what's the point of having an inner automorphisms? and different ways of thinking about it

Question

Am in my second year as a graduate student in theoretical physics. In my course on fiber formalism and connection formalism, we talk a little about inner automorphism. The definition is pretty clear, but since am not a purist of a mathematician, I need some tangible or different ways of thinking about this concept.

I can remember in linear algebra when something pretty similar occurs. It's when we want to calculate the eigenvalues of some transformation A, so we change the basis with an element g of the linear group and then apply the transformation A then the inverse of g. By doing so we get a new diagonal transformation with the eigenvalues attached on the diagonal.

My question is: inner automorphism seems very important in mathematics especially in physics, I want to know, besides the title, how mathematicians think about it? and why on principal bundles we define an adjoint representation of the structural group by an inner automorphism?

Sorry if am not rigorous in my explanations, I don't have the chance to talk to mathematicians.

Answer 1

Inner automorphisms are also known as conjugations. Given a group element $g\in G$, the function $gxg^{-1}$ is called conjugation by $g$. It's instructive to have some examples under our belt first:

• One example of this is, as you mention, changing basis in a vector space. If we represent a linear transformation by matrix $X$ with respect to one coordinate system, and by matrix $Y$ with respect to a a second, where the change-of-basis matrix is $B$, then $Y=BXB^{-1}$.
• Another example is permutations. These can be represented in two-line notation or cycle notation, for instance. The effect of $gxg^{-1}$ on $x$'s two-line notation is to apply $g$ to each element of both rows of this notation, or its effect on $x$'s cycle notation is to apply $g$ to each element of $x$'s cycles.
• Consider the symmetry group of a cube. It has 24 rotations and 24 (proper or improper) reflections. If $x$ is a rotation around an axis $\ell$, say, and $g$ is another symmetry, then $gxg^{-1}$ is a rotation (by the same angle as $x$) through the axis $g\ell$ (which we get by applying $g$ to the axis $\ell$). (One may make this statement more precise by interpreting the axes to be oriented, so angles uniquely specify rotations.) Or, if $x$ is a reflection across a plane $\Pi$, then $gxg^{-1}$ is a reflection across the plane $g\Pi$, where again we get $g\Pi$ by applying $g$ to the plane $\Pi$. (A cube also has improper reflection symmetries.)

The takeaway from these examples is this... Often, we think of a group as acting on some mathematical object, and thus on various features of that object, so the description of a group element $x$ can make reference to various features of the object (e.g. vertices, edges, faces of a cube). Then $gxg^{-1}$ has the same description as $x$, but with $g$ applied to all features referenced by $x$'s description! Thus, we can think of conjugation as "changing perspective" in some way.

One consequence is that conjugacy classes can be interpreted as all group elements that "do the same thing," or whose descriptions reference the same kinds of features in the same kind of way. So, for instance, if $x$ is a $120^{\circ}$ rotation around a cube vertex, then its conjugacy class $\{gxg^{-1}\mid x\in G\}$ consists of all $120^{\circ}$ rotations around cube vertices. If $x$ is a permutation with a certain cycle type, then its conjugacy class consists of all permutations with that cycle type. (At least, within the full symmetric group.)

It's interesting to look at an outer automorphism of e.g. $S_6$ and see how it alters cycle types of permutations. Or, look at outer automorphisms of $\mathrm{so}(8)$ (keyword "triality") and how it alters their spectral decompositions. Outer automorphisms may preserve the group's multiplication table, but they fundamentally alter what kind of "thing" a group element is.

The name for a Lie group acting on itself by conjugation (i.e. inner automorphisms) is the adjoint representation. This yields a linear action of the Lie group on its lie algebra, and differentiating again yields a representation of the lie algebra on itself (i.e. into its own algebra of endomorphisms). All of these are called the adjoint representation. Apparently physicists like to put nice mathematical objects into bundles (e.g. what physicists might call tensors mathematicians might call tensor fields) so we can extend these definitions to principal bundles.

Answer 2

One example of this is, as you mention, changing basis in a vector space. If we represent a linear transformation by matrix X with respect to one coordinate system, and by matrix Y with respect to a a second, where the change-of-basis matrix is B, then Y=BXB−1.

In this example, In physics, we search most of the time for diagonalizable matrices so from what I've understood, the point of having inner automorphisms is to search for all the representations of the Lie algebra that can be diagonalizable, i.e. all the Y that can be written $B^{-1} X B$. And the point is to search for irreducible representations by using inner automorphisms which diagonal matrices give to us.

Another example is permutations. These can be represented in two-line notation or cycle notation, for instance. The effect of $gxg^{-1}$ on x's two-line notation is to apply g to each element of both rows of this notation, or its effect on x's cycle notation is to apply g to each element of x's cycles.

On this part, Am thinking about showing that 2 major conjugation classes can be derived; one is all the permutations that are somewhat abelian (we may think of it as trivial) and the other that is not abelian, i.e. all the permutations written in cycle notation, for example, that can commute and the others that cannot commute.