Rice University

Classical and Quantum Symmetries

Why do we want to find symmetries?

Since, this image is symmetric, it suffices to know just the left half of it.

The symmetries of an object make it easier to analyze it.

Why do we want to find symmetries?

A symmetry of an object $$X$$ is an invertible property-preserving transformation of $$X$$.

The symmetries of an object make it easier to analyze it.

Since, this image is symmetric, it suffices to know just the left half of it.

flipping the picture around the red line is a symmetry of this picture

Why do we want to find symmetries?

The symmetries of an object make it easier to analyze it.

Since, this image is symmetric, it suffices to know just the left half of it.

A symmetry of an object $$X$$ is an invertible property-preserving transformation of $$X$$.

How to quantify symmetries?

A coarse measure would be to look at the size  $$|Sym(X)|$$. But this set has much more structure.

Idea - look at the collection of all symmetries

$$Sym(X) =$$ the set of symmetries of $$X$$

How to quantify symmetries?

a

b

d

c

A coarse measure would be to look at the size  $$|Sym(X)|$$. But this set has much more structure.

Idea - look at the collection of all symmetries

$$Sym(X) =$$ the set of symmetries of $$X$$

Example: Consider the symmetries of the rectangle

A coarse measure would be to look at the size  $$|Sym(X)|$$. But this set has much more structure.

How to quantify symmetries?

Idea - look at the collection of all symmetries

$$Sym(X) =$$ the set of symmetries of $$X$$

Example: Consider the symmetries of the rectangle

• We always have the transformation $$Id$$ which leaves the rectangle unchanged.

a

b

d

c

A coarse measure would be to look at the size  $$|Sym(X)|$$. But this set has much more structure.

How to quantify symmetries?

Idea - look at the collection of all symmetries

$$Sym(X) =$$ the set of symmetries of $$X$$

Example: Consider the symmetries of the rectangle

• We always have the transformation $$Id$$ which leaves the rectangle unchanged.

a

b

d

c

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b

d

c

• We can rotate it by $$180^{\circ}$$ to get

$$R_{180}$$

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$$H$$

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$$H$$

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$$V$$

$$Sym(X) = \{ Id, R_{180}, H, V \}$$

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Thus, $$VH =R_{180}$$.

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c

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$$H$$

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$$V$$

$$Sym(X) = \{ Id, R_{180}, H, V \}$$

$$H$$

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$$V$$

d

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• $$\cdot$$ is associative: $$g_1\cdot (g_2\cdot g_3) = (g_1 \cdot g_2) \cdot g_3$$
• There is an element $$e\in G$$ satisfying $$e\cdot g =g=g\cdot e$$
• For every element $$g \in G$$, there is an element $$g^{-1}$$ such that $$g\cdot g^{-1} = e = g^{-1}\cdot g$$

Groups

A Group is a set $$G$$ with a function $$\cdot:G\times G \rightarrow G$$ such that:

Example 1: $$G = Sym($$        $$) =\{Id, R_{180}, H, V \}$$

Groups

• $$\cdot$$ is associative: $$g_1\cdot (g_2\cdot g_3) = (g_1 \cdot g_2) \cdot g_3$$
• There is an element $$e\in G$$ satisfying $$e\cdot g =g=g\cdot e$$
• For every element $$g \in G$$, there is an element $$g^{-1}$$ such that $$g\cdot g^{-1} = e = g^{-1}\cdot g$$

A Group is a set $$G$$ with a function $$\cdot:G\times G \rightarrow G$$ such that:

Example 2: $$D_6 = Sym ($$     $$) =\{ Id, R_{120}, R_{120}^2, A,B,C \}$$

Example 1: $$G = Sym($$        $$) =\{Id, R_{180}, H, V \}$$

Groups

• $$\cdot$$ is associative: $$g_1\cdot (g_2\cdot g_3) = (g_1 \cdot g_2) \cdot g_3$$
• There is an element $$e\in G$$ satisfying $$e\cdot g =g=g\cdot e$$
• For every element $$g \in G$$, there is an element $$g^{-1}$$ such that $$g\cdot g^{-1} = e = g^{-1}\cdot g$$

A Group is a set $$G$$ with a function $$\cdot:G\times G \rightarrow G$$ such that:

Example 2: $$D_6 = Sym ($$     $$) =\{ Id, R_{120}, R_{120}^2, A,B,C \}$$

where $$R_{120}$$ is the $$120^{\circ}$$ rotation, $$A$$ is the flip         along the dotted line.

