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Bilinear factorization of algebras.

Introduction

A distributive law (in a bicategory) consists of two monads A and B together with a 2-cell A [cross product] B [left arrow] B [cross product] A which is compatible with the monad structures, see [2].

A distributive law A [cross product] B [right arrow] B [cross product] A is known to be equivalent to a monad structure on the composite B [cross product] A such that the multiplication commutes with the actions by B on the left and by A on the right. The monad B [cross product] A is known as a wreath product, a twisted product, or a smash product of A and B.

Given a monad R, one may ask under what conditions it is isomorphic to a wreath product of A and B. This question is known as a (strict) factorization problem and the answer is this. A monad R is isomorphic to a wreath product of A and B if and only if there are monad morphisms A [right arrow] R [left arrow] B such that composing B [cross product] A [right arrow] R [cross product] R with the multiplication R [cross product] R [right arrow] R yields an isomorphism B [cross product] A [congruent to] R. In the particular bicategory of spans this and related questions were studied in [14]. In the monoidal category (i.e. one object bicategory) of modules over a commutative ring; and also in its opposite, such questions were investigated in [7], see also [10] and [19].

In the papers [8] and [17], the notion of distributive law was generalized by weakening the compatibility conditions with the units of the monads. A so defined weak distributive law A [cross product] B [right arrow] B [cross product] A also induces an associative multiplication on B [cross product] A but it fails to be unital. However, there is a canonical idempotent on B [cross product] A. Whenever it splits, the corresponding retract is a monad, known as the weak wreath product or weak smash product of A and B, see [17] and [8].

The aim of this paper is to study the factorization problem answered by a weak wreath product. In the particular bicategory of spans, this problem has already been studied in [4]. In the paper [9] addressing questions of similar motivation, a more general notion of weak crossed product monad was considered. Such weak crossed products are not induced by weak distributive laws but by more general 1-cells in an extended bicategory of monads introduced in [3]. The factorization problem corresponding to weak crossed products is fully described in [9].

We use the term strict factorization in the same sense as in [14]. Motivated by it, when we want to stress the difference from the weak generalizations, we refer to Beck's distributive laws as strict distributive laws and to their induced wreath products as strict wreath products.

We start Section 1 by recalling from [17] the notion of weak distributive law and the corresponding construction of weak wreath product. We show that a monad R is isomorphic to a weak wreath product of some monads A and B if and only if there are monad morphisms (with trivial 1-cell parts) A [right arrow] R [left arrow] B such that composing B [cross product] A [right arrow] R [cross product] R with the multiplication R [cross product] R [right arrow] R yields a split epimorphism of B-A bimodules B [cross product] A [right arrow] R. What is more, in Theorem 1.12, for any bicategory in which idempotent 2-cells split, we prove a biequivalence of the bicategory of weak distributive laws and an appropriately defined bicategory of bilinear factorization structures. This extends [4, Theorem 3.12].

Section 2 is devoted to collecting examples of algebras over commutative rings which admit a bilinear factorization.

The algebra homomorphisms A [right arrow] R [left arrow] B in a bilinear factorization structure are not injective in general. In Paragraph 2.1 we show, however, that if R admits any bilinear factorization then it admits also a bilinear factorization with injective algebra homomorphisms [??] [right arrow] R [left arrow] [??]. In general the latter factorization is still non-strict and we characterize those cases when it happens to be strict.

In Paragraph 2.2 we consider an algebra A and an element e of it such that ea = eae for all a [member of] A (so that eA is an algebra with unit e). Assuming that there is a strict distributive law eA [cross product] B [right arrow] B [cross product] eA, we extend it to a weak distributive law A [cross product] B [right arrow] B [cross product] A. The corresponding weak wreath product is isomorphic to the strict wreath product of eA and B; hence it admits a strict factorization in terms of them.

The Ore extension of an algebra B over a commutative ring k is the wreath product of B with the algebra k[X] of polynomials of a formal variable X, see [7, Example 2.11 (1)]. In Paragraph 2.3, generalizing Ore extensions, we construct a weak wreath product of B with k[X], that we regard as a weak Ore extension of B (although it turns out to be isomorphic in a nontrivial way to a strict Ore extension of an appropriate subalgebra [??]).

For any commutative ring k, there is a bicategory Bim of k-algebras, their bimodules and bimodule maps. In Paragraph 2.4 we consider strict distributive laws in Bim. Taking a 0-cell (i.e. k-algebra) R which admits a separable Frobenius structure, we show that any distributive law over R induces a weak distributive law over k. The corresponding weak wreath product is isomorphic to the R-module tensor product. We also present a morphism between these (weak) distributive laws over the respective objects R and k. The examples in Paragraph 2.5 and Paragraph 2.6 belong to this class of examples.

In Paragraph 2.5 we start with a finite collection of strict distributive laws [A.sub.i] [cross product] [B.sub.i] [right arrow] [B.sub.i] [cross product] [A.sub.i] and construct a weak distributive law ([[direct sum].sub.i] [A.sub.i]) [cross product] ([[direct sum].sub.i] [B.sub.i]) [right arrow] ([[direct sum].sub.i] [B.sub.i]) [cross product] ([[direct sum].sub.i] [A.sub.i]). The corresponding weak wreath product is isomorphic to the direct sum of the wreath product algebras [B.sub.i] [cross product] [A.sub.i].

In Paragraph 2.6 we take a weak bialgebra H and an H-module algebra A. We show that their smash product is a weak wreath product.

In Paragraph 2.7 we present explicitly a bilinear factorization of the three dimensional noncommutative algebra of 2 x 2 upper triangle matrices with entries in a field k whose characteristic is different from 2, in terms of two copies of the commutative algebra k [direct sum] k. This example does not belong to any of the previously discussed classes.

1 Weak wreath products and bilinear factorizations

The aim of this section is to prove an equivalence between weak wreath product monads on one hand, and monads admitting a bilinear factorization on the other. As a first step to that, under the assumption that an appropriate idempotent 2-cell splits, in Theorems 1.6 and 1.8 we show that a monad (in an arbitrary bicategory) admits a bilinear factorization if and only if it is isomorphic to a weak wreath product monad. Certainly, if we stopped at this point then it would not be necessary to work in a bicategory: a monoidal hom category singled out by one object would suffice. But we aim at more. As the main result of the section, for a bicategory K in which idempotent 2-cells split, in Theorem 1.12 we prove the biequivalence of the bicategory of bilinear factorizations of monads, and the bicategory of weak distributive laws in K. Remarkably, in this way the same bicategory of weak distributive laws occurs that was introduced (in a dual form) in [5] on different grounds.

