MA PH 464 - Group Theory in Physics: Lecture Notes

Consider the set \(\mathbb{R}^*\text{,}\) with binary operation given by multiplication. It is easy to see that it satisfies the axioms of a group. Moreover, \(\mathbb{R}^*\) is a smooth manifold, and the group operations of multiplication and inversion are given by the mappings \(\alpha: (a,b) \mapsto a b\) and \(\beta: a \to \frac{1}{a}\text{,}\) for \(a,b \in \mathbb{R}^*\text{,}\) which are smooth maps. Thus \(\mathbb{R}^*\) is a one-dimensional Lie group.

Consider the one-dimensional circle \(S^1\text{,}\) which can be parameterized by an angle \(\theta \in [0,2 \pi)\text{.}\) This is a group under the operation given by addition of angles. Again, \(S^1\) is a smooth manifold, and the group operations, given by \(\alpha: (\theta_1, \theta_2) \mapsto \theta_1 + \theta_2 \text{ (mod }2 \pi \text{)}\) and \(\beta: \theta_1 \mapsto - \theta_1 \text{ (mod }2 \pi \text{)}\) are smooth maps. Thus \(S^1\) is a one-dimensional Lie group.

Let us now consider the set of matrices of the form

with \(a \in \mathbb{R}_{>0}\) and \(b \in \mathbb{R}\text{.}\) This set forms a group under matrix multiplication. Its underlying manifold is given by \(\mathbb{R}_{>0} \times \mathbb{R}\text{,}\) which is a smooth manifold. Using matrix multiplication, we see that

and hence the group operation on \(\mathbb{R}_{>0} \times \mathbb{R}\) can be written as:

which is smooth. As for inverses, given a matrix

its inverse is

Thus the group inversion on \(\mathbb{R}_{>0} \times \mathbb{R}\) can be written as:

which is also differentiable since \(a \in \mathbb{R}_{>0}\text{.}\) Therefore, this is a two-dimensional Lie group.

Consider the group \(GL(n,\mathbb{R})\) of real \(n \times n\) invertible matrices. Real invertible matrices have \(n^2\) real independent entries. One can show that this is a Lie group of dimension \(n^2\text{.}\) The details of the proof are beyond the scope of this class, but let us briefly sketch the argument. Let \(M_n(\mathbb{R})\) be the set of \(n \times n\) matrices. Then the subset of real invertible matrices \(GL(n,\mathbb{R})\) is obtained by requiring that the determinant is non-zero. But since the determinant is a polynomial map, one can show that the subset \(GL(n,\mathbb{R})\) forms an open set in \(M_n(\mathbb{R})\) (with respect to the Zarisky topology). Using this, one can conclude that \(GL(n,\mathbb{R})\) is a Lie group of dimension \(n^2\text{.}\)

Using a similar line of reasoning, one can show that \(GL(n,\mathbb{C})\) is also a Lie group, with real dimension \(2 n^2\) (since complex invertible matrices have \(n^2\) independent complex entries, that is, \(2 n^2\) independent real parameters).

The group \(SL(n,\mathbb{R})\) of real matrices with determinant one is a closed subgroup of \(GL(n,\mathbb{R})\text{,}\) and hence a Lie group. Imposing that the determinant is equal to one removes one degree of freedom, and hence the dimension of \(SL(n,\mathbb{R})\) is \(n^2-1\text{.}\)

Similarly, the group \(SL(n,\mathbb{C})\) of complex-valued matrices with determinant one is a closed subgroup of \(GL(n,\mathbb{C})\text{,}\) and hence a Lie group. Again, imposing that the determinant is equal to one fixes one constraint, but in this case it reduces the number of complex degrees of freedom by one, and hence the number of real degrees of freedom by two. So the real dimension of \(SL(n,\mathbb{C})\) is \(2 (n^2-1)\text{.}\)

Note that \(SL(n,\mathbb{R})\) and \(SL(n,\mathbb{C})\) are non-compact.

The group \(O(n)\) of real orthogonal matrices is a closed subgroup of \(GL(n,\mathbb{R})\text{,}\) and hence a Lie group. The number of independent real parameters in an \(n \times n\) orthogonal matrix can be calculated to be \(\frac{1}{2} n (n-1)\text{,}\) which is the dimension of the Lie group.

The group \(SO(n)\) of special orthogonal matrices (with determinant one) is also a closed subgroup of \(GL(n,\mathbb{R})\text{,}\) and hence a Lie group. In this case, imposing that the determinant of an orthogonal matrix \(A\) is equal to one is a discrete condition, as any orthogonal matrix must satisfy \(\det A = \pm 1\) to start with. Thus it does not reduce the number of degrees of freedom, and hence the dimension of \(SO(n)\) is also \(\frac{1}{2} n(n-1)\text{.}\)

The orthogonal groups \(O(n)\) and special orthogonal groups \(SO(n)\) are compact Lie groups.

The group \(U(n)\) of unitary matrices is a closed subgroup of \(GL(n,\mathbb{C})\text{,}\) and hence a Lie group. The number of real independent parameters in an \(n \times n\) unitary matrix can be calculated to be \(n^2\text{,}\) which is the dimension of \(U(n)\text{.}\)

The special unitary group \(SU(n)\) is also a closed subgroup of \(GL(n,\mathbb{C})\text{,}\) and hence a Lie group. Imposing the condition that the determinant is equal to one reduces the number of real degrees of freedom by one. Indeed, any unitary matrix \(A\) has \(|\det A|^2 = 1\text{,}\) which means that its determinant can be written as \(\det A = e^{i \theta}\) for some real number \(\theta\text{.}\) Imposing that \(\det A =1 \) fixes this real parameter \(\theta\text{.}\) As a result, the real dimension of \(SU(n)\) is \(n^2 - 1\text{.}\)

Just as for orthogonal groups, both \(U(n)\) and \(SU(n)\) are compact Lie groups.

