#automorphism

Niels Smits Van Burgst, Automorphism (U-Tube), 2013, Oil on Canvas, 140 x 480 cm

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Niels Smits van Burgst

Niels Smits van Burgst nasceu na Holanda em 1970 e formou-se na Real Academia de Haia em 1993. Desde então, suas pinturas foram exibidas em Nova York, Berlim, Amsterdã, Bruxelas e muitas outras cidades da Europa. Em 2013, o Museu 'de Buitenplaats' homenageou seu trabalho com uma retrospectiva. Ele ganhou o prêmio 'van Ommeren de Voogd' em 2007 e o prêmio Aku em 2011. Suas pinturas fazem parte de diversas coleções ao redor do mundo.

The Morita theory of quantum graph isomorphisms

The automorphisms of a structure are different functions that give you the same structure in the object (but maybe reärranged).

Let's look at the set of all automorphisms of a structure.

First, it will always contain at least one element, the identity map, which outputs exactly what it gets in.

Second, you might look for structure within the set, because there are many different ways of having symmetry.

It turns out, this structure is a group, with composition as the combiner, and the identity automorphism as the group identity

Theorem: The set of automorphisms form a group (Automorphisms,∘,id)

Proof: This is very similar to the proof that isomorphisms make an equivalence relation

1) Closure Assume f and g are two automorphisms We need to show g∘f is an automorphism

If x and y are elements of X, and they are related iff f(x) and f(y) are related in the same way, from f an automorphism, and f(x) and f(y) related iff g(f(x)) and g(f(y)) related in the same way

So, structure is preserved.

It is X->X, because f is X->X, and g is X->X.

It is a bijection because it is a composition of two bijections

These combine to mean it is an automorphism, so in the set.

2) The identity automorphism is the group identity. Assume f is an automorphism. For all x, elements of X f∘id(x) = f(id(x)) = f(x) id∘f(x) = id(f(x)) = f(x)

These two are the exact properties we want from an identity.

3) Associativity This flows from the associativity of function composition

4) Inverses Let's say f is an automorphism. It is a bijection X->X, so it has an inverse f-1 which is a bijection X->X

f-1 preserves structure since x and y are related in the same way f(x) and f(y) are related, and so if a and b are in X, then so are c=f-1(a) and d=f-1(b), and a=f(c) b=f(d), so a and b are related in the same way that c=f-1(a) and d=f-1(b) are related

For all x, f∘f-1(x) = f-1∘f(x) = x, so it is the group inverse.

We denote the automorphism group of model X by Aut(X)

This not only means that studying how an object X is symmetric is the same as studying Aut(X), but also that developing ideas of groups allows us to understand symmetry.

Most people have an intuitive notion of symmetry.

The mathematical objects for that intuition are shapes.

Specifically, when you flip or rotate and "get the same shape"

For example, the letter <P> has no flips or rotations.

An <M> has one flip that keeps the shape the same.

An <H> has a flip over 2 different axes and a rotation

A circle is even more "symmetric," because it can be rotated any amount, or it can be flipped around any line through its center.

The easy way to formalize this is to realize that "shape" is a type of structure, and we already have isomorphisms to see if structure is the same.

So, the solution to extending symmetry needs a special kind of isomorphism called an automorphism, which is an isomorphism f:X->X, from an object to itself.

That is, we can describe the symmetries an object has as {f : f(X)≅X}, or "the set of functions that preserve the structure." (that squiggle means “isomorphic to,” and the colon means “such that”). Notice this set always contains the identity map.

We can now count "how symmetric" an object is, by the size of that set.

We can also apply this definition to all sorts of other objects.

We can take a graph, which cares which points are attached, but not the path of the attachments.

This is the Petersen graph, which is highly symmetric.

We can look at the automorphisms of groups, like these 6 configurations of the Klein 4 group which come from just switching elements.

going the long way

What does it mean when mathematicians talk about a bijection or homomorphism?

Imagine you want to get from X to X′ but you don't know how. Then you find a "different way of looking at the same thing" using ƒ. (Map the stuff with ƒ to another space Y, then do something else over in image ƒ, then take a journey over there, and then return back with ƒ ⁻¹.)

The fact that a bijection can show you something in a new way that suddenly makes the answer to the question so obvious, is the basis of the jokes on www.theproofistrivial.com.

In a given category the homomorphisms Hom ∋ ƒ preserve all the interesting properties. Linear maps, for example (except when det=0) barely change anything--like if your government suddenly added another zero to the end of all currency denominations, just a rescaling--so they preserve most interesting properties and therefore any linear mapping to another domain could be inverted back so anything you discover over in the new domain (image of ƒ) can be used on the original problem.

All of these fancy-sounding maps are linear:

Fourier transform

Laplace transform

taking the derivative

Box-Müller

They sound fancy because whilst they leave things technically equivalent in an objective sense, the result looks very different to people. So then we get to use intuition or insight that only works in say the spectral domain, and still technically be working on the same original problem.

Pipe the problem somewhere else, look at it from another angle, solve it there, unpipe your answer back to the original viewpoint/space.

  For example: the Gaussian (normal) cumulative distribution function is monotone, hence injective (one-to-one), hence invertible.

By contrast the Gaussian probability distribution function (the "default" way of looking at a "normal Bell Curve") fails the horizontal line test, hence is many-to-one, hence cannot be totally inverted.

So in this case, integrating once ∫[pdf] = cdf made the function "mathematically nicer" without changing its interesting qualities or altering its inherent nature.

  Or here's an example from calc 101: u-substitution. You're essentially saying "Instead of solving this integral, how about if I solve a different one which is exactly equivalent?" The →ƒ in the top diagram is the u-substitution itself. The "main verb" is doing the integral. U-substituters avoid doing the hard integral, go the long way, and end up doing something much easier.

  Or in physics--like tensors and Schrödinger solving and stuff. Physicists look for substitutions that make the computation they have to do more tractable. Try solving a Schrödinger PDE for hydrogen's first electron s¹in xyz coordinates (square grid)--then try solving it in spherical coordinates (longitude & latitude on expanding shells). Since the natural symmetry of the s¹ orbital is spherical, changing basis to polar coords makes life much easier.

  Likewise one of the goals of tensor analysis is to not be tied to any particular basis--so long as the basis doesn't trip over itself, you should be free to switch between bases to get different jobs done. Terry Tao talks about something like this under the keyword "spending symmetry"--if you use up your basis isomorphism, you need to give it back before you can use it again.