Concerning examples, as long as we are interested only in objects (i.e. weak wreath product monads), the bicategorical formulation plays no role. It becomes important if we are interested also in morphisms between weak wreath product monads, possibly over different base objects. An example of this kind is presented in Example 2.4.

Throughout this section we work in a bicategory K, whose coherence isomorphisms will be omitted in our notation. The horizontal composition is denoted by g and the vertical composition is denoted by juxtaposition. Our motivating example is the one-object bicategory (i.e. monoidal category) of modules over a commutative ring (where [cross product] is the module tensor product).

1.1. Weak distributive laws. Let (A, [[mu].sub.A], [[eta].sub.A]) and (B, [[mu].sub.B], [[eta].sub.B]) be (associative and unital) monads in K on the same object, with multiplications [[mu].sub.A], [[mu].sub.B] and units [[eta].sub.A], [[eta].sub.B]. Following [8, Theorem 3.2] and [17, Definition 2.1], a 2-cell [PSI]: A [cross product] B [right arrow] B [cross product] A is said to be a weak distributive law of A over B if the following diagrams commute. (Throughout, in the labels of diagrams we omit indices A and B of n and [mu] since they can be uniquely determined from the domains and codomains of the respective arrows.)

Lemma 1.2. [17, Proposition 2.2] The third diagram in (1) is equivalent to the following two diagrams.

Proof. The following diagram shows that commutativity of the third diagram in (1) implies commutativity of the first diagram in (2),

where the region on the right commutes by the third diagram in (1). Commutativity of the second diagram in (2) is verified symmetrically. Conversely, if both diagrams in (2) commute then so does

1.3. Weak wreath product. Let [psi]: A [cross product] B [right arrow] B [cross product] A be a weak distributive law. Define [mu]: B [cross product] A [cross product] B [cross product] A [right arrow] B [cross product] A as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from the first two diagrams in (1) that [mu] is an associative multiplication. From now on, we consider B [cross product] A as an associative monad with the multiplication [mu] - possibly without a unit. (In fact, B [cross product] A can be seen to possess a preunit [[eta].sub.B] [cross product] [[eta].sub.a] in the sense discussed in [8].)

Proposition 1.4. (See [17, Proposition 2.3].) For any weak distributive law [PSI]: A [cross product] B [right arrow] B [cross product] A, define [bar.[PSI]]: B [cross product] A [right arrow] B [cross product] A by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Then [bar.[PSI]] is an idempotent endomorphism of monads (without unit), and of B-A bimodules. Moreover, [bar.[PSI]][PSI] = [PSI].

Proof. Note that [bar.[PSI]] stands in the diagonal of the diagram (3). Hence it has the equivalent forms

[bar.[PSI]] = (B [cross product] [[mu].sub.A])([PSI] [cross product] A)([[eta].sub.A] [cross product] B [cross product] A) (5) = ([[eta].sub.B] [cross product] A)(B [cross product] [PSI])(B [cross product] A [cross product] [[eta].sub.B]). (6)

Since the expression (5) is evidently a right A-module map and (6) is a left B-module map, this proves the bilinearity of [bar.[PSI]], i.e.

([[mu].sub.B] [cross product] A)(B [cross product] [bar.[PSI]]) = [bar.[PSI]]([[mu].sub.B] [cross product] A) and (B [cross product] [[mu].sub.A]) ([bar.[PSI]] [cross product] A) = [bar.[PSI]](B [cross product] [[mu].sub.A]). (7)

By commutativity of

and (5), we obtain [bar.[PSI]][PSI] = [PSI]. This implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

hence also [[bar.[PSI]].sup.2] = [PSI]. Moreover, by commutativity of

and (5), we obtain (B [cross product] [[mu].sub.A])([PSI] [cross product] A)(A [cross product] [bar.[PSI]]) = (B [cross product] [[mu].sub.A])([PSI] [cross product] A). This implies that

[mu](B [cross product] A [cross product] [bar.[PSI]]) = ([[mu].sub.B] [cross product] A)(B [cross product] B [cross product] [[mu].sub.A])(B [cross product] [PSI] [cross product] A)(B [cross product] A [cross product] [bar.[PSI]]) = [mu].

Combining it with the symmetrical counterpart, we conclude that

[mu]([bar.[PSI]] [cross product] [bar.[PSI]]) = [mu]. (9)

From (9) and (8) we get that [bar.[PSI]] is multiplicative with respect to [mu].

1.5. Splitting idempotents. Assume that the idempotent 2-cell [bar.[PSI]] associated in Proposition 1.4 to a weak distributive law [PSI] splits. That is, there is a (unique up-to isomorphism) 1-cell B [[cross product].sub.[PSI]] A and 2-cells [pi]: B [cross product] A [right arrow] B [[cross product].sub.[PSI]] A and i: B [[cross product].sub.[PSI]] A [right arrow] B [cross product] A such that [pi]i = B [[cross product].sub.[PSI]] A and i[pi] = [bar.[PSI]]. Since [bar.[PSI]] is a morphism of B-A bimodules, there is a unique B-A bimodule structure on B [[cross product].sub.[PSI]] A such that both n and i are morphisms of B-A bimodules (i.e. B [[cross product].sub.[PSI]] A is a B-A bimodule retract of B [cross product] A).

Theorem 1.6. (See [17, Theorem 2.4].) Lei [PSI]: A [cross product] B [right arrow] B [cross product] A be a weak distributive law in a bicategory K, such that the associated idempotent 2-cell [bar.[PSI]] splits. Then there is a retract monad (B [[cross product].sub.[PSI]] A, [[mu].sub.[PSI]]) of (B [cross product] A, [mu]) which is unital. Moreover, the 2-cells

[beta]:= [pi](B [cross product] [[eta].sub.A]): B [right arrow] B [[cross product].sub.[PSI]] A, [alpha] := [pi]([[eta].sub.B] [cross product] A): A [right arrow] B [[cross product].sub.[PSI]] A

are homomorphisms of unital monads such that [[mu].sub.[PSI]] (B [cross product] A): B [cross product] A [right arrow] B [[cross product].sub.[PSI]] A is equal to [pi]; and the left B- and right A-actions on B [[cross product].sub.[PSI]] A can be written as [[mu].sub.[PSI]]([beta] [cross product] (B [[cross product].sub.[PSI]] A)) and [[mu].sub.[PSI]]((B [[cross product].sub.[PSI]] A) [cross product] a), respectively.

