Coalgebra extensions and algebra coextensions of galois type

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Coalgebra extensions and algebracoextensions of galois typeTomasz Brzeziński a & Piotr M. Hajac b

a Department of Mathematics , University of York , Heslington, York, YOl5DD, U.K E-mail:b International School for Advanced Studies , Via Beirut 2-4, Trieste,34013, Italy E-mail:Published online: 27 Jun 2007.

To cite this article: Tomasz Brzeziński & Piotr M. Hajac (1999) Coalgebra extensions andalgebra coextensions of galois type, Communications in Algebra, 27:3, 1347-1367, DOI:10.1080/00927879908826498

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COMMUNICATIONS IN ALGEBRA, 27(3), 1347-1367 (1999)

COALGEBRA EXTENSIONS AND ALGEBRA COEXTENSIONS OF GALOIS TYPE

TOMASZ BRZEZINSKI~

Department of Mathematics, University of York,

Heslington, York YO1 5DD, U.K.

PIOTR M. HAJAC'

International School for Advanced Studies,

Via Beirut 2-4, 34013 Trieste, Italy.

Abstract

The notion of a coalgebra-Galois extension is defined as a natural generalisation of a Hopf-Galois extension. It is shown that any coalgebra- Galois extension induces a unique entwining map $ compatible with the right coaction. For the dual notion of an algebra-Galois coextension it is also proven that there always exists a unique entwining structure compatible with the right action.

1 Introduction

Hopf-Galois extensions can be viewed as non-commutative torsors or principal

bundles with universal differential structure. From the latter point of view a

'Lloyd's fellow. On leave from: Department of Theoretical Physics, University of L6di, Pomorska 1491153, 90-236 L6di, Poland. E-MAIL: [email protected]

'On leave from: Department of Mathematical Methods in Physics, Warsaw University, ul. Hoia 74, Warsaw, 00-682 Poland. http://info.fuw.edu.pl/KMMF/ludzie~ang.html E-MAIL: [email protected]

Copyr~ght C' 1999 hy Marcel Dekker, Inc

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1348 BRZEZINSKI AND HAJAC

quantum group gauge theory was introduced in [4] and developed in [9, 61. It

turns out that to develop gauge theory on quantum homogeneous spaces (e.g.,

the family of Podlei quantum spheres) or with braided groups (see [ll] for a

recent review), one needs to consider coalgebra bundles. Such a generalisation

of gauge theory was proposed recently in [5]. It introduces the notions of an en-

twining structure with an entwining map $, and a $-principal coalgebra bundle.

The latter can be viewed as a generalisation of a Hopf-Galois extension.

In the present paper we introduce the notion of a coalgebra-Galois extension

for arbitrary coalgebras as a natural generalisation of a Hopf-Galois extension.

It is obtained by giving up the condition that the coaction be an algebra map.

Our main result is that such defined coalgebra-Galois extension always induces a

unique compatible entwining map $. We dualise coalgebra-Galois extensions and

t,hus introduce the notion of an algebra-Galois coextension. We then prove the

dual version of the main result. Finally, we prove that if two quotient coalgebras

C/I1 and C/12 cogenerate (in the sense of Definition 5.1) the coalgebra C , then

the coinvariants of C are the same as the intersection of the coinvariants of C/I1

with the coinvariants of C / I z .

Notation. Here and below k denotes a field. All algebras are over k , asso-

ciative and unital with the unit denoted by 1. All algebra homomorphisms are

assumed to be unital. We use the standard algebra and coalgebra notation, i.e.,

A is a coproduct, m is a product, E is a counit, etc. The identity map from the

space 5' to it,self is also denoted by V. The unadorned tensor product stands for

the tensor product over k . To abbreviate notation, we identify the tensor prod-

ucts k@V and V @ k with V. For a k-algebra A we denote by MA, AM and *MA the category of right A-modules, left A-modules and A-bimodules respectively.

