What are the algebra prerequisites for Lie groups?

[kostas230]

I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract algebra course, but my background in linear algebra is pretty strong. So, I was wondering what are the algebra prerequisites for Lie groups?

Thank you. :)

 

[Incnis Mrsi]

Sorry Ī know almost no English textbooks. If your linear algebra is strong enough to understand all https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions
and most of https://en.wikipedia.org/wiki/Versor , then you already have some notion of a Lie group and can try to learn things that are needed for a concrete problem without learning the theory in depth. Common physics uses only some pieces of all theory of Lie groups and algebras.

 

[D H]

As Incnis mentioned, you don't need to know full-blown Lie theory to be able to use that theory in physics. Physicists and engineers use "Lie theory lite".

You don't really need abstract algebra (but knowing a bit of it will help). You do need a strong grip on linear algebra.

Basic concepts needed:

• 
  Matrix multiplication is in general non-commutative and therefore has a non-zero commutator $[A,B]=AB-BA$
• Matrix to some non-negative integral power: $A^0\equiv I, A^n=A^{n-1}A = A A^{n-1}$
• Matrix exponential $\exp(A) = \sum_{n=0}^{\infty} A^n/n!$
• For every invertible matrix A, there exists a matrix $B$ such that $A = \exp(B)$
• The set of $N\times N$ invertible matrices form a non-abelian group (as I wrote earlier, knowing a bit of abstract algebra does help).
• There's a mapping from the set of $N\times N$ matrices to the set of $N\times N$ invertible matrices via the matrix exponential. This, along with a few other things makes the set of $N\times N$ invertible matrices a Lie group and makes the set of $N\times N$ invertible matrices the Lie algebra for that group.

One last thing: I've found wikipedia to be an absolutely terrible source for learning this subject. There are lots of online references that are much, much better than wikipedia, and of course there are those old fashioned things called "books".

 

[Fredrik]

The book by Brian Hall requires only basic linear algebra, and a little bit about limits and continuity.

The most general definition of Lie group involves terms from differential geometry. Hall avoids this by noticing that almost all the interesting groups are matrix groups (i.e. groups whose elements are matrices) and that a matrix group is a Lie group if and only if it satisfies some simple conditions. (He doesn't prove this as a theorem, because that would require differential geometry). He then takes those conditions as the definition of a matrix Lie group.

Lie algebras are treated in a similar way.

 

[TrickyDicky]

The book by Brian Hall requires only basic linear algebra, and a little bit about limits and continuity.

The most general definition of Lie group involves terms from differential geometry. Hall avoids this by noticing that almost all the interesting groups are matrix groups (i.e. groups whose elements are matrices) and that a matrix group is a Lie group if and only if it satisfies some simple conditions. (He doesn't prove this as a theorem, because that would require differential geometry). He then takes those conditions as the definition of a matrix Lie group.

But if the OP is interested in GR he shouldn't skip the connections of groups with differential geometry which he should be at least slightly acquainted with if he is dealing with graduate GR. Furthermore, given GR nonlinearity it might be beneficial for him to get some notions outside matrix groups, with groups that don't have faithful finite dimensional matrix representation in certain cases(i.e. : SL(2,R), SL(2,C)...) and their action in manifolds.

 

[Matterwave]

The book by Brian Hall requires only basic linear algebra, and a little bit about limits and continuity.

The most general definition of Lie group involves terms from differential geometry. Hall avoids this by noticing that almost all the interesting groups are matrix groups (i.e. groups whose elements are matrices) and that a matrix group is a Lie group if and only if it satisfies some simple conditions. (He doesn't prove this as a theorem, because that would require differential geometry). He then takes those conditions as the definition of a matrix Lie group.

Lie algebras are treated in a similar way.

This sounds like the approach in "Naive Lie Theory" by John Stillwell, which I like very much. Of course, doing things this way you miss out on all the exceptional simple Lie groups! But those are much more prominent in string theory and quantum gravity and less prominent in GR or QFT.

 

[TrickyDicky]

Of course, doing things this way you miss out on all the exceptional simple Lie groups! But those are much more prominent in string theory and quantum gravity and less prominent in GR or QFT.

Less prominent? SU(2), SU(3), SL(2,C) are all simple groups and fundamental to QFT and the latter also to GR.

 

[Matterwave]

Less prominent? SU(2), SU(3), SL(2,C) are all simple groups and fundamental to QFT and the latter also to GR.

I am talking about the exceptional simple lie groups, i.e. G2, F4, E6, E7, and E8. Those are not matrix groups, and I have never seen them in QFT or GR. But of course my knowledge of either QFT or GR is not complete.

SU(2), SU(3), SL(2,C) (or really any subgroup in GL(n,C)) are all matrix Lie groups, and they can be analyzed without much differential geometry.

