Lecture 1 - Stanford Universitysporadic.stanford.edu › quantum › lecture1.pdfLecture 1 Daniel...

Introduction Bialgebras Hopf algebras Examples of Hopf algebras

Lecture 1

Daniel Bump

May 23, 2019

U V W

W V U

=

W V U

U V W


History

Quantum groups were invented in response to developments inmathematical physics. The initial motivation came from:

Solvable lattice models in statistical mechanicsQuantum integrable systems such as Heisenberg spinchains

Later important applications were found in Knot Theory andmany other important areas. Potential applications exist intopological quantum computing.

My personal interest is in unexpected connections withrepresentation theory of p-adic groups.


Origins in Physics

Solvable Lattice models are statistical mechanical systems,almost always 2 dimensional, that can be completely solved.

Historically the subject begins with Onsager’s 1944 solution ofthe 2-dimensional Ising model. Another example leadingdirectly to the invention of quantum groups is the six-vertexmodel.

Baxter invented a powerful tool for studying solvable latticemodels, the Yang-Baxter equation.

To give a feel for our subject, we will introduce this topic byexplaining the Yang-Baxter equation and hinting at how it isapplicable to Knot theory. Then we will consider how the YBEleads to quantum group territory.


Digression: the Artin Braid Group

The Braid group Bn models isotopy classes of n braids, whichwe will read from top to bottom. Braid Group (Wikipedia Link)

The braid group is generated by braids t1, · · · , tn−1. If n = 3:

t1 t2

Braids are multiplied by concatenating.

t1t2

https://en.wikipedia.org/wiki/Braid_group


The braid relations

The braid relations are:

ti ti+1ti = ti+1ti ti+1, ti tj = tj ti if |i − j | > 1.

t1t2t1 t2t1t2The braids are isotopic, hence we require the braid relation inthe algebraic presentation of the braid group

Bn = 〈t1, · · · , tn−1|braid relations〉 .


The braid group and knot theory

Clearly the braid group captures some aspects of knot theory,but it is only a beginning.

In the braid group, strands can cross but can only move in onedirection. Because of this a knot such as this trefoil knot cannotbe modeled in the braid group. The braid group does capturesome aspect of knot theory and it will be the first thing we seein quantum group theory.


The Yang-Baxter equation (braid version)

Let U,V ,W be vector spaces. We need

R ∈ End(U ⊗ V ), S ∈ End(U ⊗W ), T ∈ End(V ⊗W ).

This version of the YBE is the identity

(T ⊗ IU)(IV ⊗ S)(R ⊗ IW ) = (IW ⊗ U)(S ⊗ IV )(IU ⊗ T ) .

R ⊗ IW

IV ⊗ S

T ⊗ IU

U V W

V U W

V W U

W V U

IU ⊗ T

S ⊗ IV

IW ⊗ U

W V U

W U V

U W V

U V W


Monoidal Categories

A monoidal category is a category with a bifunctor ⊗ that isassociative, that is we have natural isomorphisms

(A⊗ B)⊗ C ∼= A⊗ (B ⊗ C)

such that all identities such as the pentagon identity

((A⊗ B)⊗ C)⊗ D

(A⊗ B)⊗ (C ⊗ D)

A⊗ (B ⊗ (C ⊗ D)

(A⊗ (B ⊗ C))⊗ D A⊗ ((B ⊗ C)⊗ D)

are satisfied. We also require an identity object I with naturalisomorphisms A⊗ I ∼= I ⊗ A ∼= A. Monoidal Category (Wikipedia Link)

https://en.wikipedia.org/wiki/Monoidal_category


Examples of Monoidal Categories

The identity object must satisfy the following axiom.

(A⊗ I)⊗ C A⊗ (I ⊗ C)

A⊗ C

∼=

∼=⊗ICIA⊗∼=

We will encounter many examples of monoidal categories. Forthe moment there are two important ones.

The category of sets. The composition functor is cartesianproduct ×, and the unit object is the set I = {1} with oneelement.The category of vector spaces over a field K . Thecomposition is tensor product, and K itself is the unitobject.


Braided Monoidal Categories (Joyal and Street)

Since this lecture is introductory we will not give the definition ofa braided monoidal category until later. Suffice it to say that it isa monoidal category with a natural morphismcA,B : A⊗ B → B ⊗ A for every pair of objects such satisfyingcertain axioms. These imply that if U,V ,W are objects then theYang-Baxter equation is satisfied:

U V W

W V U

=

W V U

U V W


Quantum Groups (Drinfeld, Jimbo)

Solutions to the Yang-Baxter equation have powerfulapplications. Conversely, such solutions have arisen in theprocess of solving practical problems. Braided monoidalcategories are a rich source of solutions to the Yang-Baxterequation, and most often a braided category is found lurkingbehind such solutions.

