III Algebras - Coalgebras, bialgebras and Hopf algebras

4Coalgebras, bialgebras and Hopf algebras

III Algebras

4 Coalgebras, bialgebras and Hopf algebras

We are almost at the end of the course. So let’s define an algebra.

Definition (Algebra). A k-algebra is a k-vector space A and k-linear maps

µ : A ⊗ A → A u : k → A

x ⊗ y 7→ xy λ 7→ λI

called the multiplication/product and unit such that the following two diagrams

commute:

A ⊗ A ⊗ A A ⊗ A

A ⊗ A A

µ⊗id

id ⊗µ

k ⊗ A A ⊗ A A ⊗ k

∼

u⊗id

id ⊗u

∼

These encode associativity and identity respectively.

Of course, the point wasn’t to actually define an algebra. The point is to

define a coalgebra, whose definition is entirely dual.

Definition (Coalgebra). A coalgebra is a k-vector space C and k-linear maps

∆ : C → C ⊗ C ε : C → k

called comultiplication/coproduct and counit respectively, such that the following

diagrams commute:

C ⊗ C ⊗ C C ⊗ C

C ⊗ C C

id ⊗∆

∆⊗id

∆

k ⊗ C C ⊗ C C ⊗ k

ε⊗id id ⊗ε

∼

These encode coassociativity and coidentity

A morphism of coalgebras

C → D

is a

-linear map such that the following

diagrams commute:

C D

C ⊗ C D ⊗ D

∆

f⊗f

C D

k k

A subspace

is a co-ideal if ∆(

)

≤ C ⊗ I

I ⊗ C

, and

(

) = 0. In this

case, C/I inherits a coproduct and counit.

A cocommutative coalgebra is one for which

τ ◦

∆ = ∆, where

V ⊗ W →

W ⊗ V given by the v ⊗ w 7→ w ⊗ v is the “twist map”.

It might be slightly difficult to get one’s head around what a coalgebra actually

is. It, of course, helps to look at some examples, and we will shortly do so. It

also helps to know that for our purposes, we don’t really care about coalgebras

per se, but things that are both algebras and coalgebras, in a compatible way.

There is a very natural reason to be interested in such things. Recall that

when doing representation theory of groups, we can take the tensor product of

two representations and get a new representation. Similarly, we can take the

dual of a representation and get a new representation.

If we try to do this for representations (ie. modules) of general algebras, we

see that this is not possible. What is missing is that in fact, the algebras

and

(

) also have the structure of coalgebras. In fact, they are Hopf algebras,

which we will define soon.

We shall now write down some coalgebra structures on kG and U(g).

Example. If G is a group, then kG is a co-algebra, with

∆(g) = g ⊗ g



(g)



We should think of the specification ∆(

) =

g ⊗ g

as saying that our groups act

diagonally on the tensor products of representations. More precisely, if

V, W

are

representations and v ∈ V, w ∈ W , then g acts on v ⊗ w by

∆(g) · (v ⊗ w) = (g ⊗ g) · (v ⊗ w) = (gv) ⊗ (gw).

Example.

For a Lie algebra

over

, the universal enveloping algebra

(

) is

a co-algebra with

∆(x) = x ⊗ 1 + 1 ⊗ x

for x ∈ g, and we extend this by making it an algebra homomorphism.

To define ε, we note that elements of U(g) are uniquely of the form

λ +

,...,i

· · · x

where {x

} is a basis of g (the PBW theorem). Then we define



λ +

,...,i

· · · x



= λ.

This time, the specification of ∆ is telling us that if

X ∈ g

and

v, w

are elements

of a representation of g, then X acts on the tensor product by

∆(X) · (v ⊗ w) = Xv ⊗ w + v ⊗ Xw.

Example. Consider

O(M

(k)) = k[X

: 1 ≤ i, j ≤ n],

the polynomial functions on

n × n

matrices, where

denotes the

th entry.

Then we define

∆(X

) =

i=1

⊗ X

and

ε(X

) = δ

These are again algebra maps.

We can also talk about

(

)) and

(

)). The formula of the

determinant gives an element

D ∈ O

(

)). Then

(

)) is given by

adding a formal inverse to

(

)), and

(

)) is obtained by

quotienting out O(GL

(k)) by the bi-ideal hD − 1i.

From an algebraic geometry point of view, these are the coordinate algebra

of the varieties M

(k), GL

(k) and SL

(k).

This is dual to matrix multiplication.

We have seen that we like things that are both algebras and coalgebras,

compatibly. These are known as bialgebras.