Proof. Equip B [[cross product].sub.[PSI]] A with the multiplication

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By (9), [pi][mu] = [[mu].sub.[PSI]] ([pi] [cross product] [pi]) and by (8), [mu](i [cross product] i) = i[[mu].sub.[PSI]]. Since i is a (split) monomorphism and [pi] is a (split) epimorphism, any of these equalities implies associativity of [[mu].sub.[PSI]]. It is also unital with [[eta].sub.[PSI]]:= [pi]([[eta].sub.B] [cross product] [[eta].sub.A]) since and symmetrically on the other side. Unitality of 6 is evident. We have i[beta][[mu].sub.B] = [bar.[PSI]](B [cross product] [[eta].sub.A])[[mu].sub.B] and by (8) and (9), i[[mu].sub.[PSI]]([beta] [cross product] [beta]) = [mu](B [cross product] [[eta].sub.A] [cross product] B [cross product] [[eta].sub.A]). Hence multiplicativity of [beta] follows by commutativity of

That [alpha] is an algebra homomorphism follows by symmetry. Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that [[mu].sub.[PSI]]([beta] [cross product] (B [[cross product].sub.[PSI]] A)) = [pi]([[mu].sub.B] [cross product] A)(B [cross product] i) as stated, and symmetrically for the right A-action. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The situation in the above theorem motivates the following notion.

1.7. Bilinear factorization structures. In an arbitrary bicategory K, consider unital monads (A, [[mu].sub.A], [[eta].sub.A]), (B, [[mu].sub.B], [[eta].sub.B]) and (R, [[mu].sub.R], [[eta].sub.R]) on the same object k. Let [alpha]: A [right arrow] R [left arrow] B: [beta] be 2-cells which are compatible with the monad structures in the sense of the diagrams

i.e. [alpha] and [beta] be morphisms of (unital) monads. (They are monad morphisms with trivial 1-cell parts in the sense of [15].) Regarding R as a left B-module via [[mu].sub.R] ([beta] [cross product] R): B [cross product] R [right arrow] R and a right A-module via [[mu].sub.R] (R [cross product] [alpha]): R [cross product] A [right arrow] R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

is a homomorphism of B-A bimodules. If n has a B-A bimodule section i, then we call the datum ([alpha]: A [right arrow] R [left arrow] B: [beta], i: R [right arrow] B [cross product] A) a bilinear factorization structure on R or, shortly, a bilinear factorization of R.

By Theorem 1.6, any weak distributive law [PSI]: A [cross product] B [right arrow] B [cross product] A for which the idempotent 2-cell [PSI] splits, determines a bilinear factorization structure ([alpha]: A [right arrow] B [[cross product].sub.[PSI]] A [left arrow] B: [beta], i: B [[cross product].sub.[PSI]] A [right arrow] B [cross product] A). We turn to proving the converse.

Theorem 1.8. For a bilinear factorization structure ([alpha]: A [right arrow] R [left arrow] B: [beta], i: R [right arrow] B [cross product] A) in an arbitrary bicategory K,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a weak distributive law of A over B such that the corresponding idempotent 2-cell [PSI] splits. Moreover, R is isomorphic to the corresponding unital monad B [cross product][PSI] A.

Proof. The assumption that i is a morphism of B-A bimodules means the equalities

i[[mu].sub.R](R [cross product] [alpha]) = (B [cross product] [[mu].sub.A])(i [cross product] A) and [[mu].sub.R]([beta] [cross product] R) = ([[mu].sub.B] [cross product] A)(B [cross product] i). (11)

Compatibility of [PSI] with the multiplication of A (i.e. the first diagram in (1)) follows by commutativity of

The top region commutes by the multiplicativity of [alpha] and the region labelled by (*) commutes since i is a section of [pi] (occurring at the bottom of this region). It follows by symmetrical considerations that [PSI] renders commutative also the second diagram in (1). As for the third one concerns, in the diagram

the region on the left commutes by the unitality of [alpha]. Commutativity of this diagram yields the equality

(B [cross product] [[mu].sub.A])([PSI] [cross product] A)([[eta].sub.A] [cross product] B [cross product] A) = i[pi].

Symmetrically,

([[mu].sub.b] [cross product] A)(B [cross product] [PSI])(B [cross product] A [cross product] [[eta].sub.B]) = i[pi]

which proves that [PSI] renders commutative the third diagram in (1), so that [PSI] is a weak distributive law.

By (5), the expression on the left hand side of (12) is [PSI] which clearly splits. The corresponding 1-cell B [[cross product].sub.[PSI]] A is defined (uniquely up-to isomorphism) via some splitting of it as [[pi].sub.[PSI]]: B [cross product] A [right arrow] B [[cross product].sub.[PSI]] A and [i.sub.[PSI]]: B [[cross product].sub.[PSI]] A [right arrow] B [cross product] A. By uniqueness up-to isomorphism of the splitting of an idempotent 2-cell, (12) implies that B [[cross product].sub.[PSI]] A and R are isomorphic 1-cells in K via the mutually inverse isomorphisms [[pi].sub.[PSI]]i: R [right arrow] B [[cross product].sub.[PSI]] A and [pi][i.sub.[PSI]]: B [[cross product].sub.[PSI]] A [right arrow] R.

Composing both equal paths in

by B [cross product] [alpha] [cross product] [beta] [cross product] A on the right, we obtain

i[[mu].sub.R]([pi] [cross product] [pi]) = ([[mu].sub.B] [cross product] [[mu].sub.A])(B [cross product] [PSI] [cross product] A), (13)

hence multiplicativity of [pi] (and i). Since [i.sub.[PSI]] is multiplicative by (8), so is [pi][i.sub.[PSI]]. Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We close this section by proving that the constructions in Theorem 1.6 and Theorem 1.8 can be regarded as the object maps of a biequivalence between appropriately defined bicategories.

The bicategory of mixed weak distributive laws was studied in [5]. Taking the dual notion, we obtain the following.