Similarly, for a k-coalgebra C we denote by MC, 'M and 'MC the category of

right C-comodules, left C-comodules and C-bicomodules respectively. Also, by

AMc (CMA) we denote the category of left (right) A-modules with the action

(p l I ) and right (left) C-comodules with the coaction A" (vA) such that

Arr 0 V p = (17p @ C ) 0 ('4 @ A") ("A 0 p~ = ( C 8 pv) 0 ("A 8 A)), i.e., Av ("A)

is right (left) A-linear. For coactions and coproducts we use Sweedler's notation

with suppressed summation sign: AA(u) = a(0) @ a ( l ) , A(c) = c(l) 8 c(2) .

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COALGEBRA-GALOIS EXTENSIONS 1349

2 C-Galois extensions

First recall the definition of a Hopf-Galois extension (see [12] for a review).

Definition 2.1 Let H be a Hopf algebra, A be a right H-comodule algebra, and B := ACOH .- .- { a E A I AA a = a @ 1). W e say that A is a ( ~ i g h t ) Hopf-Galois

extension (or H-Galois extension) of B iff the canonical left A-module right

H-comodule map can := (m @ H) o ( A BE AA) : A @B A + A B H is bzjective.

Note that in the situation of Definition 2.1, both A € 3 ~ A and A B H are

objects in AMH via the maps m A , A BE AA and m @ A , A B A respectively.

The canonical map can is a morphism in this category. Thus the extension

B A is Hopf-Galois if A A 2 A @ H as objects in AM^ by the canonical

map can.

It has recently been observed in [2] that one can view quantum embeddable

homogeneous spaces B of a Hopf algebra H , such as the family (indexed by

c E [0, co]) of quantum two-spheres of Podlea [13], as extensions by a coalgebra

C. It is known (see p.200 in [13]) that, except for the North Pole sphere ( c = O),

no other spheres of this family are quantum quotient spaces of SUq(2). They

escape the standard Hopf-Galois description. To include this important case

in Galois extension theory, one needs to generalise the notion of a Hopf-Galois

extension to the case of an algebra extended to an algebra by a coalgebra (cf. [7] for the dual picture). This is obtained by weakening the requirement that AA

be an algebra map, and leads to the notion of a coalgebra-Galois extension. (A

special case of this kind was considered in [17, p.2911.)

To formulate the definition of a coalgebra-Galois extension, first we need a

general concept of coinvariants:

Definition 2.2 ([19]) Let A be an algebra and a right C-comodule. Then

is a subalgebra of A . W e call i t the subalgebra of (right) coinvariants.

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1350 BRZEZINSKI AND H N A C

Observe that when AA is an algebra map, the above definition coincides with

the usual definition of coinvariants as elements b of A such that &(b) = b@

1. Definition 2.2 does not require the existence of a group-like element in the

coalgebra. This allows one to define coalgebra-Galois extension for arbitrary

coalgebras.

Definition 2.3 Let C be a coalgebra, A a n algebra and a right C-comodule and

let B = ACoC. W e say that A is a (right) coalgebra-Galois extension (or C-

Galois extension) of B iff the canonical left A-module right C-comodule m a p

can := ( m @ C ) o (A gB AA) : A BB A -+ A @ C is bijective.

In what follows, we will consider only right coalgebra-Galois extensions, and

skip writing "right" for brevity. The conditions of Definition 2.3 suffice to make

both A BB A and A @ C objects in AMC via the maps m @ B A, A @ B AA and

m @ A, A @ A, respectively. The canonical map is again a morphism in *M'.

The extension B A is C-Galois if can is an isomorphism in A M C .

By the reasoning as in the proof of Proposition 1.6 in [9], one can obtain an

alternative (differential) definition of a coalgebra-Galois extension:

Proposition 2.4 Let C and B C A be as above. Let R I A := Ker m denote the

tmi7iersal dzfferential calculus o n A, and Ct := K e r ~ the augmentation ideal of

C . The algebra A is a C-Galois extension if and only if the following sequence

of left A-modules is exact:

where call := (m @ C ) 0 (A @ AA).

