you mean by a direct sum of (an additive) vector space with (a multiplicative) group?

 

[mma]

If I read the table correctly, and I'm not sure it even makes sense, then $su(2)\oplus \mathfrak{E}_3=su(2)\oplus isu(2)=su(2)\oplus su(2).$

I think, elements of $su(2)$ are the traceless anti-Hermitian matrices while elements of $isu(2)$ are traceless, Hermitian ones, so $su(2)\oplus \mathfrak{E}_3=su(2)\oplus isu(2)$ consists of all traceless matrices.

 

[fresh_42]

I think, elements of $su(2)$ are the traceless anti-Hermitian matrices while elements of $isu(2)$ are traceless, Hermitian ones, so $su(2)\oplus \mathfrak{E}_3=su(2)\oplus isu(2)$ consists of all traceless matrices.

The direct sum of vector spaces makes sense, but that wasn't my question. Besides that, do you consider $su(2)$ as a real vector space? Otherwise (complex), $su(2)=isu(2).$

 

[mma]

What do you mean by a direct sum of (an additive) vector space with (a multiplicative) group?

I regard them as elements of the (additive) vector space of 2 by 2 matrices.

 

[fresh_42]

I regard them as elements of the (additive) vector space of 2 by 2 matrices.

But $SU(2)$ isn't closed under addition, nor does it have an additive neutral element!

 

[mma]

The direct sum of vector spaces makes sense, but that wasn't my question. Besides that, do you consider $su(2)$ as a real vector space? Otherwise (complex), $su(2)=isu(2).$

Oh, yes, I consider them as real vector spaces. So, $su(2)\neq isu(2).$

 

[mma]

But $SU(2)$ isn't closed under addition, nor does it have an additive neutral element!

Yes, it is a sphere in the vector space $\mathbb H$. At the ends of the green arrows are spheres in the vector space at the beginning of that arrow.

 

[fresh_42]

Yes, it is a sphere in the vector space $\mathbb H$

And what is the direct sum of a sphere and a Euclidean space?

 

[mma]

And what is the direct sum of a sphere and a Euclidean space?

Of course, it isn't a direct sum. Direct sum applies only above the main diagonal. Below it, the objects originate from the green arrow.

 

[mma]

Perhaps this is not important either, but another interesting fact for me is the multiplication table of the members of the decomposition $M(2,\mathbb C)=\mathbb RI\oplus\mathfrak E_3\oplus su(2)\oplus i\mathbb RI$.