1.9. The bicategory of weak distributive laws. The 0-cells of the bicategory Wdl (K) are weak distributive laws [PSI]: A [cross product] B [right arrow] B [cross product] A in the bicategory K. The 1-cells between them consist of monad morphisms (in the sense of [15]) [xi]: A' [cross product] V [right arrow] V [cross product] A and [zeta]: B' [cross product] V [right arrow] V [cross product] B with a common 1-cell V such that the following diagram commutes.

The 2-cells are those 2-cells on [omega]: V [right arrow] V' in K which are monad transformations (in the sense of [15]) (V, [xi]) [right arrow] (V, [xi]) and (V, [zeta]) [right arrow] (V', [zeta]'). Horizontal and vertical compositions are induced by those in K.

1.10. The bicategory of bilinear factorization structures. The 0-cells of the bicategory Bf (K) are the bilinear factorization structures ([alpha]: A [right arrow] R [left arrow] B: [beta], i: R [right arrow] B [cross product] A) in the bicategory K. The 1-cells between them are triples of monad morphisms (in the sense of [15]) [xi]: A [cross product] V [right arrow] V [cross product] A, [zeta]: B' [cross product] V [right arrow] V [cross product] B and [??]: R' [cross product] V [right arrow] V [cross product] R with a common 1-cell V such that the following diagrams commute.

The 2-cells are those 2-cells [omega]: V [right arrow] V' in K which are monad transformations (in the sense of [15]) (V, [xi]) [right arrow] (V, [xi]), (V, [zeta]) [right arrow] (V, [zeta]') and (V, [??]) [right arrow] (V, [??]'). Horizontal and vertical compositions are induced by those in K.

1.11. A pseudofunctor F: Bf(K) [right arrow] Wdl(K). The pseudofunctor F takes a bilinear factorization structure ([alpha]: A [right arrow] R [left arrow] B: [beta], i: R [right arrow] B [cross product] A) to the corresponding weak distributive law [PSI]:= i[[mu].sub.R]([alpha] [cross product] [beta]): A [cross product] B [right arrow] B [cross product] A in Theorem 1.8. It takes a 1-cell ([xi], [zeta], [??]) to ([xi], [zeta]). On the 2-cells F acts as the identity map.

The only non-trivial point to see is that ([xi], [zeta]) is indeed a 1-cell in Wdl(K) by commutativity of the following diagram.

The middle region commutes since q is a monad morphism. The bottom region commutes by commutativity of

which, in light of (10), means (V [cross product] [pi])([zeta] [cross product] A)(B [cross product] [xi]) = [??]([pi] [cross product] V).

Theorem 1.12. If idempotent 2-cells in a bicategory K split, then the pseudofunctor F: Bf (K) [right arrow] Wdl(K) in Paragraph 1.11 is a biequivalence.

Proof. First of all, F is surjective on the objects. In order to see that, take a weak distributive law [PSI]: A [cross product] B [right arrow] B [cross product] A and evaluate F on the associated bilinear factorization structure ([alpha]: A [right arrow] B [[cross product].sub.[PSI]] A [left arrow] B: [beta], i: B [[cross product].sub.[PSI]] A [right arrow] B [cross product] A) in Theorem 1.6. The resulting weak distributive law occurs in the top-right path of

Thus by commutativity of this diagram, it is equal to [PSI].

Next we show that F induces an equivalence of the hom categories. The induced functor of the hom categories is also surjective on the objects. In order to see that, take a 1-cell ([xi]: A' [cross product] V [right arrow] V [cross product] A, [zeta]: B' [cross product] V [right arrow] V [cross product] B) in Wdl(K) from the image under F of a bilinear factorization structure ([alpha]: A [right arrow] R [left arrow] B: [beta], i: R [right arrow] B [cross product] A) to the image of ([alpha]': A [right arrow] R [left arrow] B': [beta]', i': R [right arrow] B' [cross product] A); that is, from the weak distributive law [PSI]:= i[[mu].sub.R]([alpha] [cross product] [beta]) to [PSI]':= i[[mu].sub.R'] ([alpha]' [cross product] [beta]'). We show that together with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

they constitute a 1-cell in Bf (K). Unitality of [??] follows by commutativity of

The triangular region commutes by the unitality of the monad morphisms [xi] and [zeta] and the bottom left square commutes by [PSI]' ([[eta].sub.A] [cross product] [[eta].sub.B]) = i[[mu].sub.R] ([alpha]' [cross product] [beta]') ([[eta].sub.A'] [cross product] [[eta].sub.B']) = i[[mu].sub.R'] ([[eta].sub.R'] [cross product] [[eta].sub.R']) = i'[[eta].sub.R']. Multiplicativity of [??] is checked on page 233. The regions marked by (m) on page 233 commute since [xi] and [zeta] are monad morphisms. This proves that [??] is a monad morphism.

The first diagram in (15) commutes by commutativity of

The triangular region at the top left commutes by the unitality of [zeta]. The regions marked by (*) commute by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second diagram in (15) commutes by symmetrical considerations.

The functor induced by f between the hom categories acts on the morphisms as the identity map, hence it is evidently faithful. It is also full since any 2-cell [omega]: ([xi], [zeta]) [right arrow] ([xi]', [zeta]') in Wdl(K) is a 2-cell ([xi], [zeta], [??]) [right arrow] ([xi]', [zeta]', [??]') in Bf (k) by commutativity of

The regions in the middle commute since [omega] is a 2-cell in Wdl(K).

Remark 1.13. For an arbitrary bicategory k--not necessarily with split idempotents --, the pseudofunctor f in Paragraph 1.11 induces a biequivalence between Bf(K) and the full subbicategory of Wdl(K) whose 0-cells are those weak distributive laws [PSI] for which the idempotent 2-cell [bar.[PSI]] splits. It induces in particular a biequivalence between the bicategory of distributive laws in k (as a full subbicategory of Wdl(K)) and the bicategory of strict factorization structures (as a full subbicategory of Bf (K)), cf. [14].

1.14. Morphisms with trivial underlying 1-cells. For the algebraists, particularly interesting are those 1-cells in Bf (K) and Wdl(K) whose 1-cell part is trivial --these are algebra homomorphisms in the usual sense. Such 1-cells form a subcategory of the respective horizontal category.