In the Hopf-Galois case. one can generalise the above sequence to a non-universal

differential calculus in a straightforward manner:

where RH is the adR-invariant right ideal of the Hopf algebra H defining a bi-

covariant calculus on H [21], and 2 is defined by a formula fully analogous to

the formula for can. Sequence (2.2) is a starting point of the quantum-group

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gauge theory proposed in [4] and continued in [9]. The Hopf-Galois extension

describes a quantum principal bundle with the universal differential calculus.

Proposition 2.4 shows that the C-Galois extension can also be viewed as a

generalisation of such a bundle, a principal coalgebra bundle. The theory of

coalgebra bundles and connections on them (also for non-universal differential

calculus) was developed in [5]. More specifically, the theory considered in [5]

uses the notion of an entwining structure (closely connected with the theory

of factorisation of algebras considered in [lo]) and identifies coalgebra principal

bundles with C-Galois extensions constructed within this entwining structure.

The aim of this section is to show that to each C-Galois extension of Defini-

tion 2.3 there corresponds a natural entwining structure. Therefore the notions

of a C-Galois extension of Definition 2.3 and a $-principal coalgebra bundle of

[5, Proposition 2.21 are equivalent to each other provided that there exists a

group-like e E C such that A A ( l ) = 1 @ e. First we recall the definition of an

entwining structure.

Definition 2.5 Let C be a coalgebra, A a n algebra and let $ be a k-linear m a p

$ : C @ A -+ A @ C such that

where q i s the unit map q : a ct al . T h e n C and .4 are said to be entwined by

$ and the triple (A, C, $) i s called a n entwining structure.

Entwining structures can be also understood as follows. Given an algebra A and

a coalgebra C we consider A @ C as an object in AM with the structure map

m @ C . Similarly we consider C @ A as an object in CM via the map A @ A.

Then we have (see also [14, Theorem 5.21)

Proposition 2.6 Let p ~ @ c : A @ C @ A -+ A @ C and : C @ A -+ C @ A @ C be the maps mat ing A @ C a n object i n AMA and C @ '4 a n object

i n 'Me correspondingly, and such that

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Then the pazrs ( ~ 4 ~ ~ ~ AcBA) of such compatzble maps are zn one-to-one cor-

respondence wzth the entwznzng structures (A. C, $).

Proof First assume that C @ -4 E CMC and A @ C E AMA Then

Furthermore

Therefore $ satisfies conditions (2.3). Dualising the above calculation (namely,

inkrchanging A with m, C with A, E with q and AcNA with pilBC) one eas-

ily finds that @ satisfies conditions (2.4) too. Hence (A, C, T$) is an entwining

structure.

Conl,e~sely, let (A, C, $1) be an entwining structure. Define AcB A = ( C @ $) 0

(A @ .4) and ,u.~ c = (m @ C ) 0 (A @ $) . Then

( c ~ A @ n ) 0 A C , , ~

= ( C @ A @ A ) 0 ( C @ W ) o ( A @ A )

= ( c @ $ @ C ) 0 ( C @ C @ $ ) 0 ( C @ A @ A ) o ( a @ A )

= (c@$@C) o ( C @ C @ Q ) 0 ( A @ C @ A ) o ( A @ A )

= (C@y,@C)o(A@A@C)o(CB$)o(A@A)

= ( A c @ A @ ~ ) O ~ C @ A .

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We used property (2.4) to derive the second equality, and t,hen coassociativitv

of the coproduct tto derive the third one. Similarly,

Hence AcBA is a right coaction. By dualising the above argument one verifies

that p ~ ~ c is a right action. Finally, an elementary calculation shows that

Thus the bijective correspondence is established. UTo each

entwining structure (A, C , $) one can associate the category Ms($) of right

(A, C, $)-modules, introduced and studied in [3]. The objects of Mz(2I,) are

right A-modules and right C-comodules I: such that for all v E 1/ and a E 4;

The morphisms in M z ( $ ) are right A-module right C-comodule maps. Some

important categories well-studied in the Hopf algebra theory, such as the cate-

gory of right ('4, H)-Hopf modules for a right W-module algebra A [8], or the

category of right-right Yetter-Drinfeld modules [22] can be seen as examples of

MZ ($).

