Endomorphism Bialgebras of Diagrams and of Non-Commutative Algebras and Spaces

Endomorphism Bialgebras of Diagrams

and of Non-Commutative Algebras and Spaces

BODO PAREIGIS

Department of Mathematics, UniversityofMunich

Munich, Germany

Bialgebras and Hopf algebras havea very complicated structure. It is not easy to

construct explicit examples of suchandcheck all the necessary prop erties. This gets even

more complicated if wehavetoverify that something like a como dule algebra over a bialgebra

is given.

Bialgebras and como dule algebras, however, arise in a very natural way in non-commu-

tative geometry and in representation theory.Wewant to study some general principles on

how to construct such bialgebras and como dule algebras.

The leading idea throughout this pap er is the notion of an action as can b een seen most

clearly in the example of vector spaces. Given a vector space V we can asso ciate with it its

endomorphism algebra End(V ) that, in turn, de nes an action End (V ) V ! V . There is

also the general linear group GL (V ) that de nes an action GL (V ) V ! V . In the case

of the endomorphism algebra we are in the pleasant situation that End(V )isavector space

itself so that we can write the action also as End (V ) V ! V . The action of GL (V )on V

can also b e describ ed using the tensor pro duct by expanding the group GL(V ) to the group

algebra K (GL (V )) to obtain K (GL (V )) V ! V .

We are going to nd analogues of End (V )or K (GL (V )) acting on non-commutative

geometric spaces or on certain diagrams. This will lead to bialgebras, Hopf algebras, and

como dule algebras.

There are twowell-known pro cedures to obtain bialgebras from endomorphisms of cer-

tain ob jects. In the rst section we will construct endomorphism spaces in the category of

non-commutative spaces. These endomorphism spaces are describ ed through bialgebras.

In the second section we nd (co-)endomorphism coalgebras of certain diagrams of vector

spaces, graded vector spaces, di erential graded vector spaces, or others. Under additional

conditions they again will turn out to b e bialgebras.

The ob jects constructed in the rst section will primarily b e algebras, whereas in the sec-

ond section the ob jects co end(! )willhave the natural structure of a coalgebra. Nevertheless

we will show in the third section that the constructions of bialgebras from non-commutative

spaces and of bialgebras from diagrams of vector spaces, remote as they may seem, are closely

related, in fact that the case of an endomorphism space of a non-commutative space is a

sp ecial case of a co endomorphism bialgebra of a certain diagram. Some other constructions

of endomorphism spaces from the literature will also b e subsumed under the more general

construction of co endomorphism bialgebras of diagrams. We also will nd such bialgebras

coacting on Lie algebras.

This indicates that a suitable setting of non-commutative geometry might b e obtained

by considering (monoidal) diagrams of vector spaces (which can b e considered as partially

de ned algebras) as a generalization of ane non-commutative spaces. So the problem of

nding non-commutative (non-ane) schemes might b e resolved in this direction.

In the last section weshow that similar results hold for Hopf algebras acting on non-

commutative spaces resp. on diagrams. The universal Hopf algebra coacting on an algebra

is usually obtained as the Hopf envelop e of the universal bialgebra acting on this algebra.

We show that this Hopf algebra can also b e obtained as the (co-)endomorphism bialgebra of

a sp eci c diagram constructed from the given algebra.

Throughout this pap er K shall denote an arbitrary eld and stands for .

1 Endomorphisms of Non-Commutative Geometric

Spaces

In this section we discuss some simple background on non-commutative spaces and their

endomorphisms.

1.1 Ane non-commutative spaces

In algebraic geometry an ane algebraic space is given as a subset of K consisting of all

p oints satisfying certain p olynomial equalities. A typical example is the unit circle in R

given by

2 2 2

Circ(R)=f(x; y ) 2 R j x + y =1g:

Actually one is interested in circles with arbitrary radius r over any commutative R-algebra

B . They are de ned in a similar wayby

2 2 2 2

Circ(B )=f(x; y ) 2 B j x + y r =0g:

If for example r is 1 instead of 1, there are no p oints with co ecients in the eld of real

numb ers but lots of p oints with co ecients in the eld of complex numb ers.

Furthermore this de nes a functor Circ : R-Alg ! Sets from the category of all

commutative R-algebras to the category of sets, since an algebra homomorphism f : B !

0 0

B induces a map of the corresp onding unit circles Circ(f ) : Circ(B ) ! Circ(B ). The

co ordinates for the p oints whichwe consider are taken from algebra B ,thecoordinate domain,

and Circ(B ) is the set of all p oints of the space, that is the given manifold or geometric space.

One has this set for all choices of co ordinate domains B .

In general a functor X : K -Alg ! Sets with

X (B ):=f(b ;:::;b ) 2 B j p (b ;:::;b )=::: = p (b ;:::;b )=0g (1)

1 n 1 1 n r 1 n

is called an ane algebraic space. This functor is represented by the algebra

A = K [x ;:::;x ]=(p (x ;:::;x );:::;p (x ;:::;x ));

1 n 1 1 n r 1 n

hence X (B )=K -Alg (A; B ). So for the circle wehave

2 2 2

Circ(B ) R-Alg (R[x; y ]=(x + y r );B):

The Yoneda Lemma shows that this representing algebra A is uniquely determined (up

to isomorphism) by the functor X . It is usually considered as the "function algebra" of the

geometric space under consideration. Indeed there is a map A X (K ) ! K ,(a; f ) 7! f (a),

where f 2X(K ) K -Alg (A; K ).

All of this has nothing to do with the commutativity of the algebras under consideration.

Hence one can use non-commutative K -algebras A and B as well. Certain questions in

physics in fact require such algebras. Instead of representing algebras

A = K [x ;:::;x ]=(p (x ;:::;x );:::;p (x ;:::;x ))

1 n 1 1 n r 1 n

we takenow representing algebras

A = K hx ;:::;x i=(p (x ;:::;x );:::;p (x ;:::;x ))

1 n 1 1 n r 1 n

where K hx ;:::;x i is the algebra of p olynomials in non-commuting variables or, equiva-

1 n

lently, the tensor algebra of the nite-dimensional vector space with basis x ;:::;x .

1 n

DEFINITION 1.1 (B -p oints of quantum spaces)

A functor X = K -Alg(A; ) is cal ledanane non-commutative space or a quantum space

and the elements of X (B )=K -Alg(A; B ) the B -points of X .

As in (1) the B -p oints of a quantum space are elements of B ,sothat X (B ) is a subset

of B .

Well known examples of quantum spaces are the quantum planes with function algebras

[5]

2j0 1

O (A )=K hx; y i=(xy q yx); q 2 K nf0g

and [6]

2j0

O (A )=K hx; y i=(xy yx + y ):

As in (commutative) algebraic geometry one can consider the algebra A = O (X )=

K hx ;:::;x i=(p (x ;:::;x );:::;p (x ;:::;x )) (which represents the quantum space X )

1 n 1 1 n r 1 n

as the function algebra of X consisting of the (natural) functions from X (B ) to the co ordinate

algebra U (B ) where U : K -Alg ! Sets is the underlying functor. So the elements of A

are certain functions from X (B )toB .We denote the set of all natural transformations or

functions from X to U by Map (X ; U )

LEMMA 1.2 (the function algebra of a space)

Let X : K -Alg ! Sets bearepresentable functor with representing algebra A. Then there

Map(X ; U ) inducing a natural transformation A X(B ) ! B . is an isomorphism A

Pro of: The underlying functor U : K -Alg ! Sets is represented by the algebra K [x]=

K hxi.SobytheYoneda Lemma weget

Map(X ; U ) Map(K -Alg(A; );K-Alg(K [x]; )) K -Alg(K [x];A) A:

= = =

COROLLARY 1.3 (the universal prop erty of the function algebra)

Let X and A beasinLemma 1.2. If ' : C X(B ) ! B is a natural transformation then

there exists a unique map '~ : C ! A such that the fol lowing diagram commutes

C X(B )

'~ 1

@R ?

A X(B ) B:

Pro of: This follows from the Yoneda lemma.

If X and Y are quantum spaces, then we will call a natural transformation f : X!Y

simply a map of quantum spaces. So the quantum spaces form a category Q whichis

antiequivalent to the category of nitely generated K -algebras K -Alg. The set of maps from

X to Y will b e denoted by Map(X ; Y ).

If X is a quantum space, A = O (X ) its function algebra, and I A atwo-sided ideal

of A, then A ! A=I is an epimorphism which induces a monomorphism : X !X for

the asso ciated quantum spaces. In particular the B -p oints X (B )= K -Alg(A=I ; B ) can b e

identi ed with a certain subset of B -p oints in X (B ). Conversely every subspace YX (i.

e. Y (B ) X(B ) functorially for all B )isinducedby an epimorphism O (X ) !O(Y ).

(Observe, however, that not every epimorphism in the category K -Alg is surjective, e. g.

K [x] ! K (x) is an epimorphism but not a surjection.)

1.2 The commutative part of a quantum space

2j0

The quantum plane A de nes a functor from the category of algebras to the category

2j0

of sets. We call its restriction to commutative algebras the commutative part (A )

comm

of the quantum plane. In general the restriction of a quantum space X to commutative

algebras is called the commutative part of the quantum space and is denoted by X .

comm

The commutative part of a quantum space represented by an algebra A is always an ane

algebraic (commutative)scheme, since it is represented by the algebra A=[A; A], where [A; A]

denotes the two-sided ideal generated by all commutators [a; b]= ab ba for a; b 2 A.In

particular, the commutativepart X of a quantum space X is indeed a subspace of X .

comm

For a commutative algebra B the spaces X and X have the same B -p oints:

comm

X (B )= X (B ). comm

2j0

For the quantum plane A and commutative elds B the set of B -p oints consists exactly

2 2j0

of the two co ordinate axes in B since (A ) is represented by K [x; y ]=(xy qyx)

comm

K [x; y ]=(xy ) for q 6=1.

