ACTIONS IN MODIFIED CATEGORIES OF INTEREST WITH APPLICATION TO CROSSED MODULES Dedicated to Teimuraz Pirashvili on his 60th birthday

Theory an Applications of Categories, Vol. 30, No. 25, 2015, pp. 882 908. ACTIONS IN MODIFIED CATEGORIES OF INTEREST WITH APPLICATION TO CROSSED MODULES Deicate to Teimuraz Pirashvili on his 60th birthay Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU Abstract. The existence of the split extension classifier of a crosse moule in the category of associative algebras is investigate. Accoring to the equivalence of categories XAss Cat 1 -Ass we consier this problem in Cat 1 -Ass. This category is not a category of interest, it satisfies its all axioms except one. The action theory evelope in the category of interest is aapte to the new type of category, which will be calle moifie category of interest. Applying the results obtaine in this irection an the equivalence of categories we fin a conition uner which there exists the split extension classifier of a crosse moule an give the corresponing construction. 1. Introuction Categories of interest were introuce in orer to stuy properties of ifferent algebraic categories an ifferent algebras simultaneously. The iea comes from P. G. Higgins [Higgins, 1956] an the efinition is ue to M. Barr an G. Orzech [Orzech, 1972]. The categories of groups, moules over a ring, vector spaces, associative algebras, associative commutative algebras, Lie algebras, Leibniz algebras, alternative algebras, Poisson algebras, left-right non-commutative Poisson algebras are categories of interest [Orzech, 1972, Casas, Datuashvili an Lara, 2009, Casas, Datuashvili an Lara, 2014]. Note that the category of noncommutative Leibniz-Poisson algebras efine an stuie in [Casas an Datuashvili, 2006] is not a category of interest. In [Montoli, 2010] there are given new examples of categories of interest, these are associative ialgebras an trialgebras, which were efine an stuie in [Loay, 1995, Loay, 2001, Loay an Ronco, 2004]. The categories of crosse moules an precrosse moules in the category of Secon author was supporte by Ministerio e Economía y Competitivia (Spain), grant MTM2013 43687 P (European FEDER support inclue). Thir author was partially supporte by Shota Rustaveli Science Founation grant DI/27/5-103/12, D-13/2 Homological an categorical methos in topology, algebra an theory of stacks an TUBITAK grant 2221 Konuk veya akaemik izinli bilim insanı estekleme programi. The authors are grateful to the referee for his comments an suggestions. Receive by the eitors 2014-06-05 an, in revise form, 2015-06-26. Transmitte by Ronal Brown. Publishe on 2015-06-30. 2010 Mathematics Subject Classification: 08A99, 08C05, 16E99, 18B99. Key wors an phrases: split extension classifier, category of interest, associative algebra, crosse moule, cat 1 -associative algebra, equivalence of categories, actor, universal strict general actor, bimultiplier. c Y. Boyaci, J. M. Casas, T. Datuashvili an E. Ö. Uslu, 2015. Permission to copy for private use grante. 882