In Bf (K), this means monad morphisms [??]: R [right arrow] R which restrict to monad morphisms [zeta]: A [right arrow] A and [zeta]: B' [right arrow] B, i.e. for which [??][alpha]' = [alpha][xi] and [??][beta]' = [beta][zeta].

The corresponding 1-cells in Wdl(K) are pairs of monad morphisms [xi]: A' [right arrow] A and [xi]: B' [right arrow] B such that [bar.[PSI]]([zeta] [cross product] [xi])[bar.[PSI]]' = [PSI]([xi] [cross product] [zeta]).

2 Examples: Bilinear factorizations of algebras

The aim of this section is to apply the results in the previous section to the particular monoidal category--i.e. one-object bicategory--of modules over a commutative ring. (Clearly, in this bicategory idempotent 2-cells split.) More precisely, we collect here some examples of associative and unital algebras over a commutative ring k which admit a bilinear factorization. Some of these algebras admit a strict factorization as well but the most interesting ones are those which do not.

2.1. Bilinear factorization via subalgebras. The algebra homomorphisms [alpha]: A [right arrow] R [left arrow] B: [beta], occurring in a bilinear factorization of an algebra R, are not injective in general. In this paragraph we show however that, for any bilinear factorization structure ([alpha]: A [right arrow] R [left arrow] B: [beta], i: R [right arrow] B [cross product] A), there is another bilinear factorization of R with injective homomorphisms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We give sufficient and necessary conditions for the latter factorization to be strict.

Consider a weak distributive law [PSI]: A [cross product] B [right arrow] B [cross product] A, with corresponding algebra homomorphisms [alpha]: A [right arrow] B [[cross product].sub.[PSI]] A [left arrow] B: [beta] obtained by the corestrictions of [PSI](A [cross product] [eta]): A [right arrow] B [cross product] A [left arrow] B: [PSI]([eta] [cross product] B), cf. Theorem 1.6. Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [alpha] factorizes through an epimorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a monomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of algebras. Similarly, [??] factorizes through an epimorphism B [??] B and a monomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of algebras. By Theorem 1.6, i: B [[cross product].sub.[PSI]] A [right arrow] B [cross product] A is a B-A-bimodule section of [[mu].sub.[PSI]] ([beta] [cross product] [alpha]), so that

is a [??]-[??]-bimodule section of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a bilinear factorization of B [[cross product].sub.[PSI]] A via subalgebras.

By Theorem 1.8 there is a weak distributive law [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that is isomorphic to B [[cross product].sub.[PSI]] A. The weak distributive law [??] is a strict distributive law if and only if both unitality conditions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] hold. They amount to commutativity of the following diagrams.

2.2. Extension of a distributive law. Let A be an (associative and unital) algebra over a commutative ring k; and let e [member of] A such that ea = eae for all a [member of] A (so that in particular [e.sup.2] = e). Then eA is a subalgebra of A though with a different unit element e.

Assume that [PHI]: eA [cross product] B [right arrow] B [cross product] eA is a distributive law. It induces an algebra structure on B [cross product] eA with unit 1 [DELTA] e and multiplication (b' [cross product] ea')(b [cross product] ea) = b' [PHI](ea' [cross product] b)ea = b' [PHI](ea' [cross product] b)a. The maps

[alpha]: A [right arrow] B [cross product] eA, a [??] 1 [cross product] ea and [beta]: B [right arrow] B [cross product] eA, b [??] b [cross product] e

are clearly algebra homomorphisms inducing the B-A bimodule map

[pi]: B [alpha] A [right arrow] B [cross product] eA, b [cross product] a [??] b [cross product] ea.

Since n possesses a B-A bimodule section i: b [cross product] ea [??] b [cross product] ea, the datum ([alpha]: A [right arrow] B [cross product] eA [left arrow] B: [beta], i: B [cross product] eA [right arrow] B [cross product] A) is a bilinear factorization structure. Hence by Theorem 1.8 there is a weak distributive law

[PSI]: A [cross product] B [right arrow] B [cross product] A, a [cross product] b [??] [PHI] (ea [cross product] b)

such that the weak wreath product algebra B [[cross product].sub.[PSI]] A is isomorphic the the strict wreath product B [[cross product].sub.[PHI]] eA.

By the above considerations, for any element e of A satisfying ea = eae for all a [member of] A, and for any algebra B, there is a weak distributive law

A [cross product] B [right arrow] B [cross product] A, a [cross product] b [??] b [cross product] ea

such that the corresponding weak wreath product is the tensor product algebra B [cross product] eA with the factorwise multiplication. If B is the trivial k-algebra k, this gives a weak distributive law

A [congruent to] A [cross product] k [right arrow] k [cross product] A [congruent to] A, a [right arrow] ea

and the corresponding weak wreath product algebra eA.

2.3. Weak Ore extension. Recall (e.g. from [11]) that a quasi-derivation on an (associative and unital) algebra B over a commutative ring k, consists of a (unital) algebra homomorphism [alpha]: B [right arrow] B and a k-module map [delta]: B [right arrow] B such that

[delta](bb') = [sigma](b)[delta](b) + [delta](b)b', for b, b' [member of] B.

Associated to any quasi-derivation, there is an Ore extension B[X, [sigma], [delta]] of B. As a k-module it is the tensor product of B with the algebra k[X] of polynomials in a formal variable X, equipped with the B-k[X] bilinear associative and unital multiplication determined by

(1 [cross product] X)(b [cross product] 1) = [sigma](b) [cross product] X + [delta](b) [cross product] 1, for b [member of] B.

Clearly, the Ore extension is a wreath product of B and k[X] with respect to a distributive law defined iteratively, see [7, Example 2.11 (1)]. The following characterization can be found e.g. in [11, Section 1.2]. An algebra T is an Ore extension of B if and only if the following hold.

* T has a subalgebra isomorphic to B;

* there is an element X of T such that the powers of X are linearly independent over B and they span T as a left B-module;

* XB [subset or equal to] BX + B.

In what follows, we generalize the notion of a quasi-derivation on B and the corresponding construction of Ore extension of B. The resulting algebra B [X, [sigma], [delta]] will be a weak wreath product of B with k[X]. However, we also show that it is a proper Ore extension of the image of B in B [X, [sigma], [delta]].

Let B be an (associative and unital) algebra over a commutative ring k, and let p and q be elements of B such that

[p.sup.2] = p, [q.sup.2] = 0, pq = q, qp = 0, and pbp = bp, for all b [member of] B.