The main result of the paper is contained in the following:

Theorem 2.7 Let A be a C-Galois extension of B. Then there exists a unique

mop $ : C @ A -+ A @ C entwining C ,with A and such that A E Mz(yj) with

the structure maps m and AA. (The map $ is called the canonical entwining

map associated to the C-Galois extewion B C A , )

Proof Assume that B 5 A is a coalgebra-Galois extension. Then can is bijective

and there exists the translatzon map T : C -+ A @B A, r (c) := can-'(1 @ c ) . ii'e

use the notation r (c) = c(') @ d 2 ) (summation understood). Using [I. Propo-

sition 3.91 or [16, Remark 3.41 and their obvious generalisation to the present

case, for all c E C and a 6 A, one obtains that

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BRZEZINSKI AND HAJAC

Define a map $ : C @ A -+ A @ C by

]Ye show that T) entwines C and A. By definition of the translation map, we

Furthermore. by property (i) of the translation map,

Thus we have proven that the second equations in (2.3)-(2.4) hold. Next we

have

~vhere wr used the property (ii) of the translation map to derive the third equal-

ity. Hence the first of equations (2.3) holds. Similarly,

LYP used property (iii) of the translation map to derive the third equality and

thrn property (ii) to derive the fourth one. Hence C and A are entwined by $J

as required.

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COALGEBRA-GALOIS EXTENSIONS

Now, using (ii), we have for all a, a' E A

ie. A is an (A, C , $)-module with structure maps m and AA. It remains to

prove the uniqueness of the entwining map $ given by (2.5). Suppose that t.here

is an entwining map 4 such that .4 E M:($) with structure maps rn and AA.

Then, for all a E A, c E C,

where we used the definition of the translation map to obtain the last equality.

0

3 A-Galois coextensions

The dual version of a Hopf-Galois extension can be viewed as a non-commutative

generalisation of the theory of quotients of formal schemes under free actions of

formal group schemes (cf. [15]). In this section we dualise C-Galois extensions

and derive results analogous to the results discussed in the previous section.

First recall the definition of a cotensor product. Let B be a coalgebra and

M, N a right and left B-comodule respectively. The cotensor product M O B N is defined by the exact sequence

where ! is the coaction equalising map ! = & €3 N - M @ ,A. In particular,

if C is a coalgebra. I its coideal and B = C / I , then

where .ir : C -+ B = C/I is the canonical surjection. The followirig definition

[18, p.33461 dualises the concept of a Hopf-Galois extension:

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1356 BRZEZINSKI AND H N A C

Definition 3.1 Let H be a Hopf algebra, C a right N-module coalgebra with

thx action : C @ H -+ C . Then I := {pc(c, h) - ~ ( h ) c / c E C, h E H )

is a coideal in C and thus B := C / I is a coalgebra. We say that C -H B is a

(right) Hopf-Galois coextension (or H-Galois coextension) zff the canonical left

C-cornodule right H-module map cocan := (C@pc) o (A @ H ) : C@H -+ COBC

is a bijection.

IVith the help of the property A o p c = (pc @ p c ) o ( C @ flip @ H ) 0 (A @ A),

it can be directly checked that the image of the map cocan is indeed contained

in COBC. To see more clearly that Definition 3.1 is obtained by dualising

Definition 2.1, one can notice that both C @ H and COBC are objects in 'MH, which is dual to A M H . The structure maps are A @ C , C @ m and AOBC,

Cl?B/~c respectively. The canonical map cocan is a morphisrn in ' M H . The

coextension C ii B is Hopf-Galois if C @ H COBC as objects in 'MH by the

canonical map cocan.

The notion of a Hopf-Galois coextension can be generalised by replacing H

by an algebra A and weakening the condition that the action p c is a coalgebra

map. This generalisation dualises the construction of a C-Galois extension of

the previous section. First we prove

Lemma 3.2 Let A be an algebra and C a coalgebra and right A-module with an

uction pc : C @ A -t C. Then the space

is u coideal of C .