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 883 groups, respectively, are equivalent to categories of interest as well, in the sense of [Casas, Datuashvili an Lara, 2010, Casas, Datuashvili an Lara, 2007]. In the proceure of investigation of representability of actions in the category of precrosse moules in the categories of Lie algebras [Casas, Datuashvili, Lara an Uslu, 2012] an associative algebras, we recognize that the categories of cat 1 -Lie algebras an that of cat 1 -associative algebras (see Section 2 for efinitions) satisfy all axioms of category of interest except one. Consequently, we plan to introuce an stuy a new type of category of interest; namely, a category which satisfies all axioms of a category of groups with operations state in [Porter, 1987] except one, which is replace by a new axiom; this category satisfies as well two aitional axioms introuce in [Orzech, 1972] for categories of interest. The examples are mainly those categories which are equivalent to the categories of crosse moules an precrosse moules in the categories of Lie algebras, Leibniz algebras, associative an associative commutative algebras. Also, applying a result of [Borceux, Janelize an Kelly, 2005 ], the category of commutative Von Neumann regular rings is isomorphic to a category of commutative rings with a unary operation satisfying two axioms, which is a new type of category of interest as well. Therefore we ecie to give a name to this sort of a category, which coul be calle moifie category of interest. In this work our main purpose is to unify the stuy of actions in certain algebraic categories by means of this new kin of category. We escribe main notions, in particular, the notion of actor efine in [Casas, Datuashvili an Lara, 2010] in categories of interest, or, equivalently, split extension classifier efine in the more general setting of semi-abelian categories [Borceux, Janelize an Kelly, 2005], universal strict general actor, efine in [Casas, Datuashvili an Lara, 2010]. At the same time we plan to stuy concrete examples of moifie categories of interest an their equivalent ones. In this paper we fin sufficient conitions for the existence of the split extension classifier of a crosse moule in the category of associative algebras an give the corresponing construction. The analogous problem for (pre)crosse moules in the category of groups, crosse moules in the categories of Lie algebras an associative commutative algebras an in the category of Lie-Leibniz algebras were consiere in [Norrie, 1990, Casas an Lara, 1998, Arvasi an Ege, 2003, Casas, Datuashvili an Lara, 2009, Casas, Datuashvili, Lara an Uslu, 2012, Casas, Datuashvili an Lara, 2013]. Note that the results an the constructions obtaine in the cases of (pre)crosse moules in the categories of Lie an associative commutative algebras coul be obtaine in the way consiere in this paper, i.e. by application of action theory in moifie categories of interest. The analogous is true for the future investigations in the cases of precrosse moules in the category of associative algebras an (pre)crosse moules in the category of Leibniz algebras. This kin of results, like in the cases of associative algebras [Hochschil, 1947] an rings [Mac Lane, 1958], coul be applie in the cohomology an obstruction theories of the corresponing objects. The notion of crosse moule in the category of associative algebras was efine by Deecker an Lue in [Deecker an Lue, 1966]. It was use as a funamental gaget to

884 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU etermine the coefficients in low-imensional non-abelian cohomology [Lue, 1968]. On the other han, it was use for the representation of Hochschil cohomology in [Baues an Minian, 2002]. Naturally, it will be important to investigate the existence of the split extension classifier of a crosse moule in the category of associative algebras. A 0, we foun a conition uner At this vein, for a given crosse moule A : A 1 which we construct an actor of the corresponing cat 1 -associative algebra (A 1 A 0, ω 1, ω 0 ) by using the general construction of universal strict general actor of (A 1 A 0, ω 1, ω 0 ). Then applying the equivalence of the categories Cat 1 -Ass XAss of cat 1 -associative algebras (cat 1 -algebras in what follows) an crosse moules, we carry the construction of an actor of (A 1 A 0, ω 1, ω 0 ) to the category of crosse moules, which is a split extension classifier of the crosse moule A : A 1 A 0 uner the appropriate conition on it. Therefore we foun a new example of a category an iniviual objects there with representable actions an escribe the representing objects. This problem is state in [Borceux, Janelize an Kelly, 2005 ] (Problem 2). The outline of the paper is as follows: in Section 2 we recall some well-known efinitions of the category of crosse moules in the category of associative algebras an introuce some new notions, such as bimultipliers an crosse multipliers. In Section 3 we introuce a notion of moifie category of interest an actions in this category. Then we introuce notions of general an strict general properties an universal strict general actors an construct a universal strict general actor of an object in a moifie category of interest. We start Section 4 by constructing an object (A(A), ω 0, ω 1 ) for a cat 1 -algebra (A, ω 0, ω 1 ), an prove that if Ann(A) = 0 or A 2 = A, then the constructe object is an actor of (A, ω 0, ω 1 ). We finish this section by examining a particular case; namely, an actor of a cat 1 -algebra (A 1 A 0, ω 1, ω 0 ), corresponing to a given crosse moule A : A 1 A 0. Finally, in Section 5, applying this result, we prove, that if Ann(A i ) = 0 or A 2 i = A i, i = 0, 1, for a crosse moule A : A 1 A 0, then there exists the split extension classifier of this crosse moule an give the corresponing construction. 2. Crosse Moules in the Category of Associative Algebras In this section we will give some well-known efinitions an results about crosse moules in the category of associative algebras. We also efine new notions such as bimultiplier an crosse bimultiplier of a crosse moule an give some relate results, which will be neee in the rest of the paper. Let k be a fixe commutative ring with unit. All algebras in the rest of the paper will be associative algebras over k. Let A, B be associative algebras. An action (i.e. a erive action) of B on A is a pair of bilinear maps B A A, A B A