Definition

(Bialgebra)

A bialgebra is a

-vector space

and maps

µ, υ,

∆

, ε

such that

(i) (B, µ, u) is an algebra.

(ii) (B, ∆, ε) is a coalgebra.

(iii) ∆ and ε are algebra morphisms.

(iv) µ and u are coalgebra morphisms.

Being a bialgebra means we can take tensor products of modules and still

get modules. If we want to take duals as well, then it turns out the right notion

is that of a Hopf algebra:

Definition

(Hopf algebra)

A bialgebra (

H, µ, u,

∆

, ε

) is a Hopf algebra if there

is an antipode S : H → H that is a k-linear map such that

µ ◦ (S ⊗ id) ◦ ∆ = µ ◦ (id ⊗S) ◦ ∆ = u ◦ ε.

Example. kG is a Hopf algebra with S(g) = g

−1

Example. U(g) is a Hopf algebra with S(x) = −x for x ∈ U(g).

Note that our examples are all commutative or co-commutative. The term

quantum groups usually refers to a non-commutative non-co-commutative Hopf

algebras. These are neither quantum nor groups.

As usual, we write

∗

for

Hom

(

V, k

), and we note that if we have

V → W

then this induces a dual map α

∗

: W

∗

→ V

∗

Lemma.

is a coalgebra, then

∗

is an algebra with multiplication ∆

∗

(that

is, ∆

∗

⊗C

∗

) and unit ε

∗

. If C is co-commutative, then C

∗

is commutative.

However, if an algebra

is infinite dimensional as a

-vector space, then

∗

may not be a coalgebra. The problem is that (

∗

⊗ A

∗

) is a proper subspace of

(

A ⊗ A

)

∗

, and

∗

of an infinite dimensional

need not take values in

∗

⊗ A

∗

However, all is fine for finite dimensional

, or if

is graded with finite

dimensional components, where we can form a graded dual.

In general, for a Hopf algebra H, one can define the Hopf dual ,

= {f ∈ H

∗

: ker f contains an ideal of finite codimension}.

Example.

Let

be a finite group. Then (

)

∗

is a commutative non-co-

commutative Hopf algebra if G is non-abelian.

Let

{g}

be the canonical basis for

, and

{φ

}

be the dual basis of (

)

∗

Then

∆(φ

) =

⊗ φ

There is an easy way of producing non-commutative non-co-commutative Hopf

algebras — we take a non-commutative Hopf algebra and a non-co-commutative

Hopf algebra, and take the tensor product of them, but this is silly.

The easiest non-trivial example of a non-commutative non-co-commutative

Hopf algebra is the Drinfeld double, or quantum double, which is a general

construction from a finite dimensional hopf algebra.

Definition (Drinfeld double). Let G be a finite group. We define

D(G) = (kG)

∗

⊗

as a vector space, and the algebra structure is given by the crossed product

(kG)

∗

o G, where G acts on (kG)

∗

(x) = f(gxg

−1

Then the product is given by

⊗ g

)(f

⊗ g

) = f

−1

⊗ g

The coalgebra structure is the tensor of the two coalgebras (

)

∗

and

, with

∆(φ

⊗ h) =

⊗ h ⊗ φ

⊗ h.

(

) is quasitriangular, i.e. there is an invertible element

(

)

⊗ D

(

)

such that

R∆(x)R

−1

= τ(∆(x)),

where τ is the twist map. This is given by

R =

(φ

⊗ 1) ⊗ (1 ⊗ g)

(φ

⊗ 1) ⊗ (1 ⊗ g

−1

The equation

∆

−1

∆ results in an isomorphism between

U ⊗V

and

V ⊗U

for D(G)-bimodules U and V , given by flip follows by the action of R.

is non-abelian, then this is non-commutative and non-co-commutative.

The point of defining this is that the representations of

(

) correspond to the

G-equivariant k-vector bundles on G.

As we said, this is a general construction.

Theorem

(Mastnak, Witherspoon (2008))

The bialgebra cohomology

(

H, H

) for a finite-dimensional Hopf algebra is equal to

(

)

, k

where k is the trivial module, and D(H) is the Drinfeld double.

In 1990, Gerstenhaber and Schack defined bialgebra cohomology, and proved

results about deformations of bialgebras analogous to our results from the

previous chapter for algebras. In particular, one can consider infinitesimal

deformations, and up to equivalence, these correspond to elements of the 2nd

cohomology group.

There is also the question as to whether an infinitesimal deformation is

integrable to give a bialgebra structure on

V ⊗ k

[[

]], where

is the underlying

vector space of the bialgebra.

