A Quick Introduction to Symmetries, Equivalence, and Lie...

A Quick Introduction to Symmetries,Equivalence, and Lie Groups

Sunita Vatuk

July 5, 2016

Sunita Vatuk A Quick Introduction to Symmetries,Equivalence, and Lie GroupsJuly 5, 2016 1 / 39

What is Symmetry?

If asked for a definition of symmetric, a lay-person is likelyto point to something like this image, with a single mirroror line symmetry:


What is Symmetry?

If shown this image, a lay-person is likely to say that it issymmetric, but when pressed he or she may not be able todefine how. In fact, I have seen school children, secondaryschool students, and even adults look for a mirror –sometimes for quite a long time:


What is Symmetry?

If shown this image, a lay-person is also likely to say thatit is symmetric, but, most are unlikely to quickly find 5non-trivial symmetries (only a mathematician would saythere are 6!).


What is Symmetry?

In Dr. Amri’s talk, you saw symmetries of the origamipolyhedra – which moves us much closer to the generalmathematical definition of symmetry.

An informal definition that I like to start with is

“A symmetry of an object is something you can doto it behind my back without me noticing.”

Which works pretty nicely for the origami polyhedra, ifyou are unable to see colors.


What is Symmetry?

A more formal (but still intuitive) definition of symmetry:

“A symmetry on a set is an invertible function fromthe set to itself that preserves some feature of theset.”

Note all the ingredients in this definition:

1 A set (i.e., the set of vertices of a cube)

2 An invertible function (a permutation of those 8points)

3 A feature of the set that is preserved (the points mayhave been moved, but they are still the vertices of acube in the same position as the original)


What is Symmetry?

Take a minute to think about whether a permutation is asymmetry. (Maybe we can skip this slide because of thelast talk?)

1 What is the set?

2 What is the function?

3 What is preserved?


What is Symmetry?

What are the symmetries of a circle?

1 What is the set?

2 What are the functions?

3 What is preserved?


What is Symmetry?

Let’s step away from geometric objects for a moment, andlook at some algebraic expressions.

Does it make sense to talk about symmetries of theexpression

x2 + y 2?

One of you said “exchange x and y”

Does it make sense to talk about the symmetries of anequation like

x = 4?

If yes, what would those symmetries look like?


What is Symmetry?

And to push things a bit further, does it make sense totalk about the symmetries of an equation like

df (x , y)

dx= g(x , y)?


What is Symmetry?

Now that I’ve primed the pump a bit, why don’t you alltake a few minutes to come up with some symmetries. I’dlike you to try to push the envelope a bit – try out someideas on unfamiliar objects.

If you don’t get it quite right that is a good thing for therest of us, because it gives us a chance to think abouthow to fix it.


What is Equivalence?

Let’s switch gears for a moment, and dig a bit deeper intoanother familiar idea: Equivalence

I think most of you probably know the formal definition ofan Equivalence Relation:

It is a relation that is symmetric, reflexive, and transitive.

But I have found that many of my students don’t have agood intuitive understanding of equivalence, so I want usto explore the idea through some familiar (and unfamiliar)examples.

Informally, equivalence is like equality, but maybe notas strong.



We need a setwe need some way of deciding whether or not twoelements of the set are equivalent

Take a couple of minutes to think of some examples ofequivalence.

If you don’t get it quite right that is a good thing for therest of us, because it gives us a chance to think about theconcept.



Some examples: Modular arithmetic:

for example: 3 ≡ 8 mod 5

1 What is the set?

2 What is the “rule” for deciding whether two elementsare equivalent?



Some statements to get us started:

“A coffee cup is the same as a donut.”

3

4=

9

12

4ABC ∼= 4DEF

4ABC ∼ 4DEF

x3 = −27 is equivalent to x = −3

A→ B is logically equivalent to ¬B → ¬A

D3∼= S3



So let’s go through this list, and ask “what is the set?”and “what is the rule?”

1 equivalence mod 5

2 topological equivalence

3 equivalence of fractions (equals!)

4 congruence of geometric objects

5 similarity of geometric objects

6 equivalence of equations

7 logical equivalence

8 isomorphism of groups (and rings, and fields, andgraphs, and ...)



Now that we have a nice stable of examples, we can ask“why do we care about equivalence?”

A simple example:Are those two circles different? From one point of view –yes, because if you were to write down their equations,they would be different.

