Lie groups and SU(3). I’m having trouble understanding these. The trouble is that when I look for an explanation of SU(2), SU(3), SO(3) it says “look up Lie Groups” and when I look for an explanation of Lie Groups it says “look up SU(3)”. Can someone define these for me in the simplest possible terms without referring to the other? Please. Using the words “rotation”, “reflection” and “glide reflection” would help because I understand those. I also understand non-Abelian, det, and e^iθ.

https://en.wikipedia.org/wiki/Lie_group

Michael V said:

https://en.wikipedia.org/wiki/Lie_group

Just had a look at that.

Maybe it’s too basic for moll, or maybe he just dismisses TATE links.

It’s a mystery to me how these mathematical approaches to things seem to be talking about stuff I use every day, but in such a way that they are totally incomprehensible.

mollwollfumble said:

Lie groups and SU(3). I’m having trouble understanding these. The trouble is that when I look for an explanation of SU(2), SU(3), SO(3) it says “look up Lie Groups” and when I look for an explanation of Lie Groups it says “look up SU(3)”.

oops…

Its handy to have a hidden escape sequence in an endless loop.

and here we thought this was yet another “Aus Politics” thread

The Rev Dodgson said:

SCIENCE said:

and here we thought this was yet another “Aus Politics” thread

It’s a mystery to me how these mathematical approaches to things seem to be talking about stuff I use every day, but in such a way that they are totally incomprehensible.

fair point, you’re right, it really is yet another “Aus Politics” thread

The Rev Dodgson said:

Michael V said:

https://en.wikipedia.org/wiki/Lie_group

Just had a look at that.

Maybe it’s too basic for moll, or maybe he just dismisses TATE links.

It’s a mystery to me how these mathematical approaches to things seem to be talking about stuff I use every day, but in such a way that they are totally incomprehensible.

Yesterday a video popped up in my youtube suggestions on Clifford’s geometric algebra – a better way of handling complex numbers and quaternions etc. I’d never heard of William Clifford and he certainly appears to be an amazing chap. During the video it suggested that it has applications in circuit analysis and power systems in EE – so I went for a look around.

Found a paper that suggests that the way we’ve been calculating apparent power is incorrect under non-sinusoidal conditions. I then found a paper from last year which calls the original paper a joke (actually used those words) and then found another paper from last year that shows a comparison between calculation techniques and shows a large discrepancy between traditional, complex number manipulation, and the new geometric algebra way. I’m now confused :)

The problem is that I can actually follow most of the maths up to a point and the bits that elude me are bits that I once new – at least in the complex case. Up until yesterday I didn’t even realise that there were issues in the way things were calculated in this realm and that there was an academic brouhaha over the issue.

The Rev Dodgson said:

Michael V said:

https://en.wikipedia.org/wiki/Lie_group

Just had a look at that.

Maybe it’s too basic for moll, or maybe he just dismisses TATE links.

It’s a mystery to me how these mathematical approaches to things seem to be talking about stuff I use every day, but in such a way that they are totally incomprehensible.

The first line of that wikipedia article is:

> In mathematics, a Lie group (pronounced /liː/ “Lee”) is a group that is also a differentiable manifold.

Differentiable manifolds are not an easy topic. I do understand them, just, but being topology takes them totally outside the field of group theory. So I totally don’t see the connection. And that’s in the first sentence! In Wiklipedia! Wikipedia follows that straight on with:

> Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions given by the special orthogonal group SO(3).

Here’s the problem. In order to understand Lie groups, they assume that you to understand SO(3) first. In order to understand SO(3) they assume that you understand Lie groups first. This applies not just in wikipedia, but in at least half a dozen other documents that I’ve looked up.

I haven’t even been able to find out yet whether Lie groups are finite or infinite! Which is a rather important distinction.

> Much of Jacobi’s work was published posthumously in the 1860s, generating enormous interest in France and Germany. Lie’s idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations.

Like that’s useful in understanding it. Not. Going down to the start of the Overview section.

> Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its “infinitesimal group” and which has since become known as its Lie algebra.

Like that’s also useful in understanding it. Not. Lie algebra is a more advanced topic than Lie groups.

differentiable manifold

.