Proof. Let I denote the space defined in Lemma 3.2. We prove the lemma

by showing that D := {,8 E Hom(C, k) I P ( I ) = 0) is a subalgebra of the

convolution algebra Hom(C, k ) (see [20. Proposition 1.4.6 c)]). Note first that

6 E D. Furthermore,

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If 4 , , p, E D, then for any a, c, n we have

Hence pl * p2 E D, and D is a subalgebra of Hom(C, k ) , as needed. I?

Lemma 3.3 Let C and A be as above and let I denote the coideal of Lemma 3.2.

Put B := C / I , and denote b y .ir : C -t C / I the canonical surjection. Then the

action , L L ~ is left B-colinear, i.e., (T@C) o A o pc = (B@pc) o ((.ir@C)oA @ A ) ,

and ( ( C @ ~ C ) O ( A @ A))(C @ A) C COBC.

Proof: Choose a E A, c E C . The above B-colinearity condition can be writ-

ten as T(;LC(C, a)(l)) @ pc(c, a)(?) = .ir(c(l)) @ ,uc(c(~), a) It is equivalent to the

condition

which can be written as

This proves our first assertion.

As for the second assertion, observe that it can be stated as

This is true by the left B-colinearity argument applied to the last two tensorands.

0

Now we can conclude that we have a well-defined map

and can consider:

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Definition 3.4 Let A be an algebra, C a coalgebra and right A-module, and

B = C / I , where I is the coideal of Lemma 3.2. W e say that C is a (right)

algebra-Galois coextension (or A-Galois coextension) of B i f f the canonical left

C-cornodule right A-module map cocan := (C@pC)o(A @ A) : C @ A -+ C O B C

is bijective.

Again, to see that Definition 3.4 dualises the notion of a C-Galois extension one

can notice that both C @ A and CUnC are objects in 'MA, which is dual to

The struct2ure maps are A @ C, C @ m and AOBC, C U B ~ C , respectively.

The canonical map cocan is a morphism in CMA. The right coextension C +

B is A-Galois if C @ A 2 C U B C as objects in 'MA by the canonical map

cocnn. (In what follows, we consider only right coextensions and omit "right"

for brevity.)

\'cry much as in the previous section, it turns out that every .4-Galois ex-

tension is equipped with an entwining structure. More precisely, we have the

following dual version of Theorem 2.7:

Theorem 3.5 Let C be a n A-Galois coextension of B . T h e n there exists a

uniqr~e map 11! : C @ A -+ A @ C entwining C with A and such that C E M:(d)) with the structure maps 4 and pc. (The map $ is called the canonical entwining

map associated to the A-Galois coextension C -H B.)

Proof \Ve dualise the proof of Theorem 2.7. Assume that C -H B is an A-

Galois roextension. Then cocan is a bijection and there exists the cotranslation

rrmp i : COBC -+ A, i := ( ~ @ , 4 ) o cocan-I. By dualising properties of the

translat,iorl map (or directly from the definition of i), one can establish the

following properties of the cotranslation map:

(i) i o A = q o ~ ,

It follows from (iii) that

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Consequently, we can conclude from (ii) that

(Observe that ( C @ h @ C ) ( C O B C O B C ) 5 C ~ B C @ COBC. Here and below one

has to pay attention to the domains of certain mappings.)

Using the cotranslation map and noticing that ( C @ A ) (CUBC) 2 COBC@C.

one defines a map y : C @ A -+ A Q C by

We now show that 10 entwines C and A. For any c E C we find

where we used property ( i ) to derive the last equality. Furthermore,

( A @ & ) o @ = ( ( E B A ) o cocan-' @ E ) o ( C @ A) o cocan

= (€@.A) 0 cocan-' 0 cocan = &@A.