1.3 Commuting p oints

0 0

In algebraic geometry anytwopoints (b ;:::;b )and(b ;:::;b ) with co ecients in the same

1 m

1 n

co ordinate algebra B have the prop erty that their co ordinates mutually commute under the

0 0

multiplication b b = b b of B since B is commutative. This do es not hold any longer for

i i

j j

non-commutative algebras B and arbitrary quantum spaces X and Y .

DEFINITION 1.4 (commuting p oints)

0 0 0

If A = O (X ) and A = O (Y ) and if p : A ! B 2X(B ) and p : A ! B 2Y(B ) aretwo

0 0

points with coordinates in B , we say that they are commuting p oints if for al l a 2 A; a 2 A

0 0 0 0 0

we have p(a)p (a )=p (a )p(a),i. e. the images of the algebra homomorphisms p and p

commute elementwise.

Obviously it is sucient to require this just for a set of algebra generators a of A

0 0 0

and a of A . In particular if A is of the form A = K hx ;:::;x i=I and A is of the form

1 m

0 0 0

A = K hx ;:::;x i=J then the B -p oints are given by(b ;:::;b ) resp. (b ;:::;b )with

1 n 1 m

1 n

0 0

co ordinates in B , and the twopoints commute i b b = b b for all i and j .

i i

j j

The set of commuting p oints

(X?Y)(B )= X (B )?Y (B ):= f(p; q ) 2X(B ) Y(B )j p and q commuteg

is a subfunctor of XY since commutativityoftwoelements in B is preserved by algebra

homomorphisms f : B ! B .Wecallittheorthogonal product of the quantum spaces X

and Y .

LEMMA 1.5 (the orthogonal pro duct)

If X and Y are quantum spaces with function algebras A = O (X ) and A = O (Y ) then the

orthogonal product X?Y is a quantum space with (representing) function algebra O (X?Y)=

A A = O (X ) O(Y ).

Pro of: Let (f; g) 2 (X?Y)(B ) b e a pair of commuting p oints. Then there is a unique

homomorphism h : A A ! B suchthat

p p

0 0

A A A A

? @R

0 0

commutes; in fact the map is given by h(a a )=f (a)g (a ). Conversely every algebra

homomorphism h : A A ! B de nes a pair of commuting p oints bytheabove diagram.

0 0

The algebra homomorphism p : A ! A A is de ned by p(a)=a 1, and p is de ned

similarly.

This lemma shows that the set of commuting p oints

0 0 0 0

((b ;:::;b ); (b ;:::;b ))=(b ;:::;b ;b ;:::;b )

1 m 1 m

1 n 1 n

0 0

with b b b b = 0 forms again a quantum space. It is now easy to show that the category Q

i i

j j

of quantum spaces is a monoidal category with the orthogonal pro duct X?Y (in the sense of

[3]). The asso ciativity of the orthogonal pro duct arises from the asso ciativity of the tensor

pro duct of the function algebras.

The preceding lemma sheds some light on the reason, whywehave restricted our con-

siderations to commuting p oints. There is a general credo that the function algebra of a

"non-commutative" space should b e graded and have p olynomial growth that is some kind

ofaPoincare-Birkho -Witt theorem should hold. But the free pro duct of algebras (which

would corresp ond to the pro duct of the quantum spaces) grows exp onentially (with the de-

gree). Some kind of commutation relation among the elements of the function algebra is

required and this is given by letting the elements of the two function algebras A and B in

the orthogonal pro duct of the quantum spaces commute. Thisisdoneby the tensor pro duct.

1.4 The endomorphisms of a quantum space

In the category of quantum spaces wewant to nd an analogue of the endomorphism algebra

End(V )ofavector space V and of its action on V . Actually we consider a somewhat more

general situation of an action H(X ; Y )?X !Y which resembles the action Hom(V; W )

V ! W for vector spaces. The tensor pro duct of vector spaces will b e replaced bythe

orthogonal pro duct of quantum spaces ?.

DEFINITION 1.6 (the hom of quantum spaces)

Let X and Y be a quantum spaces. A (universal) quantum space H(X ; Y ) together with a map

: H(X ; Y )?X !Y, such that for every quantum space Z and every map : Z?X !Y

there is a unique map : Z!H(X ; Y ) such that the diagram

Z?X

? @R

H(X ; Y )?X Y

commutes, is cal ledahomomorphism space. E := H(X ; X ) is cal ledanendomorphism

space.

Apart from the map H(X ; Y )?X !Y, whichwe will regard as a multiplication of

H(X ; Y )onX (with values in Y ), this construction leads to further multiplications.

LEMMA 1.7 (the multiplication of homs)

If there exist homomorphism spaces H(X ; Y ), H(Y ; Z ),andH(X ; Z ) for the quantum spaces

X , Y ,andZ , then there is a multiplication m : H(Y ; Z )?H(X ; Y ) !H(X ; Z ) with respect

to the orthogonal product structureinQ.Furthermore this product is associative and unitary

if the necessary homomorphism spaces exist. Pro of: Consider the diagram

- -

(H(Y ; Z )?H(X ; Y ))?X H(Y ; Z )?(H(X ; Y )?X ) H(Y ; Z )?Y

m?1

? ?

H(X ; Z )?X Z

which induces a unique homomorphism m : H(Y ; Z )?H(X ; Y ) !H(X ; Z ), the multipli-

cation.

COROLLARY 1.8 (the endomorphism space is a monoid)

The endomorphism space E of a quantum space X is a monoid (an algebra) in the category

Q w. r. t. the orthogonal product. The space H(X ; Y ) is a left E -space and a right E -space

Y X

(an E ; E -bimodule) in Q.

Y X

Pro of: The multiplication is given in Lemma 1.7. To get the unit we consider the one p oint

quantum space J (B )=fu g with function algebra K and the diagram

J?X

= u?1

@R ?

E ?X X

which induces a unique homomorphism u : J!E , the unit for E .

X X

Standard arguments shownow that (E ;m;u) forms a monoid in the category of quantum

spaces and that the spaces H(X ; Y )are E - resp. E -spaces.

Y X

We will call a quantum monoid a space E together with E?E !E in Q whichisa

monoid w.r.t. to the orthogonal pro duct. If E acts on a quantum space X by E?X !X

such that this action is asso ciative and unitary then wecallX an E -space.

PROPOSITION 1.9 (the universal prop erties of the endomorphism space)

The endomorphism space E and the map : E ?X !X have the fol lowing universal

X X

properties:

a) For every quantum space Z and map : Z?X !X there is a unique map : Z!E

such that the diagram

Z?X

? @R

E ?X X

commutes.

b) For every quantum monoid M and map : M?X !X which makes X an M-space

there is a unique monoid map : M!E such that the diagram X

M?X

@R ?

E ?X X

commutes.

Pro of: a) is the de nition of E .

b) We consider the diagram

m ?1

M?M?X M?X

? ?1 ?1

? ?

m?1

E ?E ?X E ?X

X X X

The right triangle commutes by de nition of .Thelower square commutes since

( ?1)(1? )= ? = ?( ?1) = (1?)( ? ?1):

The horizontal pairs are equalized since X is an M-space and an E -space. So we get

( ?1)(m ?1) = (m?1)( ? ?1). By the universal prop ertya)ofE , the map is com-

patible with the multiplication. Similarly one shows that it is compatible with the unit.

It is not so clear which homomorphism spaces or endomorphisms spaces exist. Tambara

[15] has given a construction for homomorphism spaces H(X ; Y )whereX is represented bya

nite dimensional algebra. In 3.2 we shall give another pro of of his theorem. There are other

examples of such co endomorphism bialgebras e. g. for "quadratic algebras" as considered

in [5]. We will reconstruct them in 3.3.

1.5 Bialgebras and como dule algebras

With quantum homomorphism spaces and endomorphism spaces wehave already obtained

bialgebras and como dule algebras.

Let X resp. Y b e quantum spaces with function algebras A = O (X )resp. B = O (Y ).

Assume that E and E exist and have function algebras E = O (E )resp. E = O (E ).

X Y A X B Y

Then the op erations E ?E !E , E ?X !X, E ?H(X ; Y ) !H(X ; Y ), and

X X X X Y

H(X ; Y )?E !H(X ; Y ) (if the quantum space H(X ; Y ) exists) lead to algebra homo-

morphisms

E ! E E ;

A A A

A ! E A;

O (H(X ; Y )) ! E O(H(X ; Y ));

O (H(X ; Y )) !O(H(X ; Y )) E :

In particular E and E are bialgebras, and A and O (H(X ; Y )) are como dule algebras

A B

over E ,andB and O (H(X ; Y )) are como dule algebras over E . Furthermore the map

A B

H(X ; Y )?X !Y induces an algebra homomorphism B !O(H(X ; Y )) A.Write

a(A; B ):= O (H(X ; Y )):

Then wehave

Map(Z ; H(X ; Y )) Map(Z?X ; Y )

and

K -Alg(a(A; B );C) K -Alg(B; C A):

and a universal algebra homomorphism B ! a(A; B ) A. Here wehave used the notation

of Tambara [15 ] for the universal algebra.