Drinfeld formulated the notion of a quasitriangular Hopf algebra(QTHA) in 1986. The main fact is that the category of modulesof a QTHA is braided. Hence:

Quantum groups⇒ Braided categories⇒ applications

where the applications include solvable lattice models, knotinvariants such as the Jones polynomial, etc.


Deforming groups

In Drinfeld’s view, a quantum group is a deformation of a groupG, typically a Lie group. Thus the quantum group depends on aparameter q that can be thought of as e~, where ~ is Planck’sconstant. Thus in the “classical limit” q 7→ 1 we recover thegroup G.

However groups are rigid and can’t be deformed this way. Thesolution is to replace the group by another algebraic objectwhose representation theory is the same as that of G, andwhich lives in a category that does allow such deformations.

This is the category of Hopf algebras.


From Groups to Hopf Algebras

We would like to understand the notion of a group using mapsexclusively. Then we will take the axioms which (in the categoryof sets) describe a group, and transfer them to the category ofvector spaces. This will give the notion of a Hopf algebra.

We begin with the easier notion of a monoid. We willaxiomatize a monoid in a way that uses maps instead ofelements. Then transferring these axioms to the category ofvector spaces will produce the notion of an algebra (if we do itin the simplest way) or a bialgebra (if we do it a better way).

As you know a monoid is a set M with an associativemultiplication map µ : M ×M → M and a unit element 1M suchthat 1M · x = x · 1M = x where a · b is short for µ(a,b).


Slogans

A monoid in the category of vector spaces is an algebraA monoid in the category of vector spaces is a bialgebraA group in the category of vector spaces is a Hopf algebra

The meaning of these is that if we abstract the notion of amonoid by writing the definition in terms of maps rather thanelements, and transport the definition from the category of sets,we obtain a monoid. Monoid (Category Theory) (Wikipedia Link)

But if we repeat the same experiment including more usefulstructure (the diagonal) we obtain a bialgebra. And if we repeatit for groups, we obtain Hopf algebras.

https://en.wikipedia.org/wiki/Monoid_(category_theory)


Categorical notion of a monoid

The category of sets is monoidal with unit element I = {1} (theset with one element) and monoidal operation ×. Let η : I → Mmap 1 7→ 1M and µ : M ×M → M be multiplication. A monoidcan be defined in terms of these.

M ×M ×M M ×M

M ×M M

µ×1

1×µ µ

µ

commutes. Also expressing 1 · x = x · 1 = x :

M ×M

M × I M

µ

∼=

IM×η

M ×M

I ×M M

µ

∼=

η×IM


A basic monoid of vector spaces is an algebra (I)

Like the category of sets the category of vector spaces over thefield K is monoidal category with unit object K and operation ⊗.Let A be a vector space with linear maps µ : A⊗ A→ A andη : K → M. Assume commutative:

A⊗ A⊗ A A⊗ A

A⊗ A A

µ⊗1

1⊗µ µ

µ

commutes. Also expressing 1 · x = x · 1 = x :

A⊗ A

A⊗ I A

µ

∼=

IA⊗η

A⊗ A

I ⊗ A A

µ

∼=

η⊗IA


A basic monoid of vector spaces is an algebra (II)

Define a multiplication on A by a · b = µ(a⊗ b).

A⊗ A⊗ A A⊗ A

A⊗ A A

µ⊗1

1⊗µ µ

µ

means a(bc) = (ab)c, so A is a ring with unit 1A = η(1K ), and ηembeds K in the center of A. So A is a K -algebra.

A⊗ A

A⊗ I A

µ

∼=

IA⊗η

A⊗ A

I ⊗ A A

µ

∼=

η⊗IA


The diagonal

The categorical description of a monoid M can be improvedwith the diagonal map ∆ : M → M ×M. This is needed tosimulate any identity involving more than one copy of anelement. Let ε : M → I be the constant map x → 1.

M M ×M

M ×M M ×M ×M

∆

∆ 1M×∆

∆×1M

M ×M

M × I M

IM×ε∼=

∆

M ×M

I ×M M

ε×IM∼=

∆


Properties of the diagonal (I)

We took the basic properties of a monoid (associative law andunit properties), transferred them to an object in the category ofvector spaces, and obtained an algebra.