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 885 which we enote respectively as (b, a) b a, (a, b) a b, with conitions (b 1 b 2 ) a = b 1 (b 2 a) a (b 1 b 2 ) = (a b 1 ) b 2 (b 1 a) b 2 = b 1 (a b 2 ) b (a 1 a 2 ) = (b a 1 ) a 2 (a 1 a 2 ) b = a 1 (a 2 b) a 1 (b a 2 ) = (a 1 b) a 2 (2.1) for all a, a 1, a 2 A, b, b 1, b 2 B. We recall the construction of the algebra Bim(A) of bimultipliers of an associative algebra A efine by G. Hochschil an by S. Mac Lane for rings (calle bimultiplications in [Mac Lane, 1958] an multiplications in [Hochschil, 1947], from where the notion comes [Lavenhomme an Lucas, 1996]). An element of Bim(A) is a pair f = (f l, f r ) of k-linear maps from A to A with f l (a a ) = f l (a) a, f r (a a ) = a f r (a ), a f l (a ) = f r (a) a, for all a, a A. As it is well-known, Bim(A) has an associative algebra structure, efine by componentwise scalar multiplication an aition, an composition as the multiplication. Note that since the bimultipliers f l an f r are written from the left sie of an element, for the prouct of two bimultipliers we have (f l, f r ) (g l, g r ) = (f l g l, f r g r ), where (f l g l )(a) = f l (g l (a)) an (f r g r )(a) = g r (f r (a)), for all a A. The following efinition of a crosse moule is well-known from [Deecker an Lue, 1966] an it is a special case of the efinition of a crosse moule in categories of interest [Casas, Datuashvili an Lara, 2010] (cf. [Porter, 1987]). 2.1. Definition. A precrosse moule A : (A 1 A 0 ) in the category of associative algebras consists of an associative algebra homomorphism : A 1 A 0, calle bounary map, together with an action of A 0 on A 1, satisfying the conitions for all a 1 A 1, a 0 A 0. In aition, if for all a 1, a 1 A 1, then A : (A 1 (a 0 a 1 ) = a 0 (a 1 ), (a 1 a 0 ) = (a 1 ) a 0, (a 1) a 1 = a 1 a 1, a 1 (a 1) = a 1 a 1, A 0 ) is calle a crosse moule. Let A : (A 1 A 0 ) an A : (A 1 A 0) be crosse moules. A homomorphism from A to A is a pair (µ 1, µ 0 ) where µ 1 : A 1 A 1 an µ 0 : A 0 A 0 are associative algebra homomorphisms, such that µ 1 = µ 0 an µ 1 (a 0 a 1 ) = µ 0 (a 0 ) µ 1 (a 1 ), µ 1 (a 1 a 0 ) = µ 1 (a 1 ) µ 0 (a 0 ),

886 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU for all a 1 A 1, a 0 A 0. The corresponing category of crosse moules in the category of associative algebras will be enote by XAss. 2.2. Example. Let I be a two-sie ieal of an associative algebra A. Then, inc. : I A is a crosse moule with the action by conjugation (prouct) of A on I. Consequently, i : A A an inc. : 0 A are crosse moules. 2.3. Example. Let C be a singular associative algebra (i.e. an associative algebra with trivial multiplication) with an action of the associative algebra A. Then, 0 : C A is a crosse moule. Take C = A, then 0 : A A an 0 : A 0 are crosse moules. Nevertheless, if C is non-singular, then 0 : C A is a precrosse moule, in general. 2.4. Definition. Let A : (A 1 A 0 ) be a crosse moule in the category of associative algebras. A bimultiplier of A is a pair (α, β) of bimultipliers α an β of A 1, A 0, respectively, such that α l = β l, α r = β r an a) α l (a 0 a 1 ) = β l (a 0 ) a 1, b) α l (a 1 a 0 ) = α l (a 1 ) a 0, c) a 0 α l (a 1 ) = β r (a 0 ) a 1, ) α r (a 1 a 0 ) = a 1 β r (a 0 ), e) α r (a 0 a 1 ) = a 0 α r (a 1 ), f) α r (a 1 ) a 0 = a 1 β l (a 0 ), for all a 0 A 0, a 1 A 1. 2.5. Notation. In the efinition, instea of writing α l = β l, α r = β r, we may write these two equalities in one as α l,r = β l,r. In the rest of the paper we will use this notation for shortness. The set of all bimultipliers of a crosse moule A is enote by Bim(A). It can be easily checke that Bim(A) is an associative algebra with usual scalar multiplication an aition, an the multiplication efine by (α, β) (α, β ) = (α α, β β ), for all (α, β), (α, β ) Bim(A), where α α = (α l α l, α rα r) an β β = (β l β l, β rβ r). 2.6. Definition. Let A : (A 1 A 0 ) be a crosse moule an l : A 0 A 1, r : A 0 A 1 be k-linear maps such that r (a 0 a 0) = a 0 r (a 0), l (a 0 a 0) = l (a 0 ) a 0 an a 0 l (a 0) = r (a 0 ) a 0, for all a 0, a 0 A 0. Then the pair := ( l, r ) will be calle a crosse bimultiplier of A. The set of all crosse bimultipliers of a crosse moule A : (A 1 A 0 ) will be enote by BM(A).

