My first cut at this post was a fairly traditional introduction to one parameter subgroups of Lie groups. Along the way, I made this illustration showing the complicated global versus simple local behavior of one parameter subgroups:
But then, in writing down examples, I realized that the readers are likely to be intimately familiar with the multiplication of complex numbers, and so I decided to concentrate on the multiplication of nonzero complex numbers. These form a nice (real) 2-dimensional Lie group with very clear one-parameter subgroups.
So I decided to rewrite the post using simple examples only. You can get the abstract Lie theory from wikipedia, but I had to include the above illustration cause it’s too pretty to put in the bit bucket.
The nonzero complex numbers form a Lie group under multiplication. Let . Then the complex numbers of the form , where n is an integer, are closed under multiplication, include 1, and inverses, and therefore are a subgroup of the nonzero complex numbers. More generally, the set of complex numbers defined by where b is a real number form a subgroup, and we can label the members of the subgroup by the parameter b. This is an example of a 1-parameter subgroup of a Lie group. If r is equal to 1, the subgroup is the unit circle or just the element 1. Otherwise, the subgroup is a spiral or the real line, for example:
A given nonzero point in the complex plain is on an infinite number of different spirals (except for the points on the unit circle). If we take a set of 64 points distributed in a circle around 1, they will generate a set of 32 spiral subgroups. Plotting them shows how the subgroups intertwine with each other:
The behavior of these subgroups is complicated in general, but from the above you can see that their behavior near the (yellow marked) 1, is simple. In the neighborhood of unity, the subgroups look like straight lines. In order for this to happen, the Lie group operator (in this case complex multiplication), when written as an exponential relation as in , must be linear. Replacing with a more general element near unity, we will instead use where is taken to be a small complex number. Then the requirement for the subgroups near unity being approximately straight lines is that to first order in . This is generally true for all these sorts of subgroups of Lie groups. Because of the parameter b, one of these subgroups is called a one parameter subgroup. They are 1-dimensional Lie subgroups.
The one-parameter subgroups of a Lie group are Abelian, that is, the elements of one commute: ab = ba. The purpose of this blog series is to discuss the standard model of particle physics which requires non commutative Lie groups. For these, we will require Lie groups that have a minimum of 3-dimensions, that is, 3-paramater Lie groups. Before looking at that case, it is worthwhile to discuss the intermediate case, that of the 2-parameter Lie groups.
If we have an (n>2)-dimensional Lie group, we can consider a 2-dimensional submanifold neighborhood of 1. Uh, I guess I should illustrate a 2-dimensional manifold here:
Since the Lie group is a manifold, this 2-dimensional neighborhood is automatically a submanifold, but it need not be a subgroup. The submanifold certainly has 1, and we can suppose that it has inverses, but it might not be closed under the group multiplication.
So let us discuss the 2-dimensional submanifolds of a Lie group. First, let’s define a coordinate system that is accurate enough near unity. Our coordinates will be centered on the multiplicative unit, so technically we will have 1 = (0,0). But this notation is too confusing, so instead we will write the point (a,b) as , and then we get . In other words, we will treat the 2-dimensional submanifold as if it were a vector space on its Cartesian coordinates.
It’s important to note that our Cartesian coordinates only work in a neighborhood of 1 and cannot be extended to the whole Lie group. To convince yourself of this, look again at the illustration above showing the 32 intersecting spirals, the one-parameter subgroups of the nonzero complex numbers. The topology of the Cartesian coordinate system breaks down when the one-parameter subgroups wrap back around on themselves.
Using this notation, the fact that multiplication is continuous implies that the multiplication rule near unity is approximately additive in our Cartesian coordinates. That is, ignoring higher order terms like bd, we have that: . Written in (x,y) form, the multiplication rule near the origin reads as (a,b) (c,d) = (a+c,b+d). I’m omitting some unimportant mathematical details here and elsewhere.
For the example of the nonzero complex numbers treated as a 2-dimensional Lie group, there are several natural choices for a Cartesian coordinate system to use near the origin. The coordinates don’t have to be Cartesian to all orders, just Cartesian near unity. For example, we can use . The product of two such objects, (a,b)(c,d) is given exactly by . Keeping only first order in a, b, c, and d, this becomes , and converting it back into Cartesian form, the product is as advertised, linear to first order, (a,b)(c,d) = (a+c,b+d).
Addition is Abelian, so if we want to see behavior in Lie groups that is non Abelian (for example, if we wish to distinguish between the Abelian and non Abelian Lie groups), we have to go to at least 2nd order in our calculations. It turns out that in order to be non Abelian, a Lie group must have at least 3 dimensions.
In writing the Lie group multiplication using a Cartesian vector form, we’ve done a sneaky thing. While the original Lie group defined only a multiplication (the group multiplication), in writing the neighborhood of unity in parameterized Cartesian form, we introduce vectors and since vectors can be added, we have a method of adding some Lie group elements. Unfortunately, as far as a method of defining addition, this has the defect that it depends on the choice of coordinate system. That is, a change to the choice of coordinate system will change addition in second order, so our addition is not defined exactly.
But for elements of the Lie group near the origin our method of defining addition works, and it works better the closer one gets to the origin. This gives us a rough definition of the Lie Algebra, it is the Lie group in an infinitesimal neighborhood of the origin. The mathematical definition of an algebra over the reals is that it is a vector space (the real vector space of Cartesian coordinates which therefore can be added together), that has a multiplication (the group multiplication of the Lie group), along with some unimportant mathematical details.
The 2-dimensional Lie Group: U(1) x U(1)
In our first example of a 2-dimensional Lie group, we got our multiplication from the multiplication of complex numbers. Since the nonzero complex numbers form a group and are a (real) 2-manifold, this gave us an example of a 2-dimensional Lie group. Given two real numbers there are other multiplications we could define. For example, if s and t are arbitrary nonzero real numbers, then a Lie group using matrix multiplication can be defined by:
This is the Lie group U(1) x U(1).
The matrices in U(1) x U(1) are all in diagonal form. If it is possible to write a Lie group in a matrix form which is diagonal, then it is a product of U(1)s. More generally, one might have two Lie groups M and N, one m-dimensional, the other n-dimensional, both of which are written in matrix form. One can define an m+n dimensional Lie group by taking an arbitrary matrix from each Lie group, and combining them as a block diagonal matrix. The matrix product then gives a definition of multiplication for the new Lie group which is m+n dimensional, and is written as M x N.
A natural Cartesian coordinate map that works near 1 is
The reader is invited to use this rule to convert the (matrix) multiplication rule into (a,b) form to obtain (a,b) (c,d) = (a+c+ac,b+d+bd). To first order, this is just addition, as expected.
In the example of nonzero complex numbers, there was a one-parameter subgroup that was a little more special than the others. The unit circle was unique in that it was the only one parameter subgroup that intersected the other subgroups only at unity. It was also special in that it was the only subgroup that was finite in length.
The one-parameter subgroups of U(1) x U(1) intersect each other only at 1. And each of them are infinite in length. Thus the U(1) x U(1) Lie group is different from the nonzero complex number example considered above, which is called . This raises the question of how do we write down an example of the Lie group , which we leave as an exercise for the student.
While the nonzero complex numbers, and U(1) x U(1) are clearly different Lie groups, their Lie algebras are the same, and is usually written u(1) x u(1) in the mathematics literature, and, sloppily, the same as the Lie group, U(1) x U(1), in a lot of the physics literature.