I stuck one of them on the old Ford.

mollwollfumble said:

> In mathematics, a Lie group (pronounced /liː/ “Lee”) is a group that is also a differentiable manifold.

Differentiable manifolds are not an easy topic. I do understand them, just, but being topology takes them totally outside the field of group theory. So I totally don’t see the connection.

What it’s saying is that every element of a Lie group corresponds to a point of a differentiable manifold. Yes, differentiable manifolds have topology, but so do Lie groups. For example, GL(2,R) has two connected components corresponding to the positive and negative values of the determinant. Note that the zero value of the determinant does not belong to the group, and therefore separates the positive and negative values. On the other hand, GL(2,C) has one component because one can continuously travel from a positive real value to a negative real value without crossing zero by going around zero via complex values.

mollwollfumble said:

> Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions given by the special orthogonal group SO(3).

Here’s the problem. In order to understand Lie groups, they assume that you to understand SO(3) first. In order to understand SO(3) they assume that you understand Lie groups first.

SO(3) is an example of a Lie group. It is the group of all rotations in three dimensions. If you are not familiar with this group, why are you concerned with Lie groups in general? And no, you don’t need to understand the concept of Lie groups to have an understanding of SO(3). However, an understanding of the differentiable manifold corresponding to SO(3) will provide a deeper understanding of SO(3).

mollwollfumble said:

I haven’t even been able to find out yet whether Lie groups are finite or infinite! Which is a rather important distinction.

Lie groups are continuously infinite, like the points of a differentiable manifold. However, the standard definition of a Lie group has a finite number of generators, corresponding to a finite-dimensional manifold.

mollwollfumble said:

Lie algebra is a more advanced topic than Lie groups.

A Lie algebra corresponds to the elements of the Lie group in the neighbourhood of the identity element. As infinitesimals, a Lie algebra is actually simpler than the Lie group itself. For example, angular velocities in different directions combine simply as vectors, whereas finite rotations do not.

KJW said:

What it’s saying is that every element of a Lie group corresponds to a point of a differentiable manifold.

My first exposure to Lie groups was from an old textbook on “Ricci Calculus”. IIRC, according to that book, the group operation corresponds to parallel transport on the manifold.

KJW said:

KJW said:

What it’s saying is that every element of a Lie group corresponds to a point of a differentiable manifold.

My first exposure to Lie groups was from an old textbook on “Ricci Calculus”. IIRC, according to that book, the group operation corresponds to parallel transport on the manifold.

Thanks. I’ve come across Ricci calculus in Misner, Thorne and Wheeler. PS. I only uver understood the easy thread in that book. But I don’t think that’s a good introductory book for Lie groups.

mollwollfumble said:

KJW said:

KJW said:

What it’s saying is that every element of a Lie group corresponds to a point of a differentiable manifold.

My first exposure to Lie groups was from an old textbook on “Ricci Calculus”. IIRC, according to that book, the group operation corresponds to parallel transport on the manifold.

Thanks. I’ve come across Ricci calculus in Misner, Thorne and Wheeler. PS. I only uver understood the easy thread in that book. But I don’t think that’s a good introductory book for Lie groups.

The book to which I refer is the most comprehensive treatment of the subject of Ricci Calculus I’ve come across. However, it focuses on maths rather than physics. But, being an old textbook on the subject, it is full of indices.

KJW said:

mollwollfumble said:

> In mathematics, a Lie group (pronounced /liː/ “Lee”) is a group that is also a differentiable manifold.

Differentiable manifolds are not an easy topic. I do understand them, just, but being topology takes them totally outside the field of group theory. So I totally don’t see the connection.

What it’s saying is that every element of a Lie group corresponds to a point of a differentiable manifold. Yes, differentiable manifolds have topology, but so do Lie groups. For example, GL(2,R) has two connected components corresponding to the positive and negative values of the determinant. Note that the zero value of the determinant does not belong to the group, and therefore separates the positive and negative values. On the other hand, GL(2,C) has one component because one can continuously travel from a positive real value to a negative real value without crossing zero by going around zero via complex values.

mollwollfumble said:

> Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions given by the special orthogonal group SO(3).