 

[TrickyDicky]

I am talking about the exceptional simple lie groups, i.e. G2, F4, E6, E7, and E8. Those are not matrix groups, and I have never seen them in QFT or GR. But of course my knowledge of either QFT or GR is not complete.

Sorry, I misinterpreted the word exceptional(I gave it literal-like in outstanding- instead of mathematical meaning), I listed the classical simple groups. You are right that the exceptional simple groups are not used in QFT/GR.

SU(2), SU(3), SL(2,C) (or really any subgroup in GL(n,C)) are all matrix Lie groups, and they can be analyzed without much differential geometry.

For SL(2,C), being a non-compact group, you are not guaranteed to always have faithful matrix rep.

 

[Fredrik]

For SL(2,C), being a non-compact group, you are not guaranteed to always have faithful matrix rep.

Isn't the identity map on SL(2,ℂ) a faithful representation of SL(2,ℂ)?

 

[Matterwave]

For SL(2,C), being a non-compact group, you are not guaranteed to always have faithful matrix rep.

Since the special linear group is defined as the group of (real/complex) matrices of dimension n, and determinant 1... I'm not sure...how it might not have a faithful matrix representation? I don't understand how your statement could possibly be true since it it defined in terms of a matrix representation! Perhaps, do you mean it might not have a faithful, finite dimensional, UNITARY representation?

EDIT: Whoop, got sniped by Fredrik.

 

[TrickyDicky]

Since the special linear group is defined as the group of (real/complex) matrices of dimension n, and determinant 1... I'm not sure...how it might not have a faithful matrix representation? I don't understand how your statement could possibly be true since it it defined in terms of a matrix representation! Perhaps, do you mean it might not have a faithful, finite dimensional, UNITARY representation?

EDIT: Whoop, got sniped by Fredrik.

Sure, I was referring to finite dimensional rep.

 

[Matterwave]

Sure, I was referring to finite dimensional rep.

Actually the keyword in my statement was the term "unitary", which is why I capitalized it. ;) SL(2,C) is a legitimate matrix group. It has itself as a faithful, finite dimensional, representation as Fredrik mentioned. This representation just happens not to be unitary, since SL(2,C) matrices are not necessarily unitary matrices. But from a mathematical point of view, whether we have a unitary representation or not is not so important. From a QFT point of view, then obviously we want unitary representations of groups due to Wigner's theorem.

 

[TrickyDicky]

That's why I wrote you are not always(in all cases) guaranteed a faithfull matrix rep. Since we were talking about QFT unitarity is assumed.
Edit: I was actually referring to the universal covering case as pond Dragon pointed out.
And of course in QFT the unitary non-trivial matrix representations of SL(2,C) must be infinite dimensional.

 

[Pond Dragon]

Furthermore, given GR nonlinearity it might be beneficial for him to get some notions outside matrix groups, with groups that don't have faithful matrix representation(i.e. : SL(2,R), SL(2,C)...) and their action in manifolds.

I think you are referring to their universal covering groups...?

 

[TrickyDicky]

I think you are referring to their universal covering groups...?

Yes, I should have clarified that.

 

[TrickyDicky]

For the SL(2,R) case see: http://math.stackexchange.com/questions/129644/universal-cover-of-sl-2-mathbbr
SL(2,C) being simply connected is its own universal covering.

 

[TrickyDicky]

I guess thinking about groups in purely mathematical or formal terms is different than in a more physical way. SL(2,C) is of course a matrix group formally, but when thinking in more physical terms, here meaning requiring unitarity, one sees it under a different light.

 

[Fredrik]

Regardless of how we think of it, it is covered by the book. I don't see many advantages of studying a book based on differential geometry, if we're interested in representations. But I must confess that I'm not familiar with applications of Lie group theory to GR, so maybe I'm missing something. The OP mentioned that he encountered Lie groups in a presentation of how to find the Schwarzschild solution. I don't know how Lie group theory enters that picture, so I can't guarantee that Hall's book contains everything that the OP needs. But if it doesn't, I suspect that all he has to do is to read a few pages in a book like "introduction to smooth manifolds" by Lee...but I must also confess that I haven't read that chapter.

 

[TrickyDicky]

I don't see many advantages of studying a book based on differential geometry, if we're interested in representations. But I must confess that I'm not familiar with applications of Lie group theory to GR, so maybe I'm missing something. The OP mentioned that he encountered Lie groups in a presentation of how to find the Schwarzschild solution. I don't know how Lie group theory enters that picture, so I can't guarantee that Hall's book contains everything that the OP needs. But if it doesn't, I suspect that all he has to do is to read a few pages in a book like "introduction to smooth manifolds" by Lee...but I must also confess that I haven't read that chapter.