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 887 2.7. Remark. Definitions 2.4 an 2.6 are euce from the construction of bimultipliers of a semiirect prouct in Section 4. In Section 5 one can see that bimultipliers an crosse bimultipliers of a crosse moule A : (A 1 A 0 ) can be easily obtaine from certain type of bimultipliers of the semiirect prouct A 1 A 0 (Propositions 5.3 an 5.4). In the case of commutative associative algebras these efinitions coincie with the ones given in [Arvasi an Ege, 2003]. 2.8. Proposition. Let A : (A 1 A 0 ) be a crosse moule. Then BM(A) is a nonempty set. Proof. Let a 1 be a fixe element of A 1. Define ( a1 ) l (a 0 ) = a 1 a 0, ( a1 ) r (a 0 ) = a 0 a 1, for all a 0 A 0. Then by a irect calculation we fin that a1 := (( a1 ) l, ( a1 ) r ) is a crosse bimultiplier. We can enow BM(A) with aition an scalar multiplication operations, which are efine in the usual ways. Multiplication operation is efine as follows: where = (( ) l, ( ) r ), ( ) l = l l, ( ) r = r r, for all, BM(A). Note that, in an analogous way, as it is in the case of groups [Casas, Datuashvili an Lara, 2009], it can be easily seen from the results of Section 4, that the prouct of two crosse bimultipliers correspons to the composition of two bimultipliers of the semiirect prouct A 1 A 0, where β l,r : A 0 A 0 is zero (see Section 4, 4.1). The ientity element an the opposite of an element in aition are also efine in the usual ways. 2.9. Proposition. BM(A) enowe with the above efine operations is an associative algebra. Proof. We will only show that the prouct is a crosse bimultiplier. Other conitions nee straightforwar verifications. Let, BM(A). We have ( ) l (a 0 a 0) = l l(a 0 a 0) = l ( l(a 0 ) a 0) = l ( l(a 0 ) a 0) = l l(a 0 ) a 0,

888 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU an similarly, we have ( ) r (a 0 a 0) = a 0 r r(a 0), for all a 0, a 0 A 0. On the other han we have a 0 ( ) l (a 0) = a 0 l ( l(a 0)) = r (a 0 ) l(a 0) = r (a 0 ) l(a 0) = r (a 0 ) l(a 0) = r r (a 0 ) a 0, for all a 0, a 0 A 0. So the operation is well-efine, as require. 2.10. Definition. A crosse moule A : (A 1 A 0) is a crosse submoule of a crosse moule A : (A 1 A 0 ), if A 1, A 0 are subalgebras of A 1, A 0 respectively, = A 1 an the action of A 0 on A 1 is inuce by the action of A 0 on A 1. 2.11. Definition. A crosse submoule A : (A 1 A 0) of a crosse moule A : (A 1 A 0 ) is an ieal if A 1, A 0 are ieals of A 1, A 0, respectively; a 0 a 1, a 1 a 0 A 1, for all a 0 A 0, a 1 A 1 an a 0 a 1, a 1 a 0 A 1, for all a 0 A 0, a 1 A 1. This situation is enote by A A. Let A : (A 1 A 0) be an ieal of a crosse moule A : (A 1 A 0 ). Then the quotient crosse moule A/A is the crosse moule A 1 /A 1 A 0 /A 0 with the inuce bounary map an action. 2.12. Definition. A cat 1 -associative algebra (or, for shortness, cat 1 -algebra) A is an associative algebra with two aitional unary operations ω 0, ω 1 : A A such that ω 0 ω 1 = ω 1, ω 1 ω 0 = ω 0 an ker ω 0 ker ω 1 = 0 = ker ω 1 ker ω 0, where ω 0 an ω 1 are associative algebra homomorphisms. Obviously, a homomorphism between two cat 1 -algebras is an associative algebra homomorphism, which preserves the unary operations. We will enote a cat 1 -algebra by (A, ω 0, ω 1 ) an the corresponing category of such triples an homomorphisms between them by Cat 1 -Ass. Note that cat n -unitary associative algebras are efine in [Ellis, 1988]. Note that from the conitions on unary operations it follows that ω i ω i = ω i, i = 0, 1. Define a functor P : Cat 1 -Ass XAss as follows; for any object (A, ω 1, ω 0 ) in Cat 1 -Ass, P (A, ω 0, ω 1 ) is the crosse moule A 1 A 0, where A 1 = ker ω 0, A 0 = Im ω 0, the action is given by multiplication in A an = ω 1 ker ω0. Now we efine a functor S : XAss Cat 1 -Ass as follows; for any crosse moule A : (A 1 A 0 ), S(A) := (A 1 A 0, ω 0, ω 1 ), where ω 0 (a 1, a 0 ) = (0, a 0 ), ω 1 (a 1, a 0 ) = (0, (a 1 ) + a 0 ),