Theorem

(Gerstenhaber–Schack)

Every deformation is equivalent to one where

the unit and counit are unchnaged. Also, deformation preserves the existence of

an antipode, though it might change.

Theorem

(Gerstenhaber–Schack)

All deformations of

(

)) or

(

))

are equivalent to one in which the comultiplication is unchanged.

We nwo try to deform

(

)). By the previous theorems, we only have

to change the multiplication. Consider O

(k)) defined by

= qX

= X

− X

= (q

−1

− q)X

We define the quantum determinant

det

= X

− q

−1

= X

− qX

Then

∆(det

) = det

⊗ det

, ε(det

) = 1.

Then we define

O(SL

(k)) =

O(M

(k))

(det

−1)

where we are quotienting by the 2-sided ideal. It is possible to define an antipode,

given by

S(X

) = X

S(X

) = −qX

S(X

) = −q

−1

S(X

) = X

and this gives a non-commutative and non-co-commutative Hopf algebra. This

is an example that we pulled out of a hat. But there is a general construction

due to Faddeev–Reshetikhin–Takhtajan (1988) via

-matrices, which are a way

of producing a k-linear map

V ⊗ V → V ⊗ V,

where V is a fintie-dimesnional vector space.

We take a basis

, · · · , e

, and thus a basis

⊗ e

V ⊗ V

. We write

for the matrix of R, defined by

R(e

⊗ e

) =

`,m

⊗ e

The rows are indexed by pairs (

`, m

), and the columns by pairs (

i, j

), which are

put in lexicographic order.

The action of

V ⊗ V

induces 3 different actions on

V ⊗ V ⊗ V

. For

s, t ∈ {

}

, we let

be the invertible map

V ⊗ V ⊗ V → V ⊗ V ⊗ V

which

acts like

on the

th and

th components, and identity on the other. So for

example,

⊗ e

⊗ v) =

i,j

⊗ e

⊗ v.

Definition

(Yang–Baxter equation)

. R

satisfies the quantum Yang–Baxter

equation (QYBE) if

= R

and the braided form of QYBE (braid equation) if

= R

Note that

satisfies QYBE iff

Rτ

satisfies the braid equation. Solutions to

either case are R-matrices.

Example. The identity map and the twist map τ satisfies both.

Take V to be 2-dimensional, and R to be the map







q 0 0 0

0 1 0 0

0 q − q

−1

1 0

0 0 0 q







where q 6= 0 ∈ K. Thus, we have

R(e

⊗ e

) = qe

⊗ e

R(e

⊗ e

) = e

⊗ e

R(e

⊗ e

) = e

⊗ e

+ (q − q

−1

⊗ e

R(e

⊗ e

) = qe

⊗ e

and this satisfies QYBE. Similarly,

(Rτ)







q 0 0 0

0 0 1 0

0 1 q − q

−1

0 0 0 q







satisfies the braid equation.

We now define the general construction.

Definition (R-symmetric algebra). Given the tensor algebra

T (V ) =

∞

n=0

⊗n

we form the R-symmetric algebra

(V ) =

T (V )

hz − R(z) : z ∈ V ⊗ V i

Example. If R is the identity, then S

(V ) = T (V ).

Example. If R = τ, then S

(V ) is the usual symmetric algebra.

Example. The quantum plane O

) can be written as S

(V ) with

R(e

⊗ e

) = qe

⊗ e

R(e

⊗ e

) = e

⊗ e

R(e

⊗ e

) = q

−1

⊗ e

R(e

⊗ e

) = e

⊗ e

Generally, given a

which is finite-dimensional as a vector space, we can

identify (V ⊗ V )

∗

with V

∗

⊗ V

∗

We set

V ⊗ V

∗

∼

End

(

)

∼

(

). We define

and

∗

E ⊗ E →

E ⊗ E

, where

acts like

on terms 1 and 3 in

V ⊗ V

∗

⊗ V ⊗ V

∗

, and

identity on the rest; R

∗

acts like R

∗

on terms 2 and 4.

Definition

(Coordinate algebra of quantum matrices)

The coordinate algebra

of quantum matrices associated with R is

T (E)

(z) − R

∗

(z) : z ∈ E ⊗ Ei

= S

(E),

where

T = R

∗

−1

The coalgebra structure remains the same as

(

)), and for the antipode,

we write

for the image of

(

), and similarly

for

. Then we map

7→

j=1

⊗ E

7→

i=1

⊗ X

This is the general construction we are up for.

Example. We have

(k)) = A

Rτ

(V )

for







q 0 0 0

0 1 0 0

0 q − q

−1

1 0

0 0 0 q