12

10

8

6

4

2

–2

–4

–6

5 5 10



Another example, that might upset some of you a bit:In a Calculus class, students differentiate between thesetwo:

24

22

20

18

16

14

12

10

8

6

4

2

– 2

– 4

– 6

– 8

– 10

– 12

15 – 10 – 5 5 10 15

f x( ) = x2

12

10

8

6

4

2

–2

–4

–6

–8

–10

–12

–5 5 10 15

But a geometer may take the point of view that thedifference between these two is a by-product of the chosencoordinates, and so might choose to think of them as thesame (arc length, curvature).



So the lesson here is that sometimes we want to payattention to some difference (in Calculus, while calculatingderivatives) and sometimes we don’t (in differentialgeometry, when we are interested in the properties of theobject: constant curvature 1

3 , for example).

And so being able to rigorously define workable, distinctnotions of “sameness” is very important in mathematics.


What does Equivalence have to do with Symmetry?

We can define equivalence using symmetries, and wedefine symmetries using the idea of equivalence. Let’s goback to our list, and ask “what is the group oftransformations that preserves equivalence?” for a few ofthem.

1 equivalence mod 52 topological equivalence3 equivalence of fractions (equals!)4 congruence of geometric objects5 similarity of geometric objects6 equivalence of equations7 logical equivalence8 isomorphism



I also used the word “invariant” in my title. Looking atthis same list, ask yourself “Is there a function that isconstant for all equivalent elements in the set?” for a fewof them.

1 equivalence mod 52 topological equivalence3 equivalence of fractions (equals!)4 congruence of geometric objects5 similarity of geometric objects6 equivalence of equations7 logical equivalence8 isomorphism



Before we jump into the last topic that I promised to talk about – Liegroups – let’s talk about equivalence classes and what geometerssometimes think about as “normal forms.” Equivalence is onlyinteresting if there are multiple objects that are being considered thesame – i.e., if we are “modding out” by some properties, whilehanging onto others. But then we need to choose representatives foreach subset of equivalent objects. What criteria do we use for those,and why?

1 equivalence mod 5: why choose 0 through 4 instead of 255through 259?

2 equivalence of fractions: why/when do we choose 23

andwhy/when might we choose 8

12?

3 congruence of geometric objects4 similarity of geometric objects5 equivalence of equations


Who Cares?

Maybe you’re thinking “so what?” and/or “does this ideahelp me solve any interesting problems?” I was hopingthat tomorrow’s talk would take care of that, because theproblems that I was thinking about all require more timeand even more machinery to solve.If you took number theory, maybe you saw Diophantus’approach to finding all the Pythagorean triples: Take allrational points that lie in the first quadrant, above theline y = x , and on the unit circle.

3

2.5

2

1.5

1

0.5

– 0.5

– 1

– 1.5

– 2

– 2.5

– 3

– 2 – 1 1 2

f x( ) = x


Who Cares?

If we decide that right triangles are equivalent if they are similar, thenwe can choose a representative for each equivalence class. In otherwords, these special points can be used to generate all thePythagorean triples by scaling. In fact, we can generate all the righttriangles with rational side length in the plane through a set oftransformations. (Which transformations?)

3.5

3

2.5

2

1.5

1

0.5

– 0.5

– 1

– 1.5

– 2

– 2.5

– 2 – 1 1 2

f x( ) = x


Who Cares?

In general, physicists and geometers care about normalforms because the coordinates that one can easilymeasure (such as position and velocity) often do not turnout to be the “right” coordinates for expressing theequations of motion in their simplest form.

Once you have defined the properties of a system thatmatter to the problem you are trying to solve, you can

1 Find the transformations that preserve thoseproperties.

2 Search for the “right” coordinates that make theproblem easier to understand.


Lie Groups

Back to symmetries.

If we take all of the symmetries of some set, do theynecessarily form a group?(Maybe this was answered already, earlier today.)


Lie Groups

A matrix group is a group in which the elements arematrices. Think about what the conditions need to be onthe set of matrices in order to be a group.

Because this talk is informal and about building intuition,let’s stick to the real numbers.


Lie Groups

The elements of a group have inverses, so the matricesneed to be invertible. This means they need to be squarematrices – i.e., they should all be n × n matrices.

The group needs to be closed under multiplication, so nhas to be the same for all of the matrices in the group.

The n× n identity matrix needs to be included. Does thattake care of it? (Matrix multiplication is associative!)