From De nition 1.6 and Lemma 1.7 we get the following

COROLLARY 1.10 (the co endomorphism bialgebra of an algebra)

Let A be a (non-commutative) algebra. Let E be an algebraand : A ! E A bean

A A

algebra homomorphism, such that for every algebra B and algebra homomorphism : A !

B A there is a unique algebra homomorphism : E ! B such that

A E A

@R ?

B A

commutes. Then E represents the endomorphism spaceof X = K -Alg(A; -), E is a

A A

bialgebra, and A is an E -comodule algebra.

Since the category of algebras is dual to the category of quantum space, we call E the

coendomorphism bialgebra of A and its elements coendomorphisms.From Prop osition 1.9 we

obtain immediately

COROLLARY 1.11 (the universal prop erties of the co endomorphism bialgebra)

The coendomorphism bialgebra E of an algebra A and the map : A ! E A have the

A A

fol lowing universal properties:

a) For every algebra B and algebra homomorphism : A ! B A there is a unique algebra

homomorphism : E ! B such that the diagram

A E A

@R ?

B A

commutes.

b) For every bialgebra B and algebra homomorphism : A ! B A making A into a

comodule algebra there is a unique bialgebra homomorphism : E ! B such that the

diagram

A E A

@R ?

B A

commutes.

2 Co endomorphism Bialgebras of Diagrams

Wehave seen that, similar to commutative algebraic geometry, non-commutative spaces

represented by non-commutative algebras induce non-commutative endomorphism spaces

which are represented by bialgebras.

Nowwemove to a seemingly unrelated sub ject which is studied in representation theory

and Tannaka duality. The principal question here is whether a group, a monoid, or an algebra

are completely determined, if all their representations or mo dules are "known". We certainly

have to sp ecify what the term "known" should mean in this context. Certain reasons, mainly

the fundamental theorem on the structure of coalgebras and como dules, make it easier to

consider como dules rather than mo dules as representations.

The purp ose of this section is to show that for each diagram of nite dimensional vector

spaces there is an asso ciated coalgebra which b ehaves like the dual of an endomorphism

algebra. In particular we asso ciate with the trivial diagram that consists of just one nite

dimensional vector space V a coalgebra C that is the dual of the endomorphism algebra of

V V . If the diagram of vector spaces has additional End(V ) this vector space: C

= =

prop erties then the asso ciated coalgebra will have additional prop erties. This construction

renders bialgebras and como dule algebras asso ciated with certain diagrams.

If we start with a coalgebra C then we can construct the diagram of all its nite

dimensional como dules and como dule homomorphisms, and also the diagram or category

of all como dules. Then the coalgebra C can b e recovered from the underlying functor

! : C omo d- C !A as the coalgebra asso ciated with this diagram.

Each bialgebra B induces a tensor pro duct in the category of como dules C omo d-B

over B . This bialgebra can also b e recovered from the underlying functor as the bialgebra

asso ciated with the given diagram. Additional prop erties of B induce additional features of

C omo d-B and conversely.

2.1 The base category

All the structures considered in this pap er are built on underlying vector spaces over a given

eld. Certain generalizations of our constructions are straightforward whence we will use a

general category A instead of the category of vector spaces.

We assume that A is an ab elian category and a monoidal category with an asso ciative

unitary tensor pro duct : AA !A as in [8]. We also assume that the category A is

co complete ([7] and [13] p. 23) and that colimits commute with tensor pro ducts. Finally we

assume that the monoidal category A is quasisymmetric in the sense of [13 ] or braided that

is there is a bifunctorial isomorphism : X Y ! Y X such that the diagram

Y (Z X ) X (Y Z ) (Y Z ) X

(X Y ) Z (Y X ) Z Y (X Z )

and the corresp onding diagram for commute. is called a symmetry if in addition

= holds.

Y;X

X;Y

We givea fewinteresting examples of such categories.

1) The category V ec of all vector spaces over K with the usual tensor pro duct is a symmetric

monoidal category and satis es the conditions for A.

2) The category C omo d-H of como dules over a co quasitriangular or braided bialgebra H [13]

with the diagonal action of H on the tensor pro duct of como dules over K and with colimits in

V ec and the canonical H -como dule structure on these is a quasisymmetric monoidal category

and satis es the conditions for A since tensor pro ducts commute with colimits.

3) The category of N-graded vector spaces is isomorphic to the category

C omo d-H , with H = K [N] the monoid algebra over the monoid of integers. Hence the

category of graded vector spaces (with the usual graded tensor pro duct) is a symmetric

monoidal category which satis es the conditions for A.

4) The category of chain complexes of vector spaces K -C omp is isomorphic to the category

1 2

C omo d-H of como dules over the Hopf algebra H = K hx; y ; y i=(xy + yx; x ) with (x)=

x 1+y x and (y )=y y [10, 11 ]. H is a cotriangular Hopf algebra ([10] p. 373)

and K -C omp is a symmetric monoidal category with the total tensor pro duct of complexes

which satis es the conditions for A.

5) Sup er-symmetric spaces are de ned as the symmetric monoidal category C omo d- H of

como dules over the Hopf algebra H = K [Z=2Z]which is cotriangular [1].

We are going to assume throughout that the category A is symmetric, so that wecan

use the full strength of the coherence theorems for monoidal categories. Thus most of the

time we will delete all asso ciativity, unity, and symmetry morphisms assuming that our

categories are strict symmetric monoidal categories. Most of the given results hold also over

quasisymmetric monoidal categories, cf.[4], but it gets quite technical if one wants to check

all the details.

WesaythatanobjectX of a category A has a dual (X ,ev) where X is an ob ject and

ev: X X ! I is a morphism in A, if there is a morphism db: I ! X X such that

db 1 1 ev

(X ! X X X ! X )=1 ;

1 db ev 1

: (X ! X X X ! X )= 1

The category A is rigid if every ob ject of A has a dual.

The full sub category of ob jects in A having duals in the sense of [12 ] is denoted by A .

The category A then is a rigid symmetric monoidal category. Observethata vector space

has a dual in this sense i it is nite dimensional.

2.2 Cohomomorphisms of diagrams

In this subsection the category A do es not have to b e symmetric or quasisymmetric.

We consider diagrams (commutative or not) in A. They are given by the ob jects at the

vertices and the morphisms along the edges. The vertices and the (directed) edges or arrows

alone de ne the shap e of the diagram, e. g. a triangle or a square. This shap e can b e made

into a category of its own right[3,7],a diagram scheme, and the concrete diagram can then

b e considered as a functor, sending the vertices to the ob jects at the vertices and the arrows

to the morphisms of the diagram. So the diagram scheme for commutative triangles

1 2

? @R

has a total of three ob jects f1; 2; 3g and 6 morphisms f ; ; ; id ; id ; id g, identities in-

1 2 3

cluded.

Let ! : D!A b e a diagram in A. The category D and thus the diagram ! is always

assumed to b e small. We call the diagram nite if the functor ! : D!A factors through

A that is if all ob jects of the diagram haveduals.

THEOREM 2.1 (the existence of cohom of diagrams)

0 0

Let (D ;!) and (D ;! ) be two diagrams in A over the same diagram scheme and let (D ;! )

be nite. Then the set of al l natural transformations Mor (!; M ! ) is representable as a

functor in M .

Pro of: This has b een proved in various sp ecial case in [2], [10 ], [16], and [12]. We follow [12]

2.1.9 and give the explicit construction here since this construction will b e used later on. As

remarked in [12] 2.1.7 and 2.1.9 there are isomorphisms

0 0

Mor (!; M ! ) = Hom(! (X );M ! (X )) ([3] IX:5(2))

X 2D

0 0

= Hom(! (X ) ! (X ) ;M) [! (X )must have a dual!]

X 2D

= Hom( ! (X ) ! (X ) ;M)

where the integral means the end resp. co end of the given bifunctors in the sense of [3].

Hence the representing ob ject is

X 2D

! (X ) ! (X ) =

a a

0 0

di coker ! (dom(f )) ! (co d(f )) ! (X ) ! (X ) ;

f 2Mor(D ) X 2Ob(D )

whence it can b e written as a quotient of a certain copro duct. The two morphisms F

0 0

and F are comp osed of morphisms of the form 1 ! (f ) : ! (dom(f )) ! (co d(f )) !

! (dom(f )) ! (dom(f )) resp ectively

0 0

! (f ) 1:! (dom(f )) ! (co d(f )) ! ! (co d (f )) ! (co d(f )) :

DEFINITION 2.2 (cohomomorphisms)

The representing object of Mor (!; M ! ) is written as

cohom(! ;!):=

` `

0 0

di coker ! (X ) ! (X ) : ! (dom(f )) ! (co d(f ))

f 2Mor(D )

X 2Ob(D )

0 0

The elements of cohom(! ;!) arecal led cohomomorphisms.If! = ! then the representing

object of Mor (!; M ! ) is written as co end (! ), the set of co endomorphisms of ! .

It is essential for us to describ e very explicitly the equivalence relation for the di erence

cokernel. Let f : X ! Y b e a morphism in D . It induces homomorphisms ! (f ):! (X ) !

0 0 0

! (Y )and! (f ) : ! (Y ) ! ! (X ) .Wethus obtain the twocomponents of the maps F

and F D

! (X ) ! (X )

1 ! (f )

! (X ) ! (Y )

! (f ) 1

! (Y ) ! (Y )

0 0

which map elements x 2 ! (X ) ! (Y ) to two equivalentelements x ! (f ) ( )

0 0

! (f )(x) . So cohom(! ;!) is the direct sum of the ob jects ! (X ) ! (X ) for all X 2D

mo dulo this equivalence relation.