Then by a (p, q)-quasi-derivation we mean a couple of k-linear maps [sigma], [delta]: B [right arrow] B such that the following identities hold for all b, b [member of] B:

[sigma](bb') = [sigma](b)[sigma](b), [sigma](1) = [sigma](p) = p, [sigma](q) = 0, [delta](bb') = [sigma](b)[delta](b) + [delta](b)b'p, [delta](1) = [delta](p) = q, [delta](q) = 0.

So that a (1,0)-quasi-derivation coincides with the classical notion of quasi-derivation recalled above. For example, if B is the algebra of 2 x 2 upper triangle matrices of entries in k, we may take

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In terms of a (p, q)-quasi-derivation ([sigma], [delta]) on an algebra B, define a k-module map [PSI]: k[X] [cross product] B [right arrow] B [cross product] k[X] iteratively as

[PSI](1 [cross product] b):= bq [cross product] X + bp [cross product] 1, [PSI](X [cross product] b):= [sigma](b)q [cross product] [X.sup.2] + ([sigma](b) + [delta](b)q) [cross product] X + 8(b)p [cross product] 1, [PSI](Xn+1 [cross product] b):= [PSI]([X.sup.n] [cross product] [sigma](b))X + [PSI]([X.sup.n] [cross product] [delta](b)),

for n > 0 and b [member of] B. By induction in n and m, one easily checks the following properties for all b, b' [member of] B and n, m [greater than or equal to] 0.

* [PSI]([X.sup.n] [cross product] bp) = [PSI] ([X.sup.n] [cross product] b) and [PSI] ([X.sup.n] [cross product] bq) = 0;

* b[PSI] ([X.sup.n] [cross product] 1) = [PSI] (1 [cross product] b)[X.sup.n],

* (B [cross product] [mu])([PSI] [cross product] k[X])(k[X] [cross product] [PSI])([X.sup.n] [cross product] [X.sup.m] [cross product] b) = [PSI] ([X.sup.n+m] [cross product] b),

* ([mu] [cross product] k[X])(B [cross product] [PSI])([PSI] [cross product] B)([X.sup.n] [cross product] b [cross product] b') = [PSI] ([X.sup.n] [cross product] bb').

That is, [PSI] is a weak distributive law and we may regard the corresponding weak wreath product B [[cross product].sub. [PSI]] k[X] as a weak Ore extension of B.

Note however, that Y renders commutative both diagrams in (16). Hence B [cross product]Y k[X] is a strict wreath product of the subalgebras [??] = {b(q [cross product] X + p [cross product] 1)|b [member of] B} and [??], the latter having the set of powers {[PSI] [(X [cross product] 1).sup.n] = [PSI] ([X.sup.n] [cross product] 1)|n [greater than or equal to] 0} as a k-basis. In fact, by the characterization of Ore extensions recalled above, the weak Ore extension B [[cross product].sub.[PSI]] k[X] is isomorphic to an Ore extension of [??].

2.4. Distributive laws over separable Frobenius algebras. An (associative and unital) algebra R over a commutative ring k is said to possess a Frobenius structure if it is a finitely generated and projective k-module and there is an isomorphism of (say) left R-modules from R to [??]:= Hom(R, k). A more categorical characterization is this. Any k-algebra R can be regarded as an R-k bimodule; that is, a 1-cell k [right arrow] R in the bicategory Bim of k-algebras, bimodules and bimodule maps. It possesses a right adjoint, the k-R bimodule (i.e. 1-cell R [right arrow] k) R. Whenever R is a finitely generated and projective k-module, the 1-cell R: k [right arrow] R possesses also a left adjoint [??]: R [right arrow] k (with right R-action [phi] [??] r = [phi](r-)). A Frobenius structure is then an isomorphism between the right adjoint R: R [right arrow] k and the left adjoint [??]: R [right arrow] k. In technical terms, a Frobenius structure is given by an element [psi] [member of] R (called a Frobenius functional) and an element [[summation].sub.i] [e.sub.i] [cross product] [f.sub.i] [member of] R [cross product] R (called a Frobenius basis) such that [[summation].sub.i][psi]([re.sub.i]) [f.sub.i] = r = [[summation].sub.i] [e.sub.i] [psi]([f.sub.i]r), for all r [member of] R. Note that a Frobenius algebra R possesses a canonical (Frobenius) coalgebra structure with R-bilinear comultiplication r [??] [[summation].sub.i] [re.sub.i] [cross product] [f.sub.i] = [[summation].sub.i] [e.sub.i] [cross product] [f.sub.i]r and counit [psi]. For more on Frobenius algebras we refer e.g. to [1] and [16].

A separable structure on a k-algebra R is an R-bilinear section of the multiplication map R [cross product] R [right arrow] R. Categorically, this means a section of the counit of the adjunction R [??] R: R [right arrow] k.

Finally, a separable Frobenius structure on R is a Frobenius structure ([psi], [[summation].sub.i] [e.sub.i] [cross product] [f.sub.i]) such that the multiplication R [cross product] R [right arrow] R is split by the R-bilinear comultiplication r [right arrow] [[summation].sub.i] [re.sub.i] [cross product] [f.sub.i] = [[summation].sub.i] [e.sub.i] [cross product] [f.sub.i]r. In other words, a Frobenius structure ([psi], [[summation].sub.i][e.sub.i] [cross product] [f.sub.i]) such that [[summation].sub.i][e.sub.i][f.sub.i] = [1.sub.R]. Categorically, the counit of the adjunction R [??] R: R [right arrow] k is split by the unit of the adjunction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For a separable Frobenius algebra R, a right R-module M and a left R-module N, the canonical epimorphism

[pi]: M [[cross product].sub.k] N [right arrow] M [[cross product].sub.R] N, m [[cross product].sub.k] n [right arrow] m [[cross product].sub.R]n

is split by

i: M [[cross product].sub.R] N [right arrow] M [[cross product].sub.k] N, m [[cross product].sub.R] n [right arrow] [summation over (i)] m.[e.sub.i] [[cross product].sub.k] [f.sub.i].n,

naturally in M and N. Thus the image of i is isomorphic to M [[cross product].sub.R] N.