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 889 for all a 1 A 1, a 0 A 0. Note that the semiirect prouct A 1 A 0 is efine by the action of A 0 on A 1, as it is in the corresponing crosse moule. These two functors give rise to an equivalence of categories XAss Cat 1 -Ass. Precrosse moules in the category of associative algebras suggest the appropriate efinition of precat 1 -associative algebras, an we have an equivalence of categories PreXAss PreCat 1 -Ass. Similarly, for Lie an Leibniz algebras. Some etails can be foun in [Ellis, 1988, Ellis, 1993]. 3. Moifie Categories of interest We will have the main efinitions an the statements given for categories of interest in [Casas, Datuashvili an Lara, 2010, Datuashvili, 1995, Orzech, 1972] with certain moifications which we present as follows. Let C be a category of groups with a set of operations Ω an with a set of ientities E, such that E inclues the group ientities an the following conitions hol. If Ω i is the set of i-ary operations in Ω, then: (a) Ω = Ω 0 Ω 1 Ω 2 ; (b) the group operations (written aitively: 0,, +) are elements of Ω 0, Ω 1 an Ω 2 respectively. Let Ω 2 = Ω 2 \ {+}, Ω 1 = Ω 1 \ { }. Assume that if Ω 2, then Ω 2 contains efine by x y = y x an assume Ω 0 = {0}; (c) for each Ω 2, E inclues the ientity x (y + z) = x y + x z; () for each ω Ω 1 an Ω 2, E inclues the ientities ω(x + y) = ω(x) + ω(y) an ω(x y) = ω(x) ω(y). Let C be an object of C an x 1, x 2, x 3 C: Axiom 1. x 1 + (x 2 x 3 ) = (x 2 x 3 ) + x 1, for each Ω 2. Axiom 2. For each orere pair (, ) Ω 2 Ω 2 there is a wor W such that (x 1 x 2 ) x 3 = W (x 1 (x 2 x 3 ), x 1 (x 3 x 2 ), (x 2 x 3 )x 1, (x 3 x 2 )x 1, x 2 (x 1 x 3 ), x 2 (x 3 x 1 ), (x 1 x 3 )x 2, (x 3 x 1 )x 2 ), where each juxtaposition represents an operation in Ω 2. We will enote the right sie of Axiom 2 by W (x 1, x 2, x 3 ;, ). 3.1. Definition. A category of groups with operations C satisfying conitions (a) (), Axiom 1 an Axiom 2, will be calle a moifie category of interest.

890 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU 3.2. Remark. The ifference of this efinition from the original one of a category of interest is the ientity ω(x) ω(y) = ω(x y), which is ω(x) y = ω(x y) in the efinition of a category of interest. Denote by E G the subset of ientities of E which inclues the group ientities an the ientities (c) an (). We enote by C G the corresponing category of groups with operations. Thus we have E G E, C = (Ω, E), C G = (Ω, E G ) an there is a full inclusion functor C C G. We will call C G a general category of groups with operations of a moifie category of interest C. 3.3. Example. The categories Cat 1 -Ass, Cat 1 -Lie, Cat 1 -Leibniz, PreCat 1 -Ass, PreCat 1 -Lie an PreCat 1 -Leibniz are moifie categories of interest, which are not categories of interest. Also the category of commutative von Neumann regular rings is isomorphic to the category of commutative rings with a unary operation ( ) satisfying two axioms, efine in [Borceux, Janelize an Kelly, 2005 ], which is a moifie category of interest. 3.4. Definition. Let C C. A subobject of C is calle an ieal if it is the kernel of some morphism. 3.5. Proposition. Let A be a subobject of B in C. Then like in the case of categories of interest we have that A is an ieal of B if an only if the following conitions hol: 1. A is a normal subgroup of B, 2. a b A, for all a A, b B an Ω 2. Proof. Can be prove in a similar way as Theorem 1.7 in [Orzech, 1972]. 3.6. Definition. Let A, B C. An extension of B by A is a sequence 0 A i E p B 0, (3.1) in which p is surjective an i is the kernel of p. We say that an extension is split if there is a morphism s : B E such that ps = 1 B. 3.7. Definition. For A, B C we will say that we have a set of actions of B on A, whenever there is a map f : A B A for each Ω 2. A split extension of B by A, inuces an action of B on A corresponing to the operations in C. For a given split extension (3.1), we have b a = s(b) + a s(b), (3.2) b a = s(b) a, (3.3) for all b B, a A an Ω 2. Actions efine by (3.2) an (3.3) will be calle erive actions of B on A. We will often use the notation b a, where we mean both the ot an the star actions.