Example 1: $$G = Sym($$        $$) =\{Id, R_{180}, H, V \}$$

Groups

• $$\cdot$$ is associative: $$g_1\cdot (g_2\cdot g_3) = (g_1 \cdot g_2) \cdot g_3$$
• There is an element $$e\in G$$ satisfying $$e\cdot g =g=g\cdot e$$
• For every element $$g \in G$$, there is an element $$g^{-1}$$ such that $$g\cdot g^{-1} = e = g^{-1}\cdot g$$

A Group is a set $$G$$ with a function $$\cdot:G\times G \rightarrow G$$ such that:

Symmetries of Vector Spaces

Symmetries of vector spaces generalize the study of symmetries of objects and sets.

Still, why generalize?

Here's an motivation from Machine Learning

Symmetries of Vector Spaces

Symmetries of vector spaces generalize the study of symmetries of objects and sets.

Still, why generalize?

Here's an application from Machine Learning

In a black and white image of 3x3 resolution, we have 9 pixels where each black pixel corresponds to 1 and each white pixel to 0. Such an image is represented by a vector in $$\mathbb{R}^9$$. Thus to understand its symmetries, we need to study symmetries of $$\mathbb{R}^9$$.

Symmetries of Vector Spaces

Symmetries of vector spaces generalize the study of symmetries of objects and sets.

Still, why generalize?

Here's an application from Machine Learning

In a black and white image of 3x3 resolution, we have 9 pixels where each black pixel corresponds to 1 and each white pixel to 0. Such an image is represented by a vector in $$\mathbb{R}^9$$. Thus to understand its symmetries, we need to study symmetries of $$\mathbb{R}^9$$.

\begin{smallmatrix} 1 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{smallmatrix}

$$(1, 0, 1, 1, 0, 1, 0, 1, 0) \in \mathbb{R}^9$$

Symmetries of Vector Spaces

Symmetries of vector spaces generalize the study of symmetries of objects and sets.

Still, why generalize?

Here's an application from Machine Learning

In a black and white image of 3x3 resolution, we have 9 pixels where each black pixel corresponds to 1 and each white pixel to 0. Such an image is represented by a vector in $$\mathbb{R}^9$$. Thus to understand its symmetries, we need to study symmetries of $$\mathbb{R}^9$$.

\begin{smallmatrix} 1 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{smallmatrix}

$$(1, 0, 1, 1, 0, 1, 0, 1, 0) \in \mathbb{R}^9$$

A symmetry of a vector space $$V$$ is an invertible linear transformation $$f: V\rightarrow V$$ of $$V$$. We denote the set of all symmetries as $$GL(V)$$.

The set $$GL(V)$$ is quite large and we are more interested in certain subsets of it.

In the example of the pixelated picture, we are only interested in symmetries of $$\mathbb{R}^9$$ that keep the picture unchanged.

Thus, one can study nice subsets of $$GL(V)$$; but this is difficult.

Better approach: Group actions on vector spaces

An action* of a group $$G$$ on a vector space $$V$$ is defined as a map $$\phi: G \rightarrow GL(V)$$ which satisfies $$\phi(g\cdot g') = \phi(g) \phi(g')$$.

*In CS literature, the terminology '$$V$$ is $$G$$-equivariant' is used more prominently.

Developing Group Equivariant Neural Networks is a hot area these days

Cons of groups

$$X$$

$$X^{def}$$

Deform

$$Sym(X) =\{ Id, R,R^2,R^3,$$

$$A,B,C,D \}$$

$$Sym(X^{def})= \{ Id\}$$

This is inconvenient when generalizing to the quantum setting.

When trying to formalize Quantum Physics, we encounter quantum versions of vector spaces which are deformations of usual vector spaces.

$$V$$

$$V_q$$

Deform

V has symmetries given by some group $$G$$

Want some deformed group $$G_q$$ which gives us symmetries of $$V_q$$

It turns out that Hopf algebras provide the right setting for addressing this problem.

Because of the rigid nature of groups, in most examples, we are not able to find any such group $$G_q$$. Thus, we have to broaden our framework.

When trying to formalize Quantum Physics, we encounter quantum versions of vector spaces which are deformations of usual vector spaces.

Since, Hopf algebra generalize groups to address problems from Quantum Physics, they are also referred to as quantum groups.