Let R be a k-algebra. A monad A on R in Bim is given by a k-algebra homomorphism [??]: R [right arrow] A. (Then [??] induces an R-bimodule structure on A; [??] serves as the R-bilinear unit morphism; and the R-bilinear multiplication [??]: A [[cross product].sub.R] A [right arrow] A is the projection of the multiplication [mu]: A [[cross product].sub.k] A [right arrow] A of the k-algebra A.) A distributive law in Bim over R is an R-bimodule map [PHI]: A [[cross product].sub.R] B [right arrow] B [[cross product].sub.R] A which is compatible with the units and the multiplications of both R-rings A and B.

Then [PHI] induces on B [[cross product].sub.R] A the structure of a monad in Bim over R--that is, an algebra structure (b' [[cross product].sub.R] a')(b [[cross product].sub.R] a) = b' [PHI](a' [[cross product].sub.R] b)a and an algebra homomorphism [??] [[cross product].sub.R] [??]: R [right arrow] B [[cross product].sub.R] A. Moreover,

[alpha]:= [??] [[cross product].sub.R] A: A [right arrow] B [[cross product].sub.R] A [left arrow] B: B [[cross product].sub.R] [??] =: [beta]

are monad morphisms--that is, algebra homomorphisms which are compatible with the homomorphisms [??]. Composing & [[cross product].sub.k] [alpha]: B [[cross product].sub.k] A [right arrow] B [[cross product].sub.R] A [[cross product].sub.k] B [[cross product].sub.R] A with the multiplication induced by [PHI] on B [[cross product].sub.R] A, we re-obtain the canonical epimorphism [pi]: B [[cross product].sub.k] A [right arrow] B [[cross product].sub.R] A.

Whenever R is a separable Frobenius algebra, [pi] possesses a B-A bimodule section i above. That is to say, ([alpha]: A [right arrow] B [[cross product].sub.R] A [left arrow] B: [beta], i: B [[cross product].sub.R] A [right arrow] B [[cross product].sub.k] A) is a bilinear factorization structure. Hence by Theorem 1.8 there is a weak distributive law of the k-algebra A over B. Explicitly, it comes out as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

with corresponding idempotent

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence the resulting weak wreath product is isomorphic to the algebra B [[cross product].sub.R] A with the multiplication induced by [PHI].

There is a 1-cell in the bicategory Wdl(Bim) from the distributive law [PHI] (on the object R) to the weak distributive law (17) (on the object k) as follows. It is given by the k-R bimodule R and the k-R bimodule maps

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The induced functor R [[cross product].sub.R] (-) from the category of left R-modules to the category of k-modules lifts to an isomorphism, from the category of left modules over the wreath product R-ring corresponding to O, to the category of left modules over the weak wreath product k-algebra corresponding to the weak distributive law (17).

2.5. The direct sum of weak distributive laws. Assume that we have a finite collection [[PHI].sub.i]: [A.sub.i] [cross product] [B.sub.i] [right arrow] [B.sub.i] [cross product] [A.sub.i] of distributive laws between algebras over a commutative ring k. Consider the direct sum algebras A:= [[cross product].sub.i] [A.sub.i] (with multiplication [a.sub.i][a'.sub.j] = [[delta].sub.i,j][a.sub.i][a'.sub.i] and unit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and B:= [[cross product].sub.i][B.sub.i]. It is straightforward to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

is a weak distributive law.

We claim that it is of the type in Paragraph 2.4. Let R be the algebra [[cross product].sub.i]k with minimal orthogonal idempotents [p.sub.i]. Clearly, R is a separable Frobenius algebra via the Frobenius functional [psi]: R [right arrow] k, [p.sub.i] [??] 1 and the separable Frobenius basis [[summation].sub.i] [p.sub.i] [cross product] [p.sub.i] [member of] R [cross product] R. Thus we conclude that A [[cross product].sub.R] B is isomorphic to [[cross product].sub.i] ([A.sub.i] [cross product] [B.sub.i]) and B [[cross product].sub.R] A is isomorphic to [[cross product].sub.i]([B.sub.i] [cross product] [A.sub.i]). An R- distributive law is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Applying to it the construction in Paragraph 2.4 we re-obtain the weak distributive law (18).

2.6. Smash product with a weak bialgebra. Weak bialgebras are generalizations of bialgebras, see [12] and [6]. A weak bialgebra over a commutative ring k is a k-module H carrying both an (associative and unital) k-algebra structure ([mu], [eta]) and a (coassociative and counital) k-coalgebra structure ([delta], [epsilon]). The comultiplication is required to be multiplicative--equivalently, the multiplication is required to be comultiplicative. However, multiplicativity of the counit, unitality of the comultiplication and unitality of the counit are replaced by the weaker axioms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the usual Sweedler-Heynemann index convention is used for the components of the comultiplication, with implicit summation understood. In particular, we write [delta](1) = [1.sub.1] [cross product] [1.sub.2] = [1.sub.1'] [cross product] [1.sub.2']--possibly with primed indices if several copies occur.

The category of (say) right modules of a weak bialgebra over k is monoidal though not with the same monoidal structure as the category of k-modules. Indeed, if M and N are right H-modules, then there is a diagonal action (m [cross product] n) [??] h:= m [??] [h.sub.1] [cross product] n [??] [h.sub.2] on the k-module tensor product M [cross product] N but it fails to be unital. A unital H-module is obtained by taking the k-module retract

This defines a monoidal product [??] with monoidal unit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with H-action [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. With respect to this monoidal structure, the forgetful functor from the category of right H-modules to the category of k-modules is both monoidal and opmonoidal (hence preserves algebras and coalgebras) but it is not strict monoidal.

A right module algebra of a weak bialgebra H is a monoid in the category of right H-modules. That is a k-algebra A equipped with an (associative and unital) right H-action such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all a, a' [member of] A and h [member of] H. For any right H-module algebra A, there is a weak distributive law

[PSI]: A [cross product] H [right arrow] H [cross product] A, a [cross product] h [??] [h.sub.1] [cross product] a [??] [h.sub.2].

It is multiplicative in A by the H-linearity of the multiplication in A:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Multiplicativity in H follows by multiplicativity of the comultiplication in H:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to check the weak unitality condition note that for all a [member of] A

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also for all h [member of] H

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

hence h[1.sub.1] [cross product] [1.sub.2] = [h.sub.1] [epsilon]([h.sub.2][1.sub.1]) [cross product] [1.sub.2]. With these identities at hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The weak wreath product corresponding to Y is known as a weak smash product, see [13].