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 891 3.8. Definition. Given an action of B on A, a semiirect prouct A B is a universal algebra, whose unerlying set is A B an the operations are efine by for all a, a A, b, b B, Ω 2. ω(a, b) = (ω (a), ω (b)), (a, b ) + (a, b) = (a + b a, b + b), (a, b ) (a, b) = (a a + a b + b a, b b), 3.9. Theorem. An action of B on A is a erive action if an only if A B is an object of C. Proof. Let B have a erive action on A efine by the split extension (3.1). We will only show that A B satisfies the new conition. Other proofs can be foun in [Orzech, 1972]. By the efinition of semiirect prouct, we have: ω((a, b ) (a, b)) = ω(a a + a b + b a, b b) = (ω (a ) ω (a) + ω (a ) ω (b) + ω (b ) ω (a), ω (b ) ω (b)) = (ω (a ), ω (b )) (ω (a), ω (b)) = ω(a, b ) ω(a, b), for all (a, b), (a, b ) A B. 3.10. Proposition. A set of actions of B on A in C G is a set of erive actions if an only if it satisfies the following conitions: 1. 0 a = a, 2. b (a 1 + a 2 ) = b a 1 + b a 2, 3. (b 1 + b 2 ) a = b 1 (b 2 a), 4. b (a 1 + a 2 ) = b a 1 + b a 2, 5. (b 1 + b 2 ) a = b 1 a + b 2 a, 6. (b 1 b 2 ) (a 1 a 2 ) = a 1 a 2, 7. (b 1 b 2 ) (a b) = a b, 8. a 1 (b a 2 ) = a 1 a 2, 9. b (b 1 a) = b a, 10. ω(b a) = ω(b) ω(a), 11. ω(a b) = ω(a) ω(b), 12. x y + z t = z t + x y, for each ω Ω 1, Ω 2, b, b 1, b 2 B, a, a 1, a 2 A an for x, y, z, t A B whenever each sie of 12 makes sense. Proof. Follows from Definitions 3.7, 3.8 an Theorem 3.9.

892 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU 3.11. Remark. In the case of a category of interest, conition 11 is ω(a b) = ω(a) b [Casas, Datuashvili an Lara, 2010]. 3.12. Example. Let (A, ω A 1, ω A 0 ), (B, ω B 1, ω B 0 ) Cat 1 -Ass an let (B, ω B 1, ω B 0 ) have a erive action on (A, ω A 1, ω A 0 ). By Definition 3.7 we have the ientities (2.1) an also the ientities a b = b a = 0, if b ker ω B 0, a ker ω A 1 or b ker ω B 1, a ker ω A 0 ; ω B i (b) ω A i (a) = ω A i (b a), ω A i (a) ω B i (b) = ω A i (a b), ω B j (b) ω A i (a) = ω A i (ω B j (b) a), ω A i (a) ω B j (b) = ω A i (a ω B j (b)), ω B j (b) ω A i (a) = ω A j (b ω A i (a)), ω A i (a) ω B j (b) = ω A j (ω A i (a) b), i, j = 0, 1, i j, for any a (A, ω A 1, ω A 0 ), b (B, ω B 1, ω B 0 ). 3.13. Definition. A precrosse moule in a moifie category of interest C is a triple (C 1, C 0, ), where C 0, C 1 C, the object C 0 has a erive action on C 1 an : C 1 C 0 is a morphism in C with the conitions: a) (c 0 c 1 ) = c 0 + (c 1 ) c 0, b) (c 0 c 1 ) = c 0 (c 1 ), for all c 0 C 0, c 1 C 1, an Ω 2. In aition, if : C 1 C 0 satisfies the conitions c) (c 1 ) c 1 = c 1 + c 1 c 1, ) (c 1 ) c 1 = c 1 c 1, for all c 1, c 1 C 1, an Ω 2, then the triple (C 1, C 0, ) is calle a crosse moule in C. 3.14. Definition. A morphism between two (pre)crosse moules (C 1, C 0, ) (C 1, C 0, ) in C is a pair of morphisms (µ 1, µ 0 ) in C, µ 0 : C 0 C 0, µ 1 : C 1 C 1, such that a) µ 0 (c 1 ) = µ 1 (c 1 ), b) µ 1 (c 0 c 1 ) = µ 0 (c 0 ) µ 1 (c 1 ), c) µ 1 (c 0 c 1 ) = µ 0 (c 0 ) µ 1 (c 1 ), for all c 0 C 0, c 1 C 1 an Ω 2. 3.15. Definition. Let A C. The center of A is efine by Z(A) = {z A a + z = z + a, a + ω(z) = ω(z) + a, a z = 0, a ω (z) = 0, for all a A, ω Ω 1 an Ω 2 }.