Since a representable functor de nes a universal arrowweget

COROLLARY 2.3 (the universal prop erty of cohoms)

0 0

Let (D ;!) and (D ;! ) be two diagrams in A and let (D ;! ) be nite. Then there is a natural

0 0

transformation : ! ! cohom(! ;!) ! ; such that for each object M 2Aand each natural

0 0

transformation ' : ! ! M ! there is a unique morphism '~ :cohom(! ;!) ! M such

that the diagram

0 0

! cohom(! ;!) !

'~ 1

Hj ?

M !

commutes.

By [12 ] 2.1.12 the co endomorphism set co end (! ) is a coalgebra. The comultiplication

arises from the commutative diagram

! co end(! ) !

? ?

co end(! ) ! co end(! ) co end(! ) !:

Somewhat more generally one shows

PROPOSITION 2.4 (the comultiplication of cohoms)

0 00 0 00

Let (D ;!); (D ;! ); and (D ;! ) be diagrams in A and let (D ;! ) and (D ;! ) be nite. Then

thereisacomultiplication

00 0 00 0

: cohom(! ;!) ! cohom(! ;!) cohom(! ;! )

which is coassociative and counitary.

Pro of: By 2.3 the homomorphism is induced by the following commutative diagram

00 00

! cohom(! ;!) !

? ?

0 0 0 00 0 00

cohom(! ;!) ! cohom(! ;!) cohom(! ;! ) ! :

A simple calculation shows that this comultiplication is coasso ciative and counitary.

COROLLARY 2.5 (co end(! ) is a coalgebra)

0 0

Let (D ;!) and (D ;! ) be nite diagrams in A. Then co end(! ) and co end(! ) arecoalgebras,

0 0 0

al l objects ! (X ) resp. ! (X ) arecomodules, and cohom(! ;!) is a right co end(! )-comodule

and a left co end (! )-comodule.

Pro of: A consequence of the preceding prop osition is that co end (! ) and co end (! ) are coal-

gebras. The como dule structure on ! (X ) comes from : ! (X ) ! co end(! ) ! (X ). The

other como dule structures are clear.

What wehave found here is in essence the universal coalgebra C =coend(! )such that

all vector spaces and all homomorphisms of the diagram are como dules over C resp. como dule

homomorphisms. In fact an easy consequence of the last corollary is

COROLLARY 2.6 (the universal coalgebra for a diagram of como dules)

Let (D ;!) be a nite diagram in A. Then al l objects ! (X ) arecomodules over co end(! ) and

al l morphisms ! (f ) arecomodule homomorphisms. If D is another coalgebra and al l ! (X )

arecomodules over D and al l ! (f ) arecomodule homomorphisms, then there is a unique

coalgebra homomorphism ' : co end (! ) ! D such that the diagrams

(X )

! (X ) co end(! ) ! (X )

'~(X ) 1

Hj ?

D ! (X )

commute.

Pro of: '(X ):! (X ) ! D ! (X ) is in fact a natural transformation. Then the existence

and uniqueness of a homomorphism of vector spaces' ~ :coend(! ) ! D is obvious. The

fact that this is a homomorphism of coalgebras follows from the universal prop ertyof C =

co end(! )by

! C !

@ @

@R @R

C ! C C !

'~ 1

'~ 1 '~ '~ 1

? ?

! D !

@ @

@R @R ? ?

D ! D D !

1 '

where the right side of the cub e commutes bytheuniversal prop erty. Similarly one proves

that' ~ preserves the counit.

To describ e the coalgebra structure of co end(! )we denote the image of x 2 ! (X )

! (X ) in co end(! )by x . Let furthermore x denote the dual basis in ! (X ) ! (X ) .

i i

Then : ! (X ) ! co end(! ) ! (X ) is induced by the map ! (X ) ! (X ) ! co end(! )

and is given by

x x : (x )=

i i

The construction of : co end(! ) ! co end (! ) co end(! ) furnishes

( x )= x x :

i i

COROLLARY 2.7 (diagram restrictions induce morphisms for the cohoms)

0 0 0 0

Let D and D be diagram schemes and let ! : D !A and ! : D !A be nite diagrams.

0 0

Let F : D!D be a functor. Then F induces a homomorphism '~ :cohom(! F ;!F ) !

cohom(! ;!) that is compatible with the comultiplication on cohoms as describedin2.4.In

particular F inducesacoalgebra homomorphism co end(! F ) ! co end(! ).

00 0 00 0

Pro of: We obtain a morphism' ~ :cohom(! F ;! F ) ! cohom(! ;! ) from

0 00 0 00

! F cohom(! F ;! F ) ! F

'~ 1

? Hj

00 0 00

cohom(! ;! ) ! F :

that induces a commutative diagram

00 0 00 0

cohom(! F ;!F ) cohom(! F ;!F ) cohom(! F ;! F )

'~ '~ '~

? ?

00 0 00 0

cohom(! ;!) cohom(! ;!) cohom(! ;! )

PROPOSITION 2.8 (isomorphic cohoms)

0 0 0 0

Let D and D be diagram schemes and let ! : D !A and ! : D !A bediagrams such

0 0 0

that (D ;! ) is nite. Let F : D!D be a functor which is bijective an the objects and

0 0

surjective on the morphism sets. Then the map '~ : cohom(! F ;!F ) ! cohom(! ;!) is an

isomorphism.

0 0

Pro of: Since by de nition the ob jects cohom(! F ;!F ) and cohom(! ;!) represent the func-

0 0

tors Mor (! F ;M ! F ) resp. Mor (!; M ! ) it suces to show that these functors are

f f

isomorphic. But every natural transformation ' : ! F! M ! F makes the diagrams

'(X )

! F (X ) M ! F (X )

! F (f ) M ! F (f )

? ?

'(Y )

! F (Y ) M ! F (Y )

commute for all f in D . Similarly every natural transformation : ! ! ! makes the

diagrams

(X )

0 0 0

! (X ) M ! (X )

0 0 0

! (f ) M ! (X )

? ?

(Y )

0 0 0

! (Y ) M ! (Y )

0 0

commute for all f in D . Since F is bijective on the ob jects and surjective on the morphisms,

we can identify the natural transformations ' and .

2.3 Monoidal diagrams and bialgebras

If we consider additional structures like tensor pro ducts on the diagrams, we get additional

structure for the ob jects cohom(! ;!). In particular we nd examples of bialgebras and

como dule algebras. For this wehave to assume now that the category A is (quasi)symmetric.

We rst recall some de nitions ab out monoidal categories.

DEFINITION 2.9 (monoidal diagrams)

Let (D ;!) be a diagram in A. Assume that D is a monoidal category and that ! is a monoidal

functor. Then we cal l the diagram (D ;!) a monoidal diagram.

DEFINITION 2.10 (monoidal transformations)

Let (D ;!) and (D ;! ) be monoidal diagrams in A. Let C 2Abe an algebra. Anatural

transformation ' : ! ! C ! is cal led monoidal, if the diagrams

'(X ) '(Y )

0 0

! (X ) ! (Y ) C C ! (X ) ! (Y )

? ?

'(X Y )

! (X Y ) C ! (X Y )

and

K K K

? ?

'(I )

! (I ) C ! (I )

commute.

0 0

Let ! : D!A and ! : D !A b e diagrams in A.We de ne the tensor pro duct of

0 0 0 0 0 0

these diagrams by(D ;!) (D ;! )=(DD ;! ! )where(! ! )(X; Y )=! (X ) ! (Y ).

The tensor pro duct of two diagrams can b e viewed as the diagram consisting of all the tensor

pro ducts of all the ob jects of the rst diagram and all the ob jects of the second diagram;

similarly for the morphisms of the diagrams.

PROPOSITION 2.11 (the tensor pro duct of cohoms)

0 0 0 0

Let (D ;! ), (D ;! ), (D ;! ),and(D ;! ) be diagrams in A and let (D ;! ) and (D ;! )

1 1 2 2 1 2 1 2

1 2 1 2

be nite. Then

0 0 0 0

cohom(! ! ;! ! ) cohom(! ;! ) cohom(! ;! ):

1 2 1 2

Pro of: Similar to the pro of in [12 ] 2. 3. 6 we get

0 0

cohom(! ;! ) cohom(! ;! )

1 2

Z Z

X 2D Y 2D

1 2

0 0

( ! (X ) ! (X ) ) ( ! (Y ) ! (Y ) )

1 2

Z Z

Y 2D X 2D

2 1

0 0

(Y ) ) (X ) ! (Y ) ! (! (X ) !

2 1

(X;Y )2D D

1 2

0 0

(! (X ) ! (X ) ! (Y ) ! (Y ) )

1 2

(X;Y )2D D

1 2

0 0

(! ! )(X; Y ) (! ! )(X; Y )

1 2

0 0

cohom(! ! ;! ! ):

1 2

This uses the fact that colimits commute with tensor pro ducts in A.

If we lo ok at the representation of co end(! ) in De nition 2.2 then this isomorphism

0 0

identi es an element x y 2 cohom(! ;! ) cohom(! ;! ) with representative

1 2

element

0 0

(x ) (y ) 2 (! (X ) ! (X ) ) (! (Y ) ! (Y ) )

1 2

0 0

with the element (x y ) ( ) 2 cohom(! ! ;! ! ).