In the rest of this paragraph we show that the weak distributive law [PSI] above is of the kind discussed in Paragraph 2.4. Let us introduce a further map [??]: H [right arrow] H, h [right arrow] [1.sub.1] [epsilon](h[1.sub.2]). It is easy to see that for any h, hh [member of] H,

* [epsilon] [??] (h)= [epsilon](h) = [epsilon][??](h);

* [delta] [??] (h) = [1.sub.1] [cross product] [??](h)[1.sub.2] and [delta] [bar.[??]] (h) = [1.sub.1] [bar.[??]] (h) [cross product] [1.sub.2];

* [??]([??](h)h') = [??](hh') = [??](bar.[??]](h)h') and [??]([bar.[??]](h)h') = [??]([bar.[??]](hh') = [??]([bar.[??]](h)h');

* [bar.[??]](h) [??] (h') = [??](h')[bar.[??]](h);

* [??](h [??] (h')) = [??]([??](h) [??] (h')) = [1.sub.1][epsilon]([??](h) [??] (h)[i.sub.2]) = [??][(h).sub.1] [??] [(h').sub.1] [epsilon] ([??][(h).sub.2][??] [(h').sub.2]) = [??](h) [??] (h') and symmetrically, [bar.[??]](h[bar.[??]](h')) = [bar.[??]](h)[bar.[??]](h').

Note that [bar.[??]](H) possesses a separable Frobenius structure (cf. [18]) with Frobenius functional given by the restriction of e and Frobenius basis [bar.[??]] ([1.sub.2]) [cross product] [bar.[??]]([1.sub.1]) = [1.sub.2] [cross product] [bar.[??]]([1.sub.1]) (where the equality follows by [1.sub.1] [cross product] [bar.[??]]([1.sub.2]) = [1.sub.1] [cross product] [epsilon]([1.sub.2][1.sub.1'])[1.sub.2'] = [1.sub.1] [cross product] [epsilon]([1.sub.2])[1.sub.3] = [1.sub.1] [cross product] [1.sub.2]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence also the opposite algebra R: = [bar.[??]][(H).sup.op] has a separable Frobenius structure with the same Frobenius functional [epsilon] and Frobenius basis [bar.[??]]([1.sub.1]) [cross product] [1.sub.2]. Moreover,

[??] ([bar.[??]](h) [bar.[??]](h')) = [??]([??](h) [bar.[??]](h')) = [??]([bar.[??]](h') [??] (h)) = [??] (h') [??] (h)) = [??](h') [??] (h) = [??][bar.[??]](h') [??][bar.[??]](h).

That is, the restriction of [??] yields an algebra homomorphism R [right arrow] H. There is an algebra homomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as well. They induce R-actions on A and H. By [bar.[??][??] = [bar.[??]] we conclude that, for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This means that [PSI] projects to an R-distributive law

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is evidently a morphism of right R-modules. It is also a morphism of left Rmodules as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the last equality follows by [??][bar.[??]]([1.sub.1]) [cross product] [1.sub.2] = [??]([1.sub.1]) [cross product] [1.sub.2] = [1.sub.1] [cross product] [1.sub.2], and [1.sub.2] [cross product] [bar.[??]]([1.sub.1]) being a separability element for R:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Multiplicativity in both arguments is obvious. Unitality follows by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Applying the construction in Paragraph 2.4 to this R-distributive law, it yields a weak distributive law A [cross product] H -- H [cross product] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [??]([1.sub.1]) [cross product] [1.sub.2] = [1.sub.1] [cross product] [1.sub.2] = [1.sub.1] [cross product] [bar.[??]]([1.sub.2]), this is equal to [PSI].

2.7. 2x2 = 3. In this paragraph we present a bilinear factorization of the algebra T of 2 x 2 upper triangle matrices over a field k of characteristic different from 2, in terms of two copies of the group algebra k[Z.sub.2] of the order 2 cyclic group. So the attitudinizing title refers to the vector space dimensions: we obtain a 3 dimensional non-commutative algebra as a weak wreath product of two 2 dimensional commutative algebras. Note that starting from 2 dimensional algebras A and B, none of the constructions in the previous examples of the section would result in a 3 dimensional weak wreath product algebra. Hence the current example does not belong to any of the previously discussed classes.

A k-linear basis of T is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These basis elements satisfy ab = a + b - 1 and ba = -(a + b + 1). Denote the second order generator of the cyclic group [Z.sub.2] by g and consider the following algebra homomorphisms.

[alpha]: k[Z.sub.2] [right arrow] T, g [??] a and [beta]: k[Z.sub.2] [right arrow] T, g [??] b.

In terms of [alpha] and [beta], we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with values

[pi](1 [cross product] 1) = 1, [pi](1 [cross product] g) = a, [pi](g [cross product] 1) = b, [pi](g [cross product] g) = ba = -(a + b + 1).

It is straightforward to check that n has a section i: T -- kZ2 [cross product] kZ2 with values

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a homomorphism of kZ2-bimodules, with respect to the action induced by B on the first factor and the action induced by a on the second factor. This shows that T has a bilinear factorization in terms of the algebra homomorphisms a and B.

By Theorem 1.8 there is a corresponding weak distributive law

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with values

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Acknowledgements. GB thanks all members of Departamento de Algebra at Universidad de Granada for a generous invitation and for a very warm hospitality experienced during her visit in June 2010 when the work on this paper begun. Partial financial support from the Hungarian Scientific Research Fund OTKA, grant no. K68195, and from the Ministerio de Ciencia e Innovation and FEDER, grant MTM2010-20940-C02-01 is gratefully acknowledged.

References

[1] L. Abrams, Modules, comodules, and cotensor products over Frobenius algebras. J. Algebra 219 (1999), 201-213.

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Received by the editors December 2011--In revised form in May 2012.

Communicated by S. Caenepeel.

2010 Mathematics Subject Classification: 16S40, 16T05, 18C15.

Wigner Research Centre for Physics, Budapest, H-1525 Budapest 114, P.O.B. 49, Hungary

email:bohm.gabriella@wigner.mta.hu

Departamento de Algebra Universidad de Granada, E-18071 Granada, Spain

email:gomezj@ugr.es
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Author:Bohm, Gabriella; Gomez-Torrecillas, Jose
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