Here’s the problem. In order to understand Lie groups, they assume that you to understand SO(3) first. In order to understand SO(3) they assume that you understand Lie groups first.

SO(3) is an example of a Lie group. It is the group of all rotations in three dimensions. If you are not familiar with this group, why are you concerned with Lie groups in general? And no, you don’t need to understand the concept of Lie groups to have an understanding of SO(3). However, an understanding of the differentiable manifold corresponding to SO(3) will provide a deeper understanding of SO(3).

mollwollfumble said:

I haven’t even been able to find out yet whether Lie groups are finite or infinite! Which is a rather important distinction.

Lie groups are continuously infinite, like the points of a differentiable manifold. However, the standard definition of a Lie group has a finite number of generators, corresponding to a finite-dimensional manifold.

mollwollfumble said:

Lie algebra is a more advanced topic than Lie groups.

A Lie algebra corresponds to the elements of the Lie group in the neighbourhood of the identity element. As infinitesimals, a Lie algebra is actually simpler than the Lie group itself. For example, angular velocities in different directions combine simply as vectors, whereas finite rotations do not.

Thanks for brilliant answer, KJW.

Let’s start with SO(3). The group of rotations on 3-D.

To generate SO(3), take two vectors v1 and v2. Each rotation in 3-D is a rotation about v1 by angle between 0 and 2*pi followed by a rotation about v2 by angle between 0 and 2*pi, So SO(3) has two dimensions, each a real number.

That much is clear. But I totally fail to see how this ties in with quarks.

> If you are not familiar with this group, why are you concerned with Lie groups in general?

Oh. Because, quite apart from the application in quantum physaics, Lie groups have an important role in pure mathematics when they make it into the theorum enumerating all types of simple groups. I’d like to be able to understand what is and isn’t a Lie group.

KJW said:

mollwollfumble said:

> In mathematics, a Lie group (pronounced /liː/ “Lee”) is a group that is also a differentiable manifold.

Differentiable manifolds are not an easy topic. I do understand them, just, but being topology takes them totally outside the field of group theory. So I totally don’t see the connection.

What it’s saying is that every element of a Lie group corresponds to a point of a differentiable manifold. Yes, differentiable manifolds have topology, but so do Lie groups. For example, GL(2,R) has two connected components corresponding to the positive and negative values of the determinant. Note that the zero value of the determinant does not belong to the group, and therefore separates the positive and negative values. On the other hand, GL(2,C) has one component because one can continuously travel from a positive real value to a negative real value without crossing zero by going around zero via complex values.

Lie groups are continuously infinite, like the points of a differentiable manifold. However, the standard definition of a Lie group has a finite number of generators, corresponding to a finite-dimensional manifold.

mollwollfumble said:

Lie algebra is a more advanced topic than Lie groups.

A Lie algebra corresponds to the elements of the Lie group in the neighbourhood of the identity element. As infinitesimals, a Lie algebra is actually simpler than the Lie group itself. For example, angular velocities in different directions combine simply as vectors, whereas finite rotations do not.

Thanks for brilliant answer, KJW.

Let’s start with SO(3). The group of rotations on 3-D.

To generate SO(3), take two vectors v1 and v2. Each rotation in 3-D is a rotation about v1 by angle between 0 and 2*pi followed by a rotation about v2 by angle between 0 and 2*pi, So SO(3) has two dimensions, each a real number.

That much is clear. But I totally fail to see how this ties in with quarks.

So, SO(3) has only two independent parameters. Taking perpendicular axes for example, a 90 degree rotation around the x axis followed by a 90 degree rotation around the y axis generates a 90 degree rotation around the z axis. Reversing the order of x and y axes generated a 90 degree rotation about the z axis in the opposite direction. This is just the same as the cross product of vectors.

So why do websites such as https://indico.cern.ch/event/243629/attachments/415251/576988/L2.pdf say that SO(3) has three independent parameters when it actually has two? This is really fundamental.

> If you are not familiar with this group, why are you concerned with Lie groups in general?

Oh. Because, quite apart from the application in quantum physaics, Lie groups have an important role in pure mathematics when they make it into the theorum enumerating all types of simple groups. I’d like to be able to understand what is and isn’t a Lie group.