1 2

COROLLARY 2.12 (the universal prop erty of the tensor pro duct of cohoms)

Under the assumptions of Proposition 2.11 there is a natural transformation

0 0 0 0

: ! ! ! cohom(! ;! ) cohom(! ;! ) ! ! ;

1 2 1 2

0 0

such that for each object M 2Aand each natural transformation ' : ! ! ! M ! !

1 2

0 0

there is a unique morphism '~ : cohom(! ;! ) cohom(! ;! ) ! M such that the diagram

1 2

0 0 0 0

! ;! ) ! ;! ) cohom(! ! ! cohom(!

2 1 1 2

2 1 2 1

'~ 1

Pq ?

0 0

M ! !

1 2

commutes.

THEOREM 2.13 (cohom is an algebra)

0 0

Let (D ;!) and (D ;! ) be two monoidal diagrams in A and let (D ;! ) be nite. Then

0 0 0

cohom(! ;!) is an algebrainA and : ! ! cohom(! ;!) ! is monoidal.

Pro of: The multiplication on cohom(! ;!) results from the commutative diagram

0 0 0 0

! (X ) ! (Y ) cohom(! ;!) cohom(! ;!) ! (X ) ! (Y )

? ?

0 0

;!) ! (X Y ): ! (X Y ) cohom(!

and from an application of Prop osition 2.11.

Let D =(fI g; fidg) together with ! : D !Aand ! (I )=K b e the monoidal

0 0 0 0

"unit ob ject" diagram of the monoidal category Diag (A). Then (K ! K K )=(! !

co end(! ) ! )istheuniversal arrow. The unit on cohom(! ;!)isgiven by the commutative

0 0

diagram

K K K

? ?

0 0

! (I ) cohom(! ;!) ! (I ):

It is easy to verify the algebra laws and the additional claims of the theorem.

PROPOSITION 2.14 (the pro duct in the algebra cohom)

The product in cohom(! ;!) is given by

(x ) (y )=x y :

Pro of: We apply the diagram de ning the multiplication of cohom(! ;!) to an element x y

P P

x ) (y ) x y = x y x y , where for simplicityof and obtain (

i j i j i j i j

0 0 0

notation ! (X ) ! (Y )=! (X Y )and ! (X ) ! (Y )=! (X Y ) are identi ed and where

P P

0 0 0 0

x 2 ! (X ) ! (X ) and y 2 ! (Y ) ! (Y ) are dual bases. This implies the

i i j j

prop osition.

One easily proves the following corollaries

COROLLARY 2.15 ( is an algebra morphism)

Let the assumptions of Proposition2.4 be satis ed and let al l diagrams (D ;!), (D ;! ), and

(D ;! ) be monoidal. Then the morphism

00 0 00 0

: cohom(! ;!) ! cohom(! ;!) cohom(! ;! )

in 2.4 is an algebra homomorphism.

COROLLARY 2.16 (co end(! ) is a bialgebra)

0 0

Let (D ;!) and (D ;! ) be two monoidal diagrams in A and let (D ;! ) be nite. Then

0 0

co end(! ) and, if (D ;!) is nite, co end(! ) are bialgebras and cohom(! ;!) is a right

co end(! )-comodule algebra and a left co end (! )-comodule algebra.

COROLLARY 2.17 (monoidal diagram morphisms induce algebra morphisms for the co-

homs)

0 0 0 0

Let D and D be monoidal diagram schemes and let ! : D !Aand ! : D !Abe

monoidal diagrams. Let F : D!D be a monoidal functor. Then F induces algebra

0 0

homomorphism f : cohom(! F ;!F ) ! cohom(! ;!) which is compatible with the comulti-

plication on cohoms as described in 2.4. Furthermore F induces a bialgebra homomorphism

0 0

co end(! F ) ! co end(! ).

After having established the structure of an algebra on cohom(! ;!), wearenowinter-

ested in algebra homomorphisms from cohom(! ;!) to other algebras.

COROLLARY 2.18 (the universal prop erty of the algebra cohom)

0 0

Let (D ;!) and (D ;! ) be monoidal diagrams in A and let (D ;! ) be nite. Then thereis

0 0

a natural monoidal transformation : ! ! cohom(! ;!) ! , such that for each algebra

C 2A and each natural monoidal transformation ' : ! ! C ! there is a unique algebra

morphism f : cohom(! ;!) ! C such that the diagram

0 0

! cohom(! ;!) !

f 1

Hj ?

C !

commutes.

Pro of: The multiplication of cohom(! ;!)was given in the pro of of Theorem 2.13. Hence

we get the following commutative diagram

0 0 0 0

! (X ) ! (Y ) cohom(! ;!) cohom(! ;!) ! (X ) ! (Y )

@ P

@R Pq

0 0

C C ! (X ) ! (Y )

3 5

? ?

0 0

! (X Y ) cohom(! ;!) ! (X Y )

P @

? Pq @R

C ! (X Y )

where triangle 1 commutes by Corollary 2.12, square 2 commutes by the de nition of the

multiplication in Theorem 2.13, square 3 commutes since ' is a natural monoidal transfor-

;!). Thus bythe mation, and triangle 4 commutes by the universal prop erty of cohom(!

universal prop erty from Corollary 2.12 the square 5 also commutes so that the universal map

cohom(! ;!) ! C is compatible with the multiplication.

Weleave it to the reader to check that this map is also compatible with the unit, whence

it is an algebra homomorphism.

(!; C ! ) the set of natural monoidal transformations ' : ! ! We denote by Mor

C ! . It is easy to see that this is a functor in algebras C .Thus wehaveproved

COROLLARY 2.19 (cohom as a representing ob ject)

0 0

For diagrams (D ;!) and (D ;! ), with (D ;! ) nite, there is a natural isomorphism

0 0

Mor (!; M ! ) Hom(cohom(! ;!);M):

If the diagrams are monoidal then there is a natural isomorphism

0 0

Mor (!; C ! ) K -Alg(cohom(! ;!);C):

We end this section with an interesting observation ab out a particularly large "diagram"

namely the underlying functor ! : C omo d-C !A.Itsays that if all "representations" of a

coalgebra resp. bialgebra are known then the coalgebra resp. the bialgebra can b e recovered.

THEOREM 2.20 (Tannakaduality)

Let C beacoalgebra. Then C co end(! ) as coalgebras where ! : C omod-C !A is the

underlying functor. If C is a bialgebra and ! monoidal, then the coalgebras are isomorphic

as bialgebras.

For a pro of we refer to [9] Corollary 6. 4. and [10] Theorem 15 and Corollary 16.

3 Co endomorphism Bialgebras of Quantum Spaces

In this section we will establish the connection b etween the bialgebras arising as co endo-

morphism bialgebras of quantum spaces and those arising as co endomorphism bialgebras of

diagrams. It will turn out that the construction via diagrams is far more general. So by

Theorem 2.1 and Corollary 2.16 wehave established the general existence of these bialgebras.

And we can apply this to give an explicit description of co endomorphism bialgebras of nite

dimensional algebras, of ( nite) quadratic algebras, of families of quadratic algebras, and of

nite dimensional Lie algebras.

3.1 Generators and relations of the algebra of cohomomorphisms

We will construct monoidal diagrams generated by a given ( nite) family of ob jects, of

morphisms, and of commutativity conditions. Then we will calculate the asso ciated co endo-

morphism bialgebras. Since in a monoidal category there are two comp ositions, the tensor

pro duct and the comp osition of morphisms, we are constructing a free (partially de ned)

algebra from the ob jects, morphisms and relations.

Let X ;:::;X be a given set of ob jects. Then there is a free monoidal category

1 n

C [X ;:::;X ] generated bytheX ;:::;X constructed in an analogous way as the free

1 n 1 n

monoidal category on one generating ob ject in [3].

Let X ;:::;X be a given set of ob jects and let ' ;:::;' be additional morphisms

1 n 1 m

between the ob jects of the free monoidal category C [X ;:::;X ] generated by X ;:::;X .

1 n 1 n

Then there is a free monoidal category C [X ;:::;X ; ' ;:::;' ] generated by X ;:::;X ;

1 n 1 m 1 n

' ;:::;' .

1 m

If the ob jects X ;:::;X and morphisms ' ;:::;' are taken in A, then they induce

1 n 1 m

a unique monoidal functor ! : C [X ;:::;X ; ' ;:::;' ] !A.

1 n 1 m

We indicate howthevarious free monoidal categories can b e obtained. C [X ;:::;X ]is

1 n

generated as follows. The set of ob jects is given by

(O ) X ;:::;X are ob jects,

1 1 n

(O ) I is an ob ject,

(O )ifY and Y are ob jects then Y Y is an ob ject (actually this ob ject should b e written

3 1 2 1 2

as (Y Y )toavoid problems with the explicit asso ciativity conditions),

1 2

(O ) these are all ob jects.

The set of morphisms is given by

(M ) for each ob ject there is an identity morphism,

(M ) for each ob ject Y there are morphisms : I Y ! Y , : Y ! I Y , :

Y I ! Y ,and : Y ! Y I ,

(M ) for any three ob jects Y ;Y ;Y there are morphisms : Y (Y Y ) ! (Y Y ) Y

3 1 2 3 1 2 3 1 2 3

and :(Y Y ) Y ! Y (Y Y ),

1 2 3 1 2 3

(M )iff : Y ! Y and g : Y ! Y are morphisms then f g : Y Y ! Y Y is a

4 1 2 3 4 1 3 2 4

morphism,

(M )iff : Y ! Y and g : Y ! Y are morphisms then gf : Y ! Y is a morphism,

5 1 2 2 3 1 3

(M ) these are all morphisms.