When a vector space admits actions by Hopf algebras which are not coming from groups, we say they admit quantum symmetries.

Groups

Hopf algebras

• $$G$$-action on $$V$$
• $$V$$ deforms to $$V_q$$
• Do not get any non-trivial groups action on $$V_q$$
• $$H$$-action on $$V$$
• $$V$$ deforms to $$V_q$$
• Have nice deformations $$H_q$$ of $$H$$ of which act on $$V_q$$

algebras

$$\mathbb{R}[x] =$$ the set of polynomials in the variable $$x$$

• Naturally a vector space over $$\mathbb{R}$$
• Can multiply two polynomials to get a new one
• Have the polynomial $$f(x)=1$$ which satisfies $$fg=g=gf$$ for any other polynomial $$g$$

Combining elements

$$Mat_n(\mathbb{R}) =$$ the set of $$n\times n$$ matrices

• Vector space over $$\mathbb{R}$$
• Matrix multiplication
• The $$n \times n$$ identity matrix $$I_n$$ satisfies $$I_n M = M = M I_n$$

$$x+1$$

$$x$$

$$x^2+x$$

\small \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix}
\small \begin{smallmatrix} 3 & 1 \\ 1 & 0 \end{smallmatrix}
\small \begin{smallmatrix} 5 & 1 \\ 1 & 0 \end{smallmatrix}

Algebras

An algebra over $$\mathbb{R}$$ is a tuple $$(A,m,u)$$ where:​

• $$A$$ is an $$\mathbb{R}$$-vector space
• $$m=$$      $$: A\otimes A \rightarrow A$$ is a linear map, and
• $$u=$$    $$:\mathbb{R} \rightarrow A$$ is a linear map

Algebras

An algebra over $$\mathbb{R}$$ is a tuple $$(A,m,u)$$ where:​

associativity

unitality

$$f$$

$$g$$

$$h$$

$$f$$

$$g$$

$$h$$

$$gh$$

$$f(gh)$$

$$(fg)h$$

$$fg$$

$$f$$

$$f$$

$$f$$

$$f$$

$$fe$$

$$ef$$

$$1$$

$$1$$

$$e$$

$$e$$

• $$A$$ is an $$\mathbb{R}$$-vector space
• $$m=$$      $$: A\otimes A \rightarrow A$$ is a linear map, and
• $$u=$$    $$:\mathbb{R} \rightarrow A$$ is a linear map

satisfying:

Given a group $$G$$, the group algebra $$\mathbb{R}G$$ is a:

• $$\mathbb{R}$$ vector space with basis $$x_g$$ for all $$g \in G$$,
• with multiplication $$x_g x_{g'} =x_{g\cdot g'}$$, and
• unit $$x_{e}$$ where $$e$$ is the identity element of $$G$$.

By working with groups algebras, we do not lose anything. In fact,

$$G$$-actions on vector spaces

$$f: G \rightarrow GL(V)$$

$$\mathbb{R}G$$-actions on vector spaces

$$F: \mathbb{R}G \rightarrow \text{End}(V)$$

$$x_g$$

$$x_{g'}$$

$$x_{g\cdot g'}$$

$$1$$

$$x_e$$

End$$(V)=$$ the set of all linear maps $$V \rightarrow V$$

Breaking up elements into two

$$\mathbb{R}[x] =$$ the set of polynomials in the variable $$x$$

• Break up $$x^n$$ as $$\Delta(x^n) = \sum_{k=0}^n {n\choose k} x^k \otimes x^{n-k}$$, extend.

$$\mathbb{R}G =$$ the group algebra of a group $$G$$

Most natural example come from different kinds of graphs.

• Naturally a $$\mathbb{R}$$-vector space
• Break up $$x_g$$ as $$\Delta(x_g) = x_g\otimes x_g$$, then extend to $$\mathbb{R}[G]$$.

$$x_g$$

$$x_g$$

$$x_g$$

$$x^n$$

$$x^{k}$$

$$x^{n-k}$$

$$\sum_k$$

Coalgebras

An coalgebra over $$\mathbb{R}$$ is a tuple $$(C,\Delta,\varepsilon)$$ where:

