Hopf Algebras, Algebraic, Formal, and Quantum Groups 43

Home , Hopf algebra

CHAPTER 2

44 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS

3. Ane Algebraic Groups

We apply the preceding considerations to the categories K - cAlg and K - cCoalg.

Consider K - cAlg, the category of commutative K -a lgebras. Let A; B 2 K - cAlg.

Then A B is again a commutative K -a lgebra with comp onentwise multiplication.

In fact this holds also for non-commutative K -a lgebras A.5.3, but in K - cAlg we

have

Prop osition 2.3.1. The tensor product in K - cAlg is the categorical coproduct.

Proof. Let f 2 K - cAlgA; Z ;g 2 K - cAlgB; Z . The map [f; g]: A B ! Z

de ned by[f; g]a b:= f ag b is the unique algebra homomorphism such that

[f; g]a 1 = f a and [f; g]1 b=g b or such that the diagram

A B

A A B B

[f;g]

@R ?

commutes, where a=a 1 and b=1 b are algebra homomorphisms.

A B

So the category K - cAlg has nite copro ducts and also an initial ob ject K .

A more general prop erty of the tensor pro duct of arbitrary algebras was already

considered in 1.2.13.

Observe that the following diagram commutes

1 2

A A A A

1 1

A A

? @R

where r is the multiplication of the algebra and by the diagram the co diagonal of

the copro duct.

De nition 2.3.2. An ane algebraic group is a group in the category of com-

mutative geometric spaces.

By the dualitybetween the categories of commutative geometric spaces and com-

mutative algebras, an ane algebraic group is represented by a cogroup in the cate-

gory of K - cAlg of commutative algebras.

For an arbitrary ane algebraic group H we get by Corollary 2.2.7

= 2 K - cAlgH; H H ;

1 2

" = e 2 K - cAlgH; K ; and S = id 2 K - cAlgH; H :

These maps and Corollary 2.2.7 lead to

Prop osition 2.3.3. Let H 2 K - cAlg. H is a representing object for a functor

K - cAlg ! Gr if and only if H is a Hopf algebra.

3. AFFINE ALGEBRAIC GROUPS 45

Proof. Both statements are equivalent to the existence of morphisms in K - cAlg

:H ! H H " : H ! K S : H ! H

such that the following diagrams commute

H H H

coasso ciativity

? ?

H H H H H

H H H

counit

1 "

? ? Pq

" 1

H H K H H H K

= =

- -

H H K

coinverse

S id

H H H H

id S

This Prop osition says two things. First of all each commutative Hopf algebra H

de nes a functor K - cAlgH; - : K - cAlg ! Set that factors through the category

of groups or simply a functor K - cAlgH; - : K - cAlg ! Gr. Secondly each repre-

sentable functor K - cAlgH; - : K - cAlg ! Set that factors through the category of

groups is represented by a commutative Hopf algebra.

Corollary 2.3.4. An algebra H 2 K - cAlg represents an ane algebraic group if

and only if H isacommutative Hopf algebra.

The category of commutative Hopf algebras is dual to the category of ane alge-

braic groups.

In the following lemmas we consider functors represented by commutative alge-

bras. They de ne functors on the category K - cAlg as well as more generally on

K - Alg.We rst study the functors and the representing algebras. Then we use them

to construct commutative Hopf algebras.

Lemma 2.3.5. The functor G : K - Alg ! Ab de nedbyG A:=A , the

a a

underlying additive group of the algebra A,isarepresentable functor representedby

the algebra K [x] the polynomial ring in one variable x.

Proof. G is an underlying functor that forgets the multiplicative structure of

the algebra and only preserves the additive group of the algebra. Wehave to determine

natural isomorphisms natural in A G A K - AlgK [x];A. Each element a 2 A

is mapp ed to the homomorphism of algebras a : K [x] 3 px 7! pa 2 A. This is a

homomorphism of algebras since a px+ q x = pa+ q a= a px + a q x

46 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS

and a pxq x = paq a= a pxa q x. Another reason to see this is that

K [x] is the free commutative K -algebra over fxg i.e. since each map fxg! A

can b e uniquely extended to a homomorphism of algebras K [x] ! A. The map

A 3 a 7! a 2 K - AlgK [x];A is bijective with the inverse map K - AlgK [x];A 3

f 7! f x 2 A. Finally this map is natural in A since

A K - AlgK [x];A

K-AlgK [x];g

? ?

B K - AlgK [x];B

commutes for all g 2 K - AlgA; B .

Remark 2.3.6. Since A has the structure of an additive group the sets of ho-

momorphisms of algebras K - AlgK [x];A are also additive groups.

Lemma 2.3.7. The functor G = U : K - Alg ! Gr de nedbyG A:=U A,

m m

the underlying multiplicative group of units of the algebra A,isarepresentable functor

represented by the algebra K [x; x ]= K [x; y ]=xy 1 the ring of Laurent polynomials

in one variable x.

Proof. Wehave to determine natural isomorphisms natural in A G A

K - AlgK [x; x ];A. Each element a 2 G A is mapp ed to the homomorphism of

algebras a := K [x; x ] 3 x 7! a 2 A. This de nes a unique homomorphism of

algebras since each homomorphism of algebras f from K [x; x ]=K [x; y ]=xy 1

to A is completely determined by the images of x and of y but in addition the images

have to satisfy f xf y = 1, i.e. f xmust b e invertible and f y must b e the inverse

to f x. This mapping is bijective. Furthermore it is natural in A since

A K - AlgK [x; x ];A

K -AlgK [x;x ];g

? ?