The morphisms are sub ject to the following congruence conditions with resp ect to the com-

p osition and the tensor pro duct

(R ) the asso ciativity and unitary conditions of comp osition of morphisms,

(R ) the asso ciativity and unitary coherence condition for monoidal categories,

(R ) the conditions that and , and , and are inverses of each other.

The free monoidal category C [X ;:::;X ; ' ;:::;' ] for given ob jects X ;:::;X and

1 n 1 m 1 n

morphisms ' ;:::;' (where the ' ;:::;' are additional new morphisms b etween ob jects

1 m 1 m

of C [X ;:::;X ]) is obtained by adding the following to the list of conditions for generating

1 n

the set of morphisms

(M ) ' ;:::;' are morphisms.

6 1 m

If there are additional commutativity relations r ;:::;r for the morphisms expressed

1 k

by the ' , they can b e added to the de ning congruence relations to de ne the free monoidal

category C [X ;:::;X ; ' ;:::;' ; r ;:::;r ]by

1 n 1 m 1 k

(R ) r ;:::;r are in the congruence relation.

4 1 k

THEOREM 3.1 (the invariance of the co endomorphism bialgebra)

Let X ;:::;X be objects in A and let ' ;:::;' be morphisms in A between tensor

1 n 0 1 m 0

products of the objects X ;:::;X ;I.Let r ;:::;r berelations between the given morphisms

1 n 1 k

in A .De neD := C [X ;:::;X ; ' ;:::;' ] and D := C [X ;:::;X ; ' ;:::;' ; r ;:::;r ]

0 1 n 1 m 1 n 1 m 1 k

0 0

and let ! : D!Aand ! : D !Abe the corresponding underlying functors. Then

co end(! ) co end(! ) as bialgebras.

Pro of: follows immediately from Prop osition 2.8 applied to the functor F : D!D that

is the identity on the ob jects and that sends morphisms to their congruence classes so that

! = ! F .

LEMMA 3.2 (the generating set for cohom)

Let D = C [X ;:::;X ; ' ;:::;' ] be the freely generated monoidal category generatedby

1 n 1 m

the objects X ;:::;X and the morphisms ' ;:::;' .Let (D ;!) and (D ;! ) be monoidal

1 n 1 m

0 0

diagrams and let (D ;! ) be nite. Then cohom(! ;!) is generated as an algebra by the vector

spaces ! (X ) ! (X ) , i =1;:::;n.

i i

Pro of: The multiplication in cohom(! ;!)isgiven by taking tensor pro ducts of the repre-

sentatives as in Prop osition 2.14.

For ob jects X ;:::;X and morphisms ' ;:::;' generating a free monoidal category

1 n 1 m

wenow get a complete explicit description of the algebra cohom(! ;!) in terms of generators

and relations. This result resembles that of [12 ] Lemma 2.1.16 and will b e central for our

further studies.

THEOREM 3.3 (representation of the algebra cohom)

Let D = C [X ;:::;X ; ' ;:::;' ] be the freely generated monoidal category generatedby

1 n 1 m

the objects X ;:::;X and the morphisms ' ;:::;' .Let (D ;!) and (D ;! ) be monoidal

1 n 1 m

diagrams and let (D ;! ) be nite. Then

0 0

cohom(! ;!) T ( ! (X ) ! (X ) )=I

i i

! (X ) ! (X ) ) is the (free) tensor algebragenerated by the spaces ! (X ) where T (

i i i

! (X ) and where I is the two-sided ideal generated by the di erences of the images of the

' ;:::;' under the maps

1 m

a M

0 0 0

! (dom(' )) ! (co d (' )) ! (X ) ! (X ) T ( ! (X ) ! (X ) ):

i i i i

Pro of: The tensor algebra contains all the spaces of the form ! (X ) ! (X ), X 2D,

since the ob jects X are iterated tensor pro ducts of the generating ob jects X and since

the multiplication in cohom(! ;!) as describ ed in Prop osition 2.14 identi es the images of

0 0 0

(X ) ) ((! (Y ) ! (Y ) )with(! (X Y ) ! (X Y ) ). (! (X ) !

Assume that (f : X ! Y ) 2D is a morphism and that for all x 2 ! (X ) ! (Y )

the elements x ! (f ) ( ) ! (f )(x) are in I . Let Z 2D and apply f Z to x z 2

0 0 0

! (X Z ) ! (X Z ) ! (X ) ! (Y ) ! (Z ) ! (Z ) .Thenwehave

x z ! (f Z ) ( ) ! (f Z )(x z ) =

(x ! (f ) ( )) (z ) (! (f )(x) ) (z )=

(x ! (f ) ( ) ! (f (x)) ) (z ) 2 I:

Assume now that (f : X ! Y ); (g : Y ! Z ) 2Dare morphisms and that for all

0 0 0

x 2 ! (X ) ! (Y ) and all y 2 ! (Y ) ! (Z ) the elements x ! (f ) ( ) ! (f )(x)

and y ! (f ) ( ) ! (f )(y ) are in I .Thenwehave

x ! (gf ) ( ) ! (gf )(x) =

0 0 0

fx ! (f ) [! (g ) ( )] ! (f )(x) [! (g ) ( )]g +

f[! (f )(x)] ! (g ) ( ) ! (g )[! (f )(x)] g2 I:

3.2 Finite quantum spaces

Let D = C [X ; m; u] b e the free monoidal category generated by one ob ject X ,amultiplication

m : X X ! X and a unit u : I ! X . The ob jects of D are n-fold tensor pro ducts

X of X with itself. (Because of coherence theorems we can assume without of loss of

generality that wehave strict monoidal categories, i.e. that the asso ciativity morphisms and

the left and right unit morphisms are the identities.) Observethatwe do not require that m

is asso ciative or that u acts as a unit, since by Theorem 3.1 anysuch relations are irrelevant

for the construction of the co endomorphism bialgebras of diagrams over D .

Let A be in K -Alg. Let ! : D!A b e de ned by sending X to the ob ject A 2A,

and the multiplication and the unit in D to the multiplication resp. the unit of the algebra

A in A. Then ! is a monoidal functor. If A is nite dimensional, then the diagram (D ;! )

A A

is nite.

For a nite dimensional algebra A 2 K -Alg we can construct the co endomorphism

bialgebra E of A as in subsection 1.5. We also can consider the corresp onding diagram

(D ;! )inA and construct its co endomorphism bialgebra co end(! ).

A A

THEOREM 3.4 (cohomomorphisms of non-commutative spaces and of diagrams)

Let X and Y be quantum spaces with function algebras A = O (X ) and B = O (Y ).Let A be

nite dimensional. Then H(X ; Y ) exists with O (H(X ; Y )) = a(A; B ) cohom(! ;! ).

A B

Pro of: Given a quantum space Z and a map of quantum spaces Z?X !Y.LetC = O (Z ).

Then the map induces an algebra homomorphism f : B ! C A. We construct the

asso ciated diagrams (D ;! )and(D ;! ).

A B

We will shownow that there is a bijection b etween the algebra homomorphisms f :

B ! C A and the monoidal natural transformations ' : ! ! C ! .Given f we get

B A

' by

m 1

n n n n n n n

! C A = C ! (X ); '(X ):! (X )=B ! C A

A B

! C is the n-fold multiplication. This is a natural transformation since where m : C

n the diagrams

'(X X )

B B C A A

X :

f f

m 1

C C A A

m m

1 m

C A

: X

1 1

Xz ? ?

B C A

'(X )

and

'(I )

K C K

1 u

? ?

B C A

'(X )

commute. Furthermore the diagrams

r s

'(X ) '(X )

r s r s -

B B C C A A

X :

r s r s

C C A A

(r +s) (r +s)

C A

: X

? Xz ?

(r +s) (r +s)

B C A

(r +s)

'(X )

commute so that ' : ! ! C ! is a natural monoidal transformation.

B A

Conversely given a natural transformation ' : ! ! C ! we get a morphism

B A

f = '(X ):B ! C A. The diagrams

f f

B B C C A A

m 1

? ?

'(X X )

B B C A A

1 m

? ?

B C A

and

K K K

? ?

K C K

1 u

? ?

B C A

commute. Hence f : B ! C A is an algebra homomorphism. This de nes a natural

isomorphism K -Alg(B; C A) Mor (! ;C ! ). If A is nite dimensional, then the

B A

left side is represented by a(A; B ) and the right side bycohom(! ;! ) (2.19).

A B

COROLLARY 3.5 (isomorphic co endomorphism bialgebras)

co end(! ) of bialgebras such that the diagram There is a unique isomorphism E

A A

A E A

@R ?

co end(! ) A

commutes.

Pro of: If the co endomorphism bialgebra E exists, then it satis es the universal prop erty

given in Corollary 1.10.

COROLLARY 3.6 (Tambara [15 ] Thm.1.1)

Let A; B be algebras and let A be nite dimensional. Then

T (B A )=(xy x y ;(1) 1 jx; y 2 B ; 2 A ): a(A; B )

(1) (2) A

Pro of: This is an immediate consequence of Theorem 3.3 and Theorem 3.4. In fact the

map m : X X ! X induces by the construction in Theorem 2.1 the relations xy

x y for all x y 2 B B A . Similarly the map u : I ! X induces

(1) (2)

(1) 1 for all 1 2 K A .

3.3 Quadratic quantum spaces

Wenow consider quadratic algebras in the sense of Manin [5]. They are N-graded algebras A

with A = K , A generated by A with (homogeneous) relations generated by R A A .