• $$C$$ is an $$\mathbb{R}$$-vector space
• $$\Delta=$$      $$: A \rightarrow A \otimes A$$ is a linear map, and
• $$\varepsilon =$$     $$: C \rightarrow \mathbb{R}$$ is a linear map

such that $$\Delta$$ and $$\varepsilon$$ satisfy

coassociativity

counitality

Vector Spaces

Vector Spaces

Algebras

Vector Spaces

Algebras

Coalgebras

Vector Spaces

Algebras

Coalgebras

Groups

Vector Spaces

Algebras

Coalgebras

Groups

Group algebras

Vector Spaces

Algebras

Coalgebras

Groups

Group algebras

$$\mathbb{R}[x]$$

Group algebras have the structure of both an algebra and a coalgebra.

$$x_g$$

$$x_{g'}$$

$$x_{g\cdot g'}$$

$$1$$

$$x_e$$

$$x_g$$

$$x_g$$

$$x_g$$

$$x_g$$

$$\delta_{x,e}$$

However, a group had an inverse for each element and we have not used that information. Define the map:

$$S=$$     $$:\mathbb{R}G \rightarrow \mathbb{R}G$$

$$x_g \mapsto x_{g^{-1}}$$

Then we set up the definition of a Hopf algebras so that group algebras are an example.

Hopf algebras

A Hopf algebra is a tuple $$(H,m=$$        $$,u=$$     $$,\Delta=$$        $$, \varepsilon=$$    $$,$$ $$S=$$    $$:H \rightarrow H)$$ such that:

• $$(H,m,u)$$ is an algebra,
• $$(H,\Delta,\varepsilon)$$ is a coalgebra,
• Algebra and coalgebra structures are compatible
• $$S:H \rightarrow H$$ is a linear map which satisfies

$$Id_{\mathbb{R}}$$

Algebras

Coalgebras

Vector Spaces

Hopf

Algebras

Group

Algebras

Groups

Symmetries of

vector spaces

Group actions on

vector spaces

Symmetries of objects

Actions of group algebras on vector spaces

do not behave well under deformations

of the algebra

Actions of group algebras on algebras

broaden framework to consider Hopf action

Actions of Hopf

algebras on

algebras

pixelated

image example

need only certain

symmetries

equivalent

problem

in all interesting

examples, underlying vector space has algebra structure

Symmetries of

vector spaces

Group actions on

vector spaces

Symmetries of objects

Actions of group algebras on vector spaces

do not behave well under deformations

of the algebra

Actions of group algebras on algebras

broaden framework to consider Hopf action

Actions of Hopf

algebras on

algebras

pixelated

image example

need only certain

symmetries

equivalent

problem

in all interesting

examples, underlying vector space has algebra structure

H=\mathbb{R}G \curvearrowright A
G \curvearrowright A
G^{def} \curvearrowright A^{def}
H^{def} \curvearrowright A^{def}

UPSHOT:

By going to the world of Hopf algebras, we are able to find to find (quantum) symmetries of deformations of algebras.

Symmetries of

vector spaces

Group actions on

vector spaces

Symmetries of objects

Actions of group algebras on vector spaces

do not behave well under deformations

of the algebra

Actions of group algebras on algebras

broaden framework to consider Hopf action

Actions of Hopf

algebras on

algebras

pixelated

image example

need only certain

symmetries

equivalent

problem

in all interesting

examples, underlying vector space has algebra structure

H=\mathbb{R}G \curvearrowright A
G \curvearrowright A
G^{def} \curvearrowright A^{def}
H^{def} \curvearrowright A^{def}

Invariants of spaces

Hopf algebras provide us ways to distinguish 3-dimensional spaces.

The problem of distinguishing shapes comes under the field of topology. In topology, we consider two shapes to be the same if we can smoothly deform one shape into the another.

These two shapes are

considered the same

These are considered

different

No holes/punctures

One hole

No holes/punctures

For $$3$$-dimensional shapes, we can no longer eye-ball the number of holes to distinguish shapes.

Distinguishing shapes in higher dimensions is very difficult problem and many mathematicians have won the Fields medal for work in this area.

Jones polynomial

Input: $$K=$$

Output: $$V_K(t) =t+t^3-t^4$$

If two knots $$K,K'$$ have different Jones polynomials $$V_K(t), V_{K'}(t)$$, then the knots are different from each other.

In fact, the Jones polynomial is a special case of a powerful algorithm that distinguishes shapes that uses Hopf algebras as input.

Thank you for your attention!

Email : hy39@rice.edu

Slides: tinyurl.com/yg577z4h

Feel free to contact with question/comments.