But we want to add some properties of the diagonal map.Transferred to the category of vector spaces, these will give thenotion of a bialgebra.

Let τ : M → M be the map τ : (x , y)→ (y , x).

M ×M M ×M ×M ×M M ×M ×M ×M

M M ×M

∆×∆

µ

1M×τ×1M

µ×µ

∆

commutes: both maps (x , y)→ (xy , xy).


Properties of the diagonal (II)

Another couple of properties of the diagonal:

M ×M I × I

M I

ε×ε

µ ∼=

ε

I M

I × I M ×M

η

∼= ∆

η×η


Coalgebras

We may dualize the notion of an algebra by reversing thearrows. That is, instead of a multiplication map µ : M ⊗M → Mwe have a comultiplication ∆ : M → M ⊗M and instead of theunit a counit ε : M → I. The axioms are dual:

M M ⊗M

M ⊗M M ⊗M ⊗M

∆

∆ 1M⊗∆

∆⊗1M

M ⊗M

M ⊗ I M

IM⊗ε∼=

∆

M ⊗M

I ⊗M M

ε⊗IM∼=

∆


Bialgebras (I)

Let us collect all our properties and apply them to a K -vectorspace H. We have maps µ : H ⊗ H → H and ∆ : H → H ⊗ Hcalled multiplication and comultiplication, and maps η : K → H,ε : H → K called unit and counit, subject to the followingaxioms.

Associativity and coassociativity:

H ⊗ H ⊗ H H ⊗ H

H ⊗ H H

µ⊗1

1H⊗µ µ

µ

H H ⊗ H

H ⊗ H H ⊗ H ⊗ H

∆

∆ 1H⊗∆

∆⊗1H


Bialgebras (II)

Unit:H ⊗ H

H ⊗ K H

µ

∼=

1H⊗η

H ⊗ H

K ⊗ H H

µ

∼=

η⊗KH

Counit:H ⊗ H

H ⊗ I H

1H⊗ε∼=

∆

H ⊗ H

I ⊗ H H

ε⊗IH∼=

∆

The associative and unit axioms make H an algebra; thecoassociative and counit axioms make it a coalgebra.


Bialgebras (III)

The augmentation and coaugmentation axioms:

H ⊗ H I × I

H I

ε×ε

µ ∼=

ε

I H

I × I H × H

η

∼= ∆

η×η

These say that the counit is an algebra homomorphism, andthat the unit is a coalgebra homomorphism, respectively.


The Hopf axiom

A bialgebra has one more axiom. We will call this the Hopfaxiom

H ⊗ H H ⊗ H ⊗ H ⊗ H H ⊗ H ⊗ H ⊗ H

H H ⊗ H

∆⊗∆

µ

1H⊗τ⊗1H

µ⊗µ

∆

Here τ : H ⊗ H → H ⊗ H is the flip map τ(x ⊗ y) = y ⊗ x .

With this, the axioms for a bialgebra are complete. It is notablethat the axioms are unchanged if we reverse the direction ofarrows. Most of the axioms are interchanged in pairs by such achanged, but the Hopf axiom is itself unchanged.


Significance of the Hopf axiom

If A and B are algebras, so is A⊗ B with multiplication

(a⊗ b)(a′ ⊗ b′) = (aa′)⊗ (bb′).

If µA, µB and µA⊗B are the multiplications in A, B and A⊗ B thismeans

µA⊗B = (µA ⊗ µB) ◦ (1A ⊗ τ ⊗ 1B).

Hence we write the Hopf axiom

H ⊗ H (H ⊗ H)⊗ (H ⊗ H)

H H ⊗ H

∆⊗∆

µH µH⊗H

∆

In other words, ∆ : H → H ⊗ H is an algebra homomorphism.


The antipode

Remember the slogan: A group in the category of vectorspaces is a Hopf algebra.

Let G be a group, and let S : G→ G be the map S(g) = g−1.We need to express the axiom g · g−1 = g−1 · g = 1 in terms ofmaps. We have at our disposal the diagonal map, themultiplication map, the unit and counit maps, and the solution tothis problem will involve all of them.

G G ×G G ×G

K G

∆

ε

1G×S

µ

η

G G ×G G ×G

K G

∆

ε

S×1G

µ

η

Both directions are the map g 7→ 1G.


Hopf Algebras

So a Hopf algebra is a bialgebra H with a linear map S : H → Hsatisfying the additional axiom:

H H ⊗ H H ⊗ H

K H

∆

ε

1H⊗S

µ

η

H H ⊗ H H ⊗ H

K H

∆

ε

S⊗1H

µ

η

This axiom is self dual, so reversing all arrows in the definitionof a Hopf algebra does not change the definition.