0 1 1 1

The quadratic algebra corresp onding to (A ;R)is T (A )=(R)whereT (A ) is the tensor

1 1 1

algebra generated by A . The admissible algebra homomorphisms are those generated by

homomorphisms of vector spaces f : A ! B suchthat(f f )(R ) R . Thus we

1 1 A B

obtain the category QA of quadratic algebras. The dual QQ of this category is the category

of quadratic quantum spaces. We will simply denote quadratic algebras by(A; R) where we

assume R A A.

We consider the free monoidal category D = C [X; Y ; ] where : Y ! X X in

D . Then each quadratic algebra (A; R) induces a monoidal functor ! : D!A with

(A;R)

! (X )=A, ! (Y )=R,and! ( : Y ! X X )= : R ! A A.

For anytwo quadratic algebras (A; R)and(B; S), where A and R are nite dimensional,

we can construct the universal algebra cohom(! ;! ) satisfying Corollary 2.18. We

(A;R) (B;S)

show that this is the same algebra as the quantum homomorphism space hom((A; R); (B; S))

constructed by Manin [5] 4.4. that has the universal prop erty given in [5] 4. Theorem 5. In particular it is again a quadratic algebra.

THEOREM 3.7 (cohom of quadratic algebras)

Let (A; R) and (B; S) be quadratic algebras with (A; R) nite. Then

cohom(! ;! ) (B A ;S R );

(A;R) (B;S)

where R is the annihilator of R in (A A) = A A .

Pro of: By Theorem 3.3 the algebra cohom(! ;! ) is generated by the vector spaces

(A;R) (B;S)

B A and S R . It satis es the relations generated by the morphism : Y ! X X ,

which induces relations through the diagram

S R

S A A

B B A A :

0 0 0

Given an element s 2 S A A we get equivalent elements s ( )j s .

Since the map 1 is surjective, every elementin S R is equivalent to an elementin

B B A A so that we can disp ose of the generating set S R altogether. Furthermore

0 0

elements of the form s 2 B B A A are equivalent to zero if s 2 S and

induces the zero map on R that is if it is in R , so that the set of relations is induced by

S R .

Given an algebra C in A,we call an algebra homomorphism f :(A; R) ! C (B; S)

quadratic, if it satis es f (A) C B and (m B B )(f f )(R) C S that is if

R C S

? ?

f f

m 1

- -

A A C C B B C B B

commutes. The set of all quadratic homomorphisms from (A; R)toC (B; S) is denoted

q q

by K -Alg ((A; R);C (B; S)). Then one proves as in Theorem 3.4 that K -Alg ((A; R);C

Mor (B; S)) (! ;C ! ): Hence we get the following universal prop ertywhichis

(A;R) (B;S)

di erent from the one in Manin [5] 4. Theorem 5.

THEOREM 3.8 (universal prop erty of cohom for quadratic algebras)

Let (A; R) and (B; S) be quadratic algebras and let (A; R) be nite. Then there is a quadratic

algebra homomorphism :(A; R) ! cohom(! ;! ) (B; S) such that for every

(A;R) (B;S)

algebra C and every quadratic algebra homomorphism ' :(A; R) ! C (B; S) thereisa

unique algebra homomorphism '~ :cohom(! ;! ) ! C such that the diagram

(A;R) (B;S)

(A; R) cohom(! ;! ) (B; S)

(A;R) (B;S)

'~ 1

? Pq

C (B; S)

commutes.

3.4 Complete quadratic quantum spaces

The most interesting diagram for constructing como dule algebras over bialgebras is de ned

over the free monoidal category D = C [X; ] with : X X ! X X .If! : D!A is a

nite diagram over this diagram scheme with ! (X )=V and ! ()=f : V V ! V V ,

then we can de ne a quadratic algebra A := T (V )=(Im(f )). Let us furthermore de ne

(V;f )

B = B := co end(! ). Wesay that B is a bialgebrawithR-matrix.

(V;f )

LEMMA 3.9 (spaces for a bialgebra with R-matrix)

A is a B -comodule algebra.

(V;f ) (V;f )

Pro of: Certainly all vector spaces V are B -como dules and f is a como dule homo-

(V;f )

morphism. Thus Im(f ) V V is a sub como dule. Butthenonechecks easily that

A := T (V )=(Im(f )) is a como dule as well and that it is in fact a como dule algebra over

(V;f )

B .

(V;f )

Now the algebra A can b ecome very small, in fact degenerate, namely if f : V

(V;f )

V ! V V is bijective. Then A = K V where the multiplication on V is the zero map.

This happ ens in "most" cases, since for f to b e bijective it suces that det(f ) 6= 0. But the

next prop osition shows that even in the degenerate case we still haveroomtomove.

PROPOSITION 3.10 (change of the R-matrix)

Let (V; f : V V ! V V ) with V nite dimensional be given. Then for every 2 K we

have B = B .

(V;f )

(V;f id)

Pro of: By Theorem 3.3 weknowthatB = T (V V )=I .We lo ok at the relations. The

(V;f )

ideal I is generated by elements of the form

x y f ( ) f (x y )

with x; y ; 2 V; ; 2 V . But then the same ideal is also generated by elements of the form

x y (f id )( ) (f id)(x y ) ;

since the -terms simply cancel. Thus B = B .

(V;f )

(V;f id)

COROLLARY 3.11 (the sp ectrum of quantum spaces for an R-matrix)

Let (V; f : V V ! V V ) with V nite dimensional be given. Then for every 2 K the

algebra A is a B -comodule algebra. It is non-degenerate if and only of is an

(V;f )

(V;f id)

eigenvalue of f .

Pro of: Instead of changing f in the de nition of B bya multiple of the identity,wecan

(V;f )

as well change it in the de nition of A .

(V;f )

of como dule algebras. So for B wehave obtained a one-parameter family A

(V;f )

(V;f id)

Since V is nite dimensional and f has only nitely manyeigenvalues, all but nitely many

of these como dule algebras are degenerate by Lemma 3.9.

EXAMPLE 3.12 (two parameter quantum matrices)

Let us take V = Kx Ky two dimensional and f : V V ! V V given by the matrix

(with q; p 6=0)

0 1

1 0 0 0

B C

0 0 q 0

B C

R = B C :

1 1 1

@ A

0 p 1 q p 0

0 0 0 1

The bialgebra generated by this matrix is generated by the elements a = x ; b = x ; c =

y ; d = y , where ; is the dual basis to x; y . The relations are

1 1 1 1 1 1

ac = q ca; bd = q db; ad q cb = da qbc; ab = p ba; cd = p dc; ad p bc = da pcb:

From this follows qbc = pcb. This is the two parameter version of a quantum matrix bialgebra

constructed in [6] Chap. 4, 4.10. The matrix R has twoeigenvalues = 1 (of multiplicity

1 1

three) and = q p (of multiplicity one) which lead to algebras

yx) A = K hx; y i=(xy q

and

2 2

A = K hx; y i=(x ;y;xy+ py x):

These are the quantum plane with parameter q and the dual quantum plane with parameter

EXAMPLE 3.13 (two further quantum 2 2-matrices)

In we replace the generating matrix in previous example by

0 1

1 0 0 0

B C

0 0 q 0

B C

R = B C ;

@ A

0 1 1 q 0

0 1 q 1

then the relations for the corresp onding bialgebra are given by

1 1 1

ac = q ca; bd = q db; ad q cb = da qbc;

2 2

ba = ab + b ;cb+ cd + d = da db + dc; ad b(c d)=da (c + d)b:

The bialgebra coacts on the algebras

A = K hx; y i=(xy q yx)

and

2 2

A = K hx; y i=(x ;xy+ yx; xy y ):

Finally if wetake

1 0

1 0 0 0

C B

0 0 1 1

C B

C ; R = B

A @

0 1 0 1

0 1 1 0

then the bialgebra satis es the relations

2 2

ca = ac + c ;bc+ bd + d = da + db dc; ad c(b d)=da (b + d)c;

2 2

ba = ab + b ;cb+ cd + d = da db + dc; ad b(c d)=da (c + d)b:

The nondegenerate algebras it coacts on are

A = K hx; y i=(yx xy y )

and

2 2

A = K hx; y i=(x ;xy+ yx; xy y ):

The endomorphism f : V V ! V V mayhavemorethantwo eigenvalues thus

inducing more than two non-degenerate como dule algebras. In general the following holds

THEOREM 3.14 (realization of algebras as sp ectrum of quantum spaces)

Let V be a nite dimensional vector space and let (V; R ), i =1;:::;n be quadratic algebras.

Let ;:::; 2 K bepairwise distinct. Assume that therearesubspaces W V V V

1 n i i

L L

n k

such that V V = V and R = V W . Then there is a homomorphism

i i j i

i=1 j =1;j 6=i

f : V V ! V V such that the non-degenerate algebras for B areprecisely the (V; R )=

f i

(V; Im(f id) = A for al l i =1;:::;n.

(V;f id)

Pro of: We form the Jordan matrix

1 0

C B

R =

A @

with

0 1

B C

. B C

B C

. .

B C

. .

J =

. .

B C

B . C

@ A

having dim(V )entries and dim(W )entries 1. Then the induced homomorphism f :

i i i

V V ! V V with resp ect to a suitable basis through the W and V satis es R =

i i i

Im(f id).

3.5 Lie algebras

Similar to our considerations ab out nite quantum spaces, let D = C [X ; m] b e a free monoidal

category on an ob ject X with a multiplication m : X X ! X . Then every nite

dimensional Lie algebra g induces a diagram (D ;!)with! (X )=g and ! (m) the Lie

bracket. Essentially the same arguments as in Lemma 3.9 show that the bialgebra co end(! )

makes g a como dule Lie algebra (the Lie multiplication is a como dule homomorphism) and

its universal enveloping algebra a como dule bialgebra.