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 893 3.16. Definition. If A is an ieal of B, then Z(B, A) = {b B a + b = b + a, a + ω(b) = ω(b) + a, a b = 0, a ω (b) = 0, for all a A, ω Ω 1 an Ω 2 } is calle the centralizer of A in B. 3.17. Lemma. In a moifie category of interest, Z(B, A) is an ieal of B. Proof. Follows from Definition 3.16 The efinition of split extension classifier (object which represents actions), is formulate in [Borceux, Janelize an Kelly, 2005] for semi-abelian categories in terms of categorical notions of internal object action an semiirect prouct. Categories of interest are semi-abelian categories. Accoring to [Bourn an Janelize, 1998] in this special case these notions coincie with the ones given in [Orzech, 1972]. Analogous situation we have in the cases of moifie categories of interest an categories equivalent to them. Therefore the efinition of a split extension classifier for moifie categories of interest has the following form. Consier the category of all split extensions with fixe kernel A; thus the objects are 0 A C s C 0 an the arrows are the triples of morphisms (1 A, γ, γ ) ) between the extensions, which commute with the section homomorphisms as well. By efinition, an object [A] is a split extension classifier for A if there exists a erive action of [A] on A, such that the corresponing extension s 0 A A [A] [A] 0 is a terminal object in the above efine category. 3.18. Proposition. Let C be a moifie category of interest an A be an object in C. An object B C is a split extension classifier for A in the sense of [Borceux, Janelize an Kelly, 2005] if an only if it satisfies the following conition: B has a erive action on A such that for all C in C an a erive action of C on A there is a unique morphism ϕ : C B, with c a = ϕ(c) a, c a = ϕ(c) a, for all Ω 2, a A an c C. Proof. Analogous to the one for categories of interest [Casas, Datuashvili an Lara, 2010]. The object B in C satisfying the above state conition will be calle an actor of A an enote by Act(A). The corresponing universal acting object, which represents actions in the sense of [Borceux, Janelize an Kelly, 2005, Borceux, Janelize an Kelly, 2005 ], in the categories equivalent to moifie categories of interest will be calle a split extension classifier an enote by [A], as it is generally in semi-abelian categories. 3.19. Remark. As a consequence of this proposition, an actor of an object is unique up to isomorphism.