Thus we have proven that the second conditions of (2.3) and (2.4) are fulfilled

by II,. To prove the first of equations (2.3). we compute

( m @ C ) o ( A @ * ) o ( ! b @ A )

= (mo(A@i) @ C ) 0 ( A @ C @ A) 0 (i @ cocan) o ( C @ A 8 A ) o (cocan @ A )

= ( m o ( i @ i ) @ C ) 0 (CB3 8 A) 0 ( C B 2 @ c o a n ) o ( C @ A 8 A) 0 (cocan @ A )

= ( m o ( i @ i ) @ C ) o ( C a 3 @ A) 0 (C @ ( C @ c o c a n ) o ( A @ A ) ) o (cocan @ A )

= ( m o (i@i) @ C ) 0 (CB3 @ A) 0 (C @ ( A @ C ) ococan) o (cor,un @ 4 )

= ( m o ( i @ i ) o ( C @ A @ C ) @ C ) 0 (CM2 @ A ) o ( C @ cocan) o (cocan @ A ) .

Taking advantage of (3.6), we obtain

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BRZEZINSKI AND HMAC

On the other hand, for any c E C, a E A, we have

( ( E @ C B 2 ) 0 ( C @ A) 0 cocan)(c @ a ) = pc(c , a ) ( l ) @ pc(c , a ) ( q

= ( A 0 p c ) ( c @ a ) . (3.8)

Conlhining this with (3.7) yields

as desired. Here we used the property t,hat pc is an action to derive the penul-

t,irnate equality. To prove the first of equations (2.4), first we observe that

(A @ C ) o (cocnn) = ( C @ cocan) o ( A @ A). (3.9)

Hence we obtain

To finish the calculation, we note that ( C @ i ) o (A @ C ) = cocan-'. Indeed,

thanks to (3 .9 ) , we have

(C g~ i ) o ( A @ C ) o cocan = ( C @ i ) 0 (C @ cocan) (A €9 A )

= ( C @ E @ A ) ~ ( A @ A )

= C @ A .

Hence

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as needed. Thus we have proved that (A, C, $) is an entwining structure.

The next step is to show that C E M:($), i.e., A o pc = ( p c @ C) o (C @

$) 0 (A @ .A). With the help of (3.9), property (ii) of the cotranslation map i

and then (3.8), we compute:

(PC @ C) 0 (C @ $) 0 (A @ A)

=(p~@C)o(C@i@C)o(C'~@A)o(C@cocan)o([email protected])

= (pCo(c@i) C) 0 (CB2 A) 0 (A, C) 0 cocan

= ( p c o ( C @ i ) o ( ~ @ ~ ) @ C) o (C a) o cocan

= ( E @ cg2) 0 (C @ a) c ~ C O n

=aop, . As for the uniqueness of 1/), suppose that there exists another entwining map 711

such that C E MZ(4). Then

4 Galois entwining structures

Theorem 2.7 allows one to riew a $-principal bundle as a coalgebra-Galois es-

tension. More precisely, recall from [5] the following

Definition 4.1 Let (A,C,d]) be an entwining structure, and let e E C be a

group-like element. Then B := {b E A 1 $(e 8 b) = b @ e ) is an algebm,

and rue say that A(B, C: v > e) is u coalgebra $-principal bundle iff the map

can$ : A BB A -t A @ C, a @ a' ct n$(e 8 a') is bijective.

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Proposition 4.2 For a given entwining structure (A , C, $'I) and a group-like

e E C? the following statements are equivalent:

(1) .4(B, C, $, e) is a coalgebra $-principal bundle.

(2) Th,ere exists a unique coaction LA : A -+ A @ C such that B C A is a

coalgebra-Galois extension of B by C , 7 ) is the canonical entwining map, and

A,4(l) = 1 €3 e .