Again co end(! ) has a universal prop erty with resp ect to its coaction on g .Toshow

0 0

this let g and g b e Lie algebras. Wesay that a linear map f : g ! C g is multiplicative if the diagram

f f

0 0

g g C C g g

[ ; ] m [ ; ]

? ?

g C g

commutes. Then the set of all multiplicative maps Mult(g ;C g ) is certainly a functor in

C .

THEOREM 3.15 (the universal bialgebra coacting on a Lie algebra)

0 0 0

Let g and g be Lie algebras and let g be nite dimensional. Then Mult(g ;C g ) is a

representable functor with representing object cohom(! ;! ).

g g

) g is universal. In particular the multiplicative map g ! cohom(! ;!

g g

Pro of: It is analogous to the pro of of Theorem 3.4. In particular we get an isomorphism

Mult(g ;C g ) Mor (! ;C ! ).

g g

We compute now a concrete example of a universal bialgebra coacting on a Lie algebra.

EXAMPLE 3.16 (the universal bialgebra coacting on a three-dimensional Lie algebra of

upp er triangular matrices)

Let g = K fx; y ; z g b e the three dimensional Lie algebra with basis x; y ; z and Lie bracket

[x; y ] = 0 and [x; z ]= [z; y]= z . Then g is a Lie coalgebra with dual basis ; ; and

cobracket ( )=( ) = 0 and ( )= ( ) ( ). By Theorem 3.3 the

universal bialgebra co end (! ) for the diagram induced by g is generated by the elements of

the matrix

1 1 0 0

x x x a b c

C C B B

: y y y = d e f M =

A A @ @

z z z g h i

The comultiplication is describ ed in the text after Corollary 2.6 and is given by(M )=

M M . With some straightforward computations one gets the relations as

(a b)c = c(a b); (d e)f = f (d e); (a b)f = c(d e);f(a b)= (d e)c;

(a b)i = i(a b)=0; (d e)i = i(d e)= 0;g=0;h=0:

EXAMPLE 3.17 (the universal bialgebra coacting on sl (2))

Let g = K fx; y ; z g = sl (2) b e the three dimensional Lie algebra with basis x; y ; z and Lie

bracket [x; y ]= z ,[z; x]= x and [y; z]=y .Theng is a Lie coalgebra with dual basis ; ;

and cobracket ( )= ,( )= ,and( )= .

The universal bialgebra co end(! ) for the diagram induced by g again is generated bythe

elements of the matrix

0 1 0 1

a b c x x x

B C B C

M = d e f = y y y :

@ A @ A

g h i z z z

The comultiplication is given by(M )=M M . Here one gets the relations as

ab = ba; ac = ca; bc = cb; de = ed; df = fd; ef = fe; gh = hg ; g i = ig ; hi = ih;

a = ai cg = ia gc; b = ch bi = hc ib; c = bg ah = gb ha;

d = fg di = gf id; e = ei fh = ie hf ; f = dh eg = hd ge;

g = cd af = dc fa; h = bf ce = fb ec; i = ae db = ea bd:

4 Automorphisms and Hopf Algebras

In this last short section wewant to extend our techniques of using diagrams for determining

bialgebras to Hopf algebras.

Hopf algebras arise as function rings of ane algebraic groups which act as group of

automorphisms on algebraic varieties. The problem in non-commutative geometry is that the

de nition of an automorphism group is not that clear. So one de nes the coautomorphism

Hopf algebra of a space X to b e the Hopf envelop e of the co endomorphism bialgebra of X .

The construction of the Hopf envelop e H of a bialgebra B was given in a pap er of Takeuchi

[14].

Hopf algebras also arise as co endomorphism algebras of rigid diagrams [17]. So we rst

study some prop erties of rigid monoidal categories and then show that coautomorphism Hopf

algebras can also b e obtained from diagrams. One of the main theorems in this context is

THEOREM 4.1 (co endomorphism Hopf algebras of rigid diagrams)

(a) Let H be a Hopf algebra. Then the category of right H-comodules that are nite di-

mensional as vector spaces, is rigid, and the underlying functor ! : C omod-H !Vecis

monoidal and preserves dual objects (up to isomorphism).

(b) Let (D ;!) be a nite monoidal diagram and let D be rigid. Then the coendomorphism

bialgebra co end (! ) is a Hopf algebra.

Pro of: [12 ] and [16 ].

LEMMA 4.2 (the rigidization of a monoidal category)

Let D be a smal l monoidal category. Then there exists a unique (left-)rigidization (up to

isomorphism), i.e. a (left-)rigid smal l monoidal category D and a monoidal functor :

D!D such that for every (left-) rigid smal l monoidal category E and monoidal functor

: D!E there is a unique monoidal functor : D !E such that

D D

? @R

commutes.

Pro of: The construction follows essentially the same way as the construction of a free

monoidal category over a given ( nite) set of ob jects and morphisms in section 3.1.

COROLLARY 4.3 (the rigidization of a diagram)

Each nite monoidal diagram ! : D!A has a unique rigidization ! : D !A such

that ! = ! .

Pro of: A nite monoidal diagram is by de nition a monoidal diagram in the rigid category

A . 0

PROPOSITION 4.4 (extending monoidal transformations to the rigidization)

Given a nite monoidal diagram (D ;!) and its (left-)rigidization (D ;! ). Given a (left-

)Hopf algebra K and a monoidal natural transformation f : ! ! K ! . Then there

is a unique extension g : ! ! K ! , a monoidal natural transformation, such that

(g : ! ! K ! )=(f : ! ! K ! ).

Pro of: For X 2Dlet X 2D b e its dual. Let M := ! (X ), M := ! (X )andev:

M M ! K be the evaluation. M is a K -como dule. We de ne a como dule structure

on M . Since M is nite-dimensional (or more generally has a left dual in A) there is a

canonical isomorphism M K Hom(M; K ). Then (m ) 2 M K is de ned by

X X

m (m) m := m (m ) S (m ):

(0) (1)

It is tedious but straightforward to check that this is a right como dule structure on M .We

check the coasso ciativity:

X X

m (m) S (m ) S (m ) = m (m ) m S (m )

(0) (1)

(0)(0) (0)(1) (1) (0) (1)

= m (m ) m m

(0)(0) (1) (0)(1)

= m (m ) (m )

(0) (1)

= m (m) (S (m ))

(0) (1)

= m (m) S (m ) S (m ):

(0) (1) (2)

Then the evaluation ev: M M ! k satis es

X X

m (m ) m m = m (m ) S (m )m = m (m) 1

(0) (1) (0) (1) (2) K

(0) (1) (0)

hence it is a como dule homomorphism.

We need a unique K -como dule structure on M such that ev b ecomes a como dule

homomorphism. Let M havesuch a como dule structure with (m )= m m and

(0) (1)

let m 2 M with (m)= m m then weget

(0) (1)

m = m (m) 1 (m ) m m

(1) (0)

from the fact that ev is a como dule homomorphism. Hence we get

X X

m (m) m = m (m ) m "(m )

(0) (1)

(0) (1) (0) (1)

= m (m ) m m S (m )

(0) (1) (2)

(0) (1)

= m (m ) S (m ):

(0) (1)

But this is precisely the induced como dule-structure on M by the antipode S of K given

ab ove.

For iterated duals and tensor pro ducts of ob jects the K -como dule structure arises from

iterating the pro cess given ab ove resp. from using the multiplication of K to give the tensor

pro duct a como dule structure.

THEOREM 4.5 (the Hopf envelop e of a co endomorphism bialgebra)

Given a monoidal diagram (D ;!) and its rigidization (D ;! ).Let B := co end(! ), a bial-

), a Hopf algebra. Then there is a bialgebra homomorphism gebra, and H := co end(!

: B ! H such that for every Hopf algebra K and every bialgebra homomorphism

f : B ! K there is a unique bialgebra homomorphism g : H ! K such that

B H

? @R

commutes.

Pro of: The monoidal natural transformation ! ! H ! induces a unique homomorphism

: B ! H such that

! = ! B !

@R ?

H !

The homomorphism f : B ! K induces a monoidal natural transformation (f 1) :

! ! B ! ! K ! , whichmay b e extended to ! ! K ! . Hence there is a unique

g : H ! K such that

! H !

g 1

@R ?

K !

commutes. Then the following diagrams commute

! B !

J Z

? Z~

f 1

H !

g 1

? ? J^

K !

and

B H

? @R

We close with a few observations and applications of this result.

Every bialgebra B is the co endomorphism bialgebra of the diagram of all nite-dimen-

sional B -como dules by [10]. So the construction with diagrams given ab ove gives the Hopf

envelop e of an arbitrary bialgebra.

If a nite-dimensional non-commutative algebra A is given then the rigidization of the

nite monoidal diagram ! : D!A with D := C [X ; m; u] generated by A as in 3.2 has a

co endomorphism bialgebra H whichistheHopfenvelop e of the co endomorphism bialgebra

of A. A similar remark holds for quadratic algebras.

In the category of quantum spaces the endomorphism quantum space E of a quantum

space X can b e restricted to the "automorphism" quantum space A by using the homomor-

phism from the representing bialgebra of E to its Hopf envelop e. Thus A acts also on X .

The existence and the construction in the relevant cases can b e obtained from asso ciated

rigid monoidal diagrams as in the ab ove theorem.

References

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Preprint 1991.

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