Let \(x \in \mathbb{R}\text{,}\) and consider the group of rescalings

parameterized by a non-zero real number \(a \in \mathbb{R}^*\text{.}\) These transformations form an abelian group, with the operation being composition. The parameter space is \(\mathbb{R}^*\text{.}\) Composing two transformations with parameters \(a\) and \(b\text{,}\) we get:

and hence the group operation on parameter space is the mapping \(\alpha: \mathbb{R}^* \times \mathbb{R}^* \to \mathbb{R}^*\) given by

The underlying parameter space is thus the group \(\mathbb{R}^*\) with operation given by multiplication, which is a one-dimensional Lie group. It is Example 4.1.2.

As a variation of the previous example, we consider the group of transformations:

with \(a_1 \in \mathbb{R}_{>0}\) and \(b_1 \in \mathbb{R}\text{.}\) These transformations form a non-abelian group. Its parameter space is two-dimensional, and given by \(M=\mathbb{R}_{>0} \times \mathbb{R}\text{.}\) Composing two transformations with parameters \((a_1,b_1)\) and \((a_2, b_2)\text{,}\) we get:

So the group operation corresponds to the mapping \(\alpha: M \times M \to M\) on parameter space given by:

This is nothing else than the group operation on the two-dimensional Lie group constructed in Example 4.1.4.

Let the target space be an \(n\)-dimensional real vector space \(V\text{.}\) The linear invertible transformations \(L: V \to V\) form the general linear group \(GL(V)\text{,}\) which we have already studied. If \(V = \mathbb{R}^n\text{,}\) then we get the general linear group \(GL(n,\mathbb{R})\text{.}\) Note that, while the dimension of the target space is \(n\text{,}\) the dimension of the Lie group (which is the dimension of the parameter space for \(n \times n\) real invertible matrices) is \(n^2\text{.}\)

If we choose the complex vector space \(V = \mathbb{C}^n\) for target space, then we get the group \(GL(n,\mathbb{C})\text{.}\) The real dimension of the target space here is \(2 n\text{,}\) while the dimension of the Lie group is \(2 n^2\text{.}\)

One can restrict to general linear transformations in \(\mathbb{R}^n\) that preserve length and angles. Those are given by orthogonal transformations, and form the Lie group \(O(n)\text{,}\) studied in Example 4.1.9. One can also think of orthogonal transformations as symmetries of a real \(n\)-dimensional sphere, since they preserve its defining equation

Note that while the dimension of the target space is \(n\text{,}\) the dimension of the Lie group \(O(n)\) is \(\frac{1}{2} n (n-1)\text{,}\) as orthogonal \(n \times n\) matrices depend on \(\frac{1}{2}n (n-1)\) real independent parameters.

We can also further restrict to special orthogonal transformations, which form the group \(SO(n)\text{.}\) These consists in all elements of \(O(n)\) that are continuously connected to the identity. Geometrically, it can be understood as the group of rotations in \(\mathbb{R}^n\text{.}\)

We note that the underlying manifold structure of \(SO(n)\) (that is, of its parameter space) is not obvious to see in general. But for \(SO(2)\) it is clear. Rotations in two dimensions are parameterized in terms of a single angle \(\theta \in [0,2 \pi)\text{.}\) Thus the parameter space is the circle \(S^1\text{,}\) with the group operation given by addition of angles. We recover the one-dimensional Lie group of Example 4.1.3! So instead of thinking of \(SO(2)\) as the group of \(2 \times 2\) special orthogonal matrices, or as the group of rotations in \(\mathbb{R}^2\text{,}\) we can also think of it as the circle \(S^1\) with group operation given by addition of angles. This also makes it clear that it is a compact group, since the circle is compact.

For \(SO(3)\text{,}\) things are already more complicated. Rotations in \(\mathbb{R}^3\) are parameterized by three angles. So its parameter space is three-dimensional, which is indeed the dimension of the Lie group. But if you remember how to parameterize rotations in three dimensions with angles, it is not so straightforward. It turns out that the manifold structure of the parameter space here is \(\mathbb{R} \mathbb{P}^3\text{,}\) the real projective three-dimensional space. This example already higlights the fact that finding the manifold structure of the underlying parameter space of a matrix Lie group is not in general straightforward.

The unitary group \(U(n)\text{,}\) studied in Example 4.1.10 can be realized as the group of transformations in \(\mathbb{C}^n\) that preserve the inner product. One could also think \(U(n)\) as the group of symmetries of a “complex \(n\)-dimensional sphere”.

As for the the orthogonal group, the underlying manifold structure of \(U(n)\) and \(SU(n)\) is not so obvious to see. But for \(SU(2)\) we can find the parameter space directly. Any \(2 \times 2\) special unitary matrix \(A\) can be parametrized as:

for \(\alpha, \beta \in \mathbb{C}\) such that \(|\alpha|^2 + |\beta|^2 = 1\text{.}\) Writing \(\alpha = a + i b\) and \(\beta = c + i d\text{,}\) the equation \(|\alpha|^2 + |\beta|^2 = 1\) becomes

which is the equation of a three-sphere \(S^3\) in \(\mathbb{R}^4\text{.}\) Thus, the parameter space (or underlying manifold structure) of \(SU(2)\) is the three-sphere \(S^3\text{,}\) which is a compact manifold.