Proof ( 1 ) =+ ( 2 ) . By [5, Proposition 2.21, A is a right C-comodule with the

coaction A,4 : a t+ $(e @ a). Using the second of conditions (2 .3) one verifies

that Aj l ( l ) = 1 @ e. The first of conditions (2 .3 ) implies that A E M z ( $ ) with

tlir structure maps AA and m,. Therefore, by [3, Lemma 3.31, B as defined in

Definition 4.1 is identical with the set {b E A / V a 6 A. A A ( b a ) = b A A ( a ) )

considered in Definition 2.2. Hence can = can* and B A is a C-Galois ex-

tension. By Theorem 2.7, $ is the associated canonical entwining map. Assume

noJv that there exists another coaction A; satisfying condition ( 2 ) . Then, by

Theorem 2.7. .4 E with the structure maps 4; and m. Consequently,

as A',(l) = 1 @J e , we have

(2) + (1). Since $) is the canonical entwining map, '4 E M : ( ~ " ) by The-

orem 2.7, so that [3, Lemma 3.31 implies that B as defined in Definition 2.2

is ident,ical with the set {b E A / $(e 8 b) = b @ e) as required in Defini-

tion 4.1. Furthermore, since the normalisation condition L*(l) = 1 @ e implies

that r(e) = 1 @ B 1 , we have $(e @ a) = e(1)(e(2)a)(o) @ ( e ( 2 ) a ) ( l ) = AA (a), for all

a E >A. Therefore canq = can and can, is bijective as required.

Remark 4.3 .4 slightly different definition of a C-Galois extension was pro-

posed in [2]. Let A he an algebra, C a coalgebra and e a group-like element of

C . One assumes that 4 @ C E M A , A E M C , and the action and coaction are

such that 3 . 4 o m = p ~ ~ c o (AA @.4) and pABC(a @ e , a t ) = aa1(o) @ a'(l) for any

n , at E -4. Then B := {b E A / A A ( b ) = b 8 e ) is an algebra and the canonical

map ccrn : '4 BB .4 + A @ C is well-defined. B c A is a C-Galois extension if the

canonical map can is a bijection. One easily finds, however, that given A and

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C sat,isfying the above conditions, B = {b E A / tin E A; AA(hrc) = bA.4(u)):

so that '4 E BM" via the maps p.4 = rn and AA. Hence, by Theorern 2.7, this

definition of a C-Galois extension is equivalent to the one introduced in [5] and

in Definition 2.3 provided that A A ( l ) = 1 @ e. 0

Remark 4.4 In the Hopf-Galois case, the formula for 4 becomes quite simple.

$(h@a) = aio) 8 hai l ) . If the Hopf algebra H has a bijective antipode, y is an

isomorphism, and its inverse is given by w-'(a@ h) := hS-l(a(1)) 8 q g ) . (In fact,

$ is an isomorphism zf and only zf H has a bijective antipode [3, Theorem 6.51.)

Furthermore, the coaction *A : A -t Hop @ A,

makes A a left Hop-comodule algebra, where Hop stands for the Hopf a lgeb~a

with the opposite multiplication. One can define the following left version of the

canonical map. can^, : A B B -4 + Hop 8.4, can~(a @ g a') := a(-l) @ U ( ~ ) U ' . It is

straightforward to verify that $wanI, = can. Since @ and can are isomorphisms.

we can immediately conclude that so is can^ . 0

In parallel to the theory of coalgebra $-principal bundles, the notion of a dual

@-principal bundle was introduced in [ 5 ] . This is recalled in the following

Definition 4.5 Let (A , C , $) be an entwining structure, K : A + k an algebra

homomorphism (character), and

L := span{((^ 8 C) o $ ) ( c 8 a ) - m(a) I a E A. ( E C ) .

Then B := C/I , zs a coalgebra, and we say that C(B, -4. +, K) is a dual v j -

principal bundle iff the nmp cocand, = (C @ K 8 C) o (C 8 tb) o [A 9 A) : C 8 .-I -+ CUBC is bijective.

Using Theorem 3.5 we can relate dual $-principal bundles with -4-Galois coes-

tensions.

Proposition 4.6 For a given entwining .structure (A , C , 4) and an algebra map

K : A -+ k , the following statements are equivalent:

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(1) C ( B : A: 41, K) is a dual $-principal bundle.