B K - AlgK [x; x ];B

for all g 2 K - AlgA; B commute.

Remark 2.3.8. Since U A has the structure of a multiplicative group the sets

K - AlgK [x; x ];A are also groups.

Lemma 2.3.9. The functor M : K - Alg ! K - Alg with M A the algebraof

n n

n n-matrices with entries in A is representable by the algebra K h x ;x ;::: ;x i,

11 12 nn

the non commutative polynomialring in the variables x .

Proof. The p olynomial ring K h x i is free over the set fx g in the category of

ij ij

non commutative algebras, i.e. for each algebra and for each map f : fx g! A ij

3. AFFINE ALGEBRAIC GROUPS 47

there exists a unique homomorphism of algebras g : K hx ;x ;::: ;x i! A such

11 12 nn

that the diagram

fx g K hx i

ij ij

? @R

commutes. So each matrix in M A de nes a unique a homomorphism of algebras

K hx ;x ;::: ;x i! A and conversely.

11 12 nn

Example 2.3.10. 1. The ane algebraic group called additive group

G : K - cAlg ! Ab

with G A:= A from Lemma 2.3.5 is represented by the Hopf algebra K [x]. We

determine comultiplication, counit, and antip o de.

By Corollary 2.2.7 the comultiplication is = 2 K - cAlgK [x]; K [x]

1 2

G K [x] K [x]. Hence K [x]

x= x+ x=x 1+1 x:

1 2

G K hence The counit is " = e =02 K - cAlgK [x]; K

a K

"x= 0:

G K [x] hence The antipode is S =id 2 K - cAlgK [x]; K [x]

K[x]

S x= x:

2. The ane algebraic group called multiplicative group

G : K - cAlg ! Ab

with G A:=A = U A from Lemma 2.3.7 is represented by the Hopf algebra

K [x; x ]= K [x; y ]=xy 1. We determine comultiplication, counit, and antip o de.

By Corollary 2.2.7 the comultiplication is

1 1 1 1 1

= 2 K - cAlgK [x; x ]; K [x; x ] K [x; x ] G K [x; x ] K [x; x ]:

1 2 m

Hence

x= x x=x x:

1 2

The counit is " = e =12 K - cAlgK [x; x ]; K G K hence

K m

"x= 1:

1 1 1

The antipode is S =id 2 K - cAlgK [x; x ]; K [x; x ] G K [x; x ]

K[x;x ]

hence

S x= x :

3. The ane algebraic group called additive matrix group

M : K - cAlg ! Ab ; n

48 2. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS

with M A the additive group of n n-matrices with co ecients in A is represented

by the commutative algebra M = K [x j1 i; j n] Lemma 2.3.9. This algebra

must b e a Hopf algebra.

+ + + + + +

The comultiplication is = 2 K - cAlgM ;M M M M M .

1 2

n n n n n n

Hence

x = x + x =x 1+1 x :

ij 1 ij 2 ij ij ij

+ +

The counit is " = e = 0 2 K - cAlgM ; K M K hence

n n

"x = 0:

+ + + +

2 K - cAlgM ;M M M hence The antipode is S =id

n n n n

S x =x :

ij ij

4. The matrix algebra M A also has a noncommutativemultiplication, the ma-

trix multiplication, de ning a monoid structure M A. Thus K [x ] carries another

coalgebra structure which de nes a bialgebra M = K [x ]. Obviously there is no

antip o de.

M M M . The comultiplication is = 2 K - cAlgM ;M M

1 2

n n n n n n

Hence x = x x = x x or

ij 1 ij 2 ij ij ij

x = x x :

ik ij jk

M K hence The counit is " = e = E 2 K - cAlgM ; K

n n

"x = :

ij ij

5. Let K b e a eld of characteristic p. The algebra H = K [x]=x carries the

structure of a Hopf algebra with x= x 1+1 x, "x=0,and S x=x.

To show that is well de ned wehave to showx = 0. But this is obvious by

the rules for computing p-th p owers in characteristic p.Wehavex 1+1 x =

p p

x 1+1 x =0.

Thus the algebra H represents an ane algebraic group:

A:=K - cAlgH; A fa 2 Aja =0g:

The group multiplication is the addition of p-nilp otent elements. So wehave the

group of p-nilpotent elements.

6. The algebra H = K [x]=x 1 is a Hopf algebra with the comultiplication

x= x x, the counit "x = 1, and the antipode S x= x . These maps are

n n n n

well de ned since wehave for example x =x x = x x =1 1. Observe

that this Hopf algebra is isomorphic to the group algebra K C of the cyclic group C

n n

of order n.

Thus the algebra H represents an ane algebraic group:

A:= K - cAlgH; A fa 2 Aja =1g;

= n

3. AFFINE ALGEBRAIC GROUPS 49

that is the group of n-th roots of unity. The group multiplication is the ordinary

multiplication of ro ots of unity.

7. The linear groups or matrix groups GLnA, SLnA and other such groups

are further examples of ane algebraic groups. We will discuss them in the section

on quantum groups.

Problem 2.3.1. 1. The construction of the general linear group

GLnA= fa 2 M Aja invertibleg

ij n ij

de nes an ane algebraic group. Describ e the representing Hopf algebra.

2. The sp ecial linear group SLnA is an ane algebraic group. What is the

representing Hopf algebra?

3. The real unit circle S R carry the structure of a group by the addition of

1 2 2

angles. Is it p ossible to make S with the ane algebra K [c; s]=s + c 1 into an

ane algebraic group? Hint: How can you add two p oints x ;y and x ;y on

1 1 2 2

the unit circle, such that you get the addition of the asso ciated angles?