Examples

Since we derived the notion of a Hopf algebra by abstractingthe notion of a group, it is not surprising that Hopf algebras canbe derived from a group by various procedures. There are twoways of obtaining a Hopf algebra from a finite group G.

The group algebra K [G] has comultiplication ∆(g) = g ⊗ g on abasis element g ∈ G, and it is a Hopf algebra.

Dually, the ring O(G) of functions on G is a Hopf algebra. Themultiplication is pointwise (so this algebra is commutative). Thecomultiplication is obtained by identifying

O(G ×G) = O(G)⊗O(G),

so to define ∆(f ) we need to describe a function on G ×G.This is the function ∆(f )(g,h) = f (gh).


Lie groups

If G is an affine algebraic group over C, then the group G(C) ofcomplex points of G is a Lie group. Due to the theory ofChevalley we know that every semisimple Lie group such asSLn is an algebraic group in this way.

There are Two Hopf algebras functorially associated with G. Letg be the Lie algebra of G.

The universal enveloping algebra U(g)

The affine algebra or coordinate ring O(G).

Both admit quantum deformations.


The enveloping algebra (I)

The Universal enveloping algebra U = U(g) of a Lie algebra gis characterized by the following universal property. First, it isan associative ring containing g as a vector subspace, and ifX ,Y ∈ g then

[X ,Y ] = XY − YX

where the bracket operation is in g, and the multiplications onthe right are in U(g).

Universal property If A is any associative algebra and ρ : g→ Ais a linear map such that

ρ([X ,Y ]) = ρ(X )ρ(Y )− ρ(Y )ρ(X )

then ρ extends to an algebra homomorphism U → A.


The enveloping algebra (II)

Elements of g can be identified with left invariant vector fieldson G(C). These are differential operators. Indeed if X ∈ g, if f isa smooth function on G(C) we may define Xf to be the functionobtained by differentiating f along this vector field. So theenveloping algebra may be defined to be the ring of leftinvariant differential operators generated by g.

The enveloping algebra may be also identified with theconvolution ring of distributions on G concentrated at theidentity.

It has a comultiplication, counit and antipode making it a Hopfalgebra. If X ∈ g then

∆(X ) = X ⊗ 1 + 1⊗ X , ε(X ) = 0, S(X ) = −X .


The enveloping algebra of SL2

The Lie algebra of SL2(C) consists of 2× 2 complex matrices oftrace zero. As a vector space, it is 3 dimensional generated by:

X =

(0 10 0

), Y =

(0 01 0

), H =

(1 00 −1

)with bracket operations

[X ,Y ] = H, [H,X ] = 2X , [H,Y ] = −2Y .

So the enveloping algebra is the associative algebra generatedby three elements X ,Y ,H subject to the relations

XY − YX = H, HX − XH = 2X , HY − YH = −2Y .

Caution: The multiplication here is not matrix multiplication, butmultiplication in the enveloping algebra.


Function algebras

The other type of Hopf algebra associated with algebraicgroups is the ring O = O(G(C)) of rational (polynomial)functions. Unlike the enveloping algebra this is a commutativealgebra, and the multiplication is encoded in thecomultiplication.

If V is any affine algebraic variety and O(V ) is its affine algebrawe may identify

O(V × V ) ∼= O(V )⊗O(V )

and so the multiplication morphism G ×G→ G corresponds toan algebra homomorphism ∆ : O → O⊗O. The inverse map isrealized as an antipode and O is a Hopf algebra.


Dual pairing

Let us contrast U = U(G) with O = O(G).

The universal enveloping algebra is cocommutative, i.e.τ ◦∆ = ∆ if τ : U ⊗ U → U ⊗ U is the flip map x ⊗ y → y ⊗ x .The multiplication encodes the multiplication of the group.

The function algebra is commutative, and the comultiplicationencodes the multiplication of the group.

If we identify U(G) as the algebra of distributions on G that areconcentrated at the identity, there is a pairingU(G)×O(G)→ C, namely we may apply a distribution to afunction to obtain a complex number. We often encounter Hopfalgebras in duality this way.


Quantum Groups

Both U(G) and O(G) have deformations that involve aparameter q, and these are the subject of our topic.

The specialization q → 1 should recover the classicalnon-deformed theory, while the specialization q → 0 leads tothe theory of crystal bases.

Even the case G = SL(2) is enough for significant applicationssuch as the Jones polynomial in knot theory, or the six-vertexmodel in statistical mechanics.

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