I had a look at Straumann chapter on Schwarzschild and it actually just mentions in a couple of paragraphs 3D rotational symmetry group in Lorentz manifolds to justify the 2-spheres foliation of Schwarzschild spacetime. I don't think any knowledge about groups besides a very basic familiarity with groups is really necessary to follow it.
Towards the end of the book group theory is used in GR's spinor analysis in a short appendix where a working knowledge of SL(2,C) is probably required.
Barring this I'm not familiar either with other applications of group theory in GR outside the obvious fact that isometry groups are Lie groups.
In any case I've always found that learning even if just a bit about Lie groups(the basic ones: rotations, SU(2) and the Lorentz group), really pays off in terms of my understanding of differential geometry.

 

[lpetrich]

I am talking about the exceptional simple lie groups, i.e. G2, F4, E6, E7, and E8. Those are not matrix groups, ...

What gives you that idea? I ask that because the theory of Lie-algebra representations implies that *every* Lie algebra can be realized as matrices, and from them, *every* Lie group. That includes those exceptional ones.

But do you mean groups of matrices with some simple property? Like being unitary or orthogonal or symplectic. There isn't such a simple property associated with any representation of the exceptional ones, as far as I know.

 

[Matterwave]

What gives you that idea? I ask that because the theory of Lie-algebra representations implies that *every* Lie algebra can be realized as matrices, and from them, *every* Lie group. That includes those exceptional ones.

But do you mean groups of matrices with some simple property? Like being unitary or orthogonal or symplectic. There isn't such a simple property associated with any representation of the exceptional ones, as far as I know.

I may have misremembered something I read in a Lie group book I read. I read that these exceptional lie groups are based on octonions, which have a non-associative product and therefore can't be represented by matrices. But now that I think about it, group multiplication must be associative, so I'm not sure exactly what I was thinking.

 

[lpetrich]

John Baez's Octonions relates the exceptional Lie algebras to the octonions. Is that the sort of thing that you were thinking of?

For G2, it's simple. G2 is the octonions' automorphism group. For the others, it's more complicated.

Matrix R is an automorphism of binary operation * if (R.x)*(R.y) = R.(x*y) for all x and y. In general, that's difficult to solve, but it's easier in the continuous case: R ~ 1 + ε*L. Then, we test for (L.x)*y + x*(L.y) = L.(x*y), which will be linear in L. The possible L's form the automorphisms' Lie algebra.

 

[Matterwave]

John Baez's Octonions relates the exceptional Lie algebras to the octonions. Is that the sort of thing that you were thinking of?

For G2, it's simple. G2 is the octonions' automorphism group. For the others, it's more complicated.

Matrix R is an automorphism of binary operation * if (R.x)*(R.y) = R.(x*y) for all x and y. In general, that's difficult to solve, but it's easier in the continuous case: R ~ 1 + ε*L. Then, we test for (L.x)*y + x*(L.y) = L.(x*y), which will be linear in L. The possible L's form the automorphisms' Lie algebra.

I thought for sure there are some Lie groups which are not matrix Lie groups...otherwise why go through the whole left-invariant vector fields derivation of the Lie bracket when one can just use the matrix commutator?

But I am not very familiar with the non-classical Lie groups. I have only seen the exceptional Lie groups in passing.

 

[aleazk]

I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract algebra course, but my background in linear algebra is pretty strong. So, I was wondering what are the algebra prerequisites for Lie groups?

Thank you. :)

Hi, kostas. A 'spherically symmetric' spacetime (like the Schwarzschild solution) is one whose isometry group contains a subgroup isomorphic to SO(3). This is, of course, a Lie group. So, if you really, really want to understand this in a very precise and rigurous way you would need to know some Lie group theory. But I think it would be a long detour for a GR course, and also unnecessary. If you know about Killing vector fields, check the discussion of sphericall symmetry in Carroll's book ("Spacetime and geometry, etc."), starting at page 197 (in the context of Birkhoff's theorem). He talks about the Killing fields of the usual 2-sphere, Frobenius' theorem and foliations, and some other things, and I think that's enough motivation for the 'emergence' of the usual line element of the 2-sphere in the spherically symmetric spacetime (at least that's how I convinced myself when I was studying the topic, and without knowing Lie groups!). Actually, in these considerations using the Killing fields of the usual 2-sphere, etc., you are actually using things related to Lie groups/algebras, and what Carroll does explicitly is what's explained very succinctly in page 120 of Wald ("General Relativity").

If you want to know why the Killing fields have precisely these commutation relations in the 2-sphere, you would need to know some things about Lie groups using the differential geometry definition, and for later studying the action of these groups in a manifold and how the Lie algebra of the group can be realized as the generators of the diffeos in this action. Using that you can understand better the relation between SO(3), its Lie algebra and the commutation relations of the Killing fields of this example. Although I'm not very sure of the details of this. Some of these things can be found in, e.g., Isham's 'Modern Differential Geometry for Physicists'.