Note: Like permutation groups, matrix groups are usuallynon-commutative since matrix multiplication isnon-commutative. (Can you think of a matrix group thatis commutative?)


Lie Groups

There are finite matrix groups and infinite matrix groups.Just for practice, what matrix group is generated by(

0 −11 0

)?

. Take a minute to do the calculation.

Is this group familiar?


Lie Groups

You can also think of the group of 4 matrices(0 −11 0

),

(−1 00 −1

),

(0 1−1 0

),

(1 00 1

)geometrically or algebraically.

1 The cyclic group of rotations by π2 , π, 3π

2 , and 0.

2 The 4throots of unity {i , -1, −i ,1} – i.e., as 4 pointsin the plane.

3 Z4


Lie Groups

So now we are ready to define Lie groups. Lie groups aregroups on which we can do calculus. The groupmultiplication and the function taking an element to itsinverse are both infinitely differentiable. Most Lie groupsare matrix groups, especially those that are important inscience, so all of the examples we will look at will bematrix groups.


Lie Groups

Example: the symmetries of a circle around the origin.

We have rotations, which are parameterized by realnumbers α – the angle of rotation, counter-clockwise.(

cos(α) − sin(α)sin(α) cos(α)

)And we have reflections over lines through the origin,which are also parameterized by real numbers α. (Theline of reflection makes an angle of α

2 w.r.t. the x-axis.)(cos(α) sin(α)sin(α) − cos(α)

)Sunita Vatuk A Quick Introduction to Symmetries,Equivalence, and Lie GroupsJuly 5, 2016 32 / 39

Lie Groups

You can think about the differentiability in two ways – youcan either think about the matrices or you can think ofthe parameter α as representing the real line wrappedaround the circle.

This is the orthogonal group written O(R2). It is exactlythose transformations that preserve Euclidean distance.Note that we can’t include translations in this group,because the zero vector has length zero, so anytransformation that moves the origin won’t preservedistance.


Lie Groups

Question: If the dot product < u, u > is preserved, doesthat mean distance is preserved?

Question: If distance is preserved, does that mean anglesare preserved?

Question: If distance is preserved, does that mean the dotproduct is preserved?


Lie Groups

Question: If the dot product < u, u > is preserved, does that meanlength/distance is preserved?Yes, because length of a vector is defined as the square root of thedot product of the vector with itself.

Question: If distance is preserved, does that mean angles arepreserved? Yes, because of the Euclidean theorem of congruence oftriangles (SSS).

Question: If distance is preserved, does that mean the dot product ispreserved?Yes, because if d(u, v) = d(A(u),A(v)) > that means< (u − v), (u − v) >=< A(u − v),A(u − v) > and< (u − v), (u − v) >=< u, u > −2 < u, v > + < v , v >and < A(u − v),A(u − v) >=< A(u)− A(v),A(u)− A(v) >=< A(u),A(u) > −2 < A(u),A(v) > + < A(v),A(v) >

Notice that we are using the fact that the transformation A is linear,so A(u − v) = A(u)− A(v) and that the dot product is also linear.


Lie Groups

Having answered those questions, we see that anotherway to think about the O(R2) is as the set oftransformations that preserves the dot product.

Why would we want to reformulate it that way?

Before I can answer that, I need to rewrite this all inmatrix notation:< u, v >= uTvSo for a matrix M corresponding to transformation A,< Mu,Mv >= (Mu)T (Mv) = uTMTMv =uT (MTM)v = uTv .But that must mean that MTM = In,which also means that MT InM = In.


Lie Groups

That turns out to be a lovely way to think about thisgroup of matrices.

Everybody take a moment to calculate MTM = In for thematrix

M =

(a bc d

)

and decide what that means for the vectors(ac

)and

(bd

).


Lie Groups

It means

(ac

)and

(bd

)are of length 1 and are

orthogonal! That explains the name of the group.


Lie Groups

Another reason to put it in this language is to allow forother notions of distance. For example, in specialrelativity, time and space are mixed together intospacetime, and instead of the Euclidean dot product,there is the Lorentzian metric. So the set of matrices thatpreserve Lorenztian distance satisfy another equation:

MT

(1 00 −1

)M =

(1 00 −1

)So instead of the symmetries of a circle, we getsymmetries of a hyperbola.There is a whole class of Lie Groups that preserve otherinner products.


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