(2) There exists a unique action p c : C @ A -+ C such that C -. B is

an A-Galois coextension of B by A, 11, is the canonical entwining map,, and

E o p ~ = E @ n .

Proof (1) + (2). By [5, Proposition 2.61, C is a right .4-module with the action

p c = (KBC) o q . Taking advantage of the second of conditions (2.4) one verifies

that E o = E @ n. The first of conditions (2.4) implies that C E M s ( $ ) with

structure maps A and pc. lising this fact, explicit form of kc, and (2.4) one can

show that I& = I, the latter being defined in Lemma 3.2. Hence cocan = cocnn4

and C ii B is an A-Galois coextension. By Theorem 3.5, $ is the associated

canonical entwining map. Assume now that there exists another action ,& of

A on C satisfying conditions (2). Then, by Theorem 3.5, C E MAC($) with the

structure maps A and &. Consequently. since E o = E @ K , we have

(2) + (1). Since z) is the canonical entwining map, C E M z ( $ ) by Theo-

rem 3.5. The normalisation condition ~ o p c = E @ K implies that ( E @ E ) ococm =

E @ h: and? consequentl3.; K o i = E @ E. This, in turn, leads to the equality

/ I C = ( K @ C ) o $. Using this equality, t,he fact that C E MAC($), and (2.4) one

shows that I, = I. Hence cocan* = cocan and cocan$ is bijective, as required.

0

5 Appendix

TVe say that two subgroups GI and Gz of a group G generate G if any element of

G can be wi t t en as a finite length word whose letters are elements of GI or G 2 .

The (dual) coalgebra version of this concept is given in the following definition:

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Definition 5.1 Let C be a coalgebra and Ii , I2 zts cozdeals. Let denote the

composzte map

where A,(c) := ql) 8 . . . 8 c(,+l), (i) := (i, , . . . , i n ) E (1, 2 ) X n is a finite multi-

index, and each ni, is a canonical surjection. We say that the quotient coalgebras

C / I i and C/12 cogenerate C iff n(,),,w, Ker = 0, where Mf i s the space of

all finite multi-indices. W e write then ((?/I1) . ( C / I z ) = C .

Ohserve that the above construction is closely related to the wedge construc-

tion of [20]. In the group situation it is clear that what is invariant under both

generating subgroups is invariant under the whole group, and vice-versa. Below

is the (dual) coalgebra version of this classical phenomenon.

Proposition 5.2 Let C be a coalgebra, II and I2 coideals of C , and A a right

C-comodule. Then , defining coinvariants us i n Definition 2.2, we have:

( C / I 1 ) . (C/12) = C +. '4°C = ~ c o ( C / I l ) n A ~ o ( C / I ~ ) ,

Proof. Clearly, we always have A""" A ~ ~ ( ~ I ~ ~ ) n ilC0(C112). Assunie now that,

there exists b E A ~ ~ ( ~ / * ~ ) n Aco(C/12) such that 2, @ AcoC. Then there also exists

a E A such that 0 # A . ~ ( b a ) - b A A ( a ) =: xJEr f, @ h,,, where { f n ) a t A is a basis

of A and {hj),EJ is a non-empty set which does not contain zero. Furthermore.

for any (i) E Mf, we have:

Consequently, by the linear independence of f,, E J', me have b ~ ( ~ ) ( h ~ ) = 0 for

j E J'. Hence, as this is true for any ( 2 ) E Mf. we obtain n ,,,,,, E<erp(l) # 0,

as needed. 0

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ACKNOWLEDGEMENTS

This work mas supported by the EPSRC grant GRlK02244, the University of

L6di grant 5051582 and the Lloyd's Tercentenary Foundation (T.B.), and by the

Ii.ATO postdoctoral fellowship and KBN grant 2 P301 020 07 (P.N.H.). The

authors are very grateful to Mitsuhiro Takeuchi for providing a definition of

coinvariants that does not require the existence of a group-like element, and his

suggestion to use it in this paper.

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Received: October 1997

Revised: April 1998

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Documents

Coalgebra extensions and algebra coextensions of galois type