894 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU 3.20. Definition. Let A, B C. We will say that a set of actions of B on A is strict if for any two elements b, b B, from the conitions b a = b a, ω(b) a = ω(b ) a, b a = b a an ω(b) a = ω(b ) a, for all a A, ω Ω 1 an Ω 2, it follows that b = b. 3.21. Remark. In the case of a category of interest, the conition ω(b) a = ω(b ) a is always satisfie. 3.22. Example. For any object A C we have an action of A on itself efine by a a = a + a a, a a = a a, for all a, a A, Ω 2, where on the left sie enotes the action an on the right sie the operation in A. This action is calle an action by conjugation of A on itself. 3.23. Example. Let A C an Z(A) = 0. Then the action by conjugation of A on itself is a strict action. 3.24. Proposition. Let A, B C. A set of erive actions of B on A is strict if an only if in the corresponing split extension (3.1), we have Im(s) Z(E, A) = 0. Proof. Follows from Definitions 3.16 an 3.20. 3.25. Example. Let A C. If Act(A) exists, then the erive action of Act(A) on A is strict. 3.26. Remark. Let A C. If Act(A) exists, then by Definition 3.18 an Example 3.22, there is a unique morphism β : A Act(A) in C, etermine by β(a) a = a a, an β(a) a = a a, for all a, a A. β : A Act(A) is a crosse moule an a terminal object in the category of crosse moules with the same omain A. Up to isomorphism, there is a unique crosse moule with this property. 3.27. Definition. We will say that an object G C G has a general actor property to the object A C (for shortness GA(A)-property) if G has a set of actions on A C, which is a set of erive actions in C G an for any object C C an a erive action of C on A in C, there exists in C G a unique morphism ϕ : C G such that c a = ϕ(c) a, for all c C, a A an Ω 2. 3.28. Definition. We will say that an object G C G has a strict general actor property to the object A C (for shortness SGA(A)-property) if G has GA(A)-property an the action of G on A is strict. Below we formulate a conition on objects with GA(A)-property, which will be use in the efinition of universal strict general actor of an object in C. Conition 1. Let A C an {B j } j J enote the set of all objects in C which have erive actions on A. Let G be an object in C G with GA(A)-property an ϕ j : B j G, j J, enote the corresponing unique morphism such that b j a = ϕj (b j ) a, for all

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 895 b j B j, a A, Ω 2. The elements of G of the type ϕ i (b i ), i J satisfy the following equality: (ϕ i (b i ) ϕ j (b j )) a = W (ϕ i (b i ), ϕ j (b j), a;, ) for any b i B i, b j B j,, Ω 2, i, j J an a A. 3.29. Definition. A universal strict general actor of an object A, enote by USGA(A), is an object in C G with SGA(A)-property an with Conition 1, such that for any object G with SGA(A)-property an with Conition 1 there exists a unique morphism η : USGA(A) G in the category C G, with ηψ j = ϕ j, for any j J, where ϕ j : B j G an ψ j : B j USGA(A) enote the corresponing unique morphisms with the appropriate properties from the efinition of general actor property. By Corollary 3.33 a universal strict general actor is the unique object (up to isomorphism) satisfying the corresponing properties. We establish the following two statements without proofs, since they are the same as the ones of Proposition 3.8 an Theorem 3.9 in the case of categories of interest [Casas, Datuashvili an Lara, 2010]. 3.30. Proposition. Let C be a moifie category of interest an A C. If an actor Act(A) exists, then the unique morphism η : USGA(A) Act(A) is an isomorphism with x a = η(x) (a), for all x USGA(A), a A. 3.31. Theorem. Let C be a moifie category of interest an A C. A has an actor if an only if the semiirect prouct A USGA(A) is an object in C. If it is the case, then Act(A) = USGA(A). Now we will show the existence of USGA(A) for any object A in a moifie category of interest C. The construction given here is similar to that given for the case of categories of interest in [Casas, Datuashvili an Lara, 2010] with some moifications. Let A C; consier all split extensions of A in C E j : 0 A i j C j p j Bj 0, j J. When B j = B k = B, for j k, in this case the corresponing extensions erive ifferent actions of B on A. Let {b j, b j b j B j, Ω 2} be the set of functions efine by the action of B j on A. For any element b j B j enote b j = {b j, b j, Ω 2}. Let B = {b j b j B j, j J}. Thus each element b j B, j J is a special type of a function b j : Ω 2 Maps(A A), efine by Ω 2. b j (+) = b j, b j ( ) = b j : A A,

896 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU We efine the operation, b i b k, Ω 2, for the elements of B accoring to Axiom 2. ((b i b k )( )) (a) = W (b i, b k, a;, ), an ((b i b k )(+))(a) = a. We efine the operation of aition by ((b i + b k )(+))(a) = b i (b k a), ((b i + b k )( ))(a) = b i a + b k a. For a unary operation ω Ω 1 we efine (ω(b k )(+))(a) = ω(b k ) (a), (ω(b k )( ))(a) = ω(b k ) (a), ω(b b ) = ω(b) ω (b ) (in the case of categories of interest, this conition is ω(b b ) = ω(b) b ) ω(b 1 + + b n ) = ω(b 1 ) + + ω(b n ), (( b k )(+))(a) = ( b k ) a, ( b) (a) = a (( b k )( ))(a) = (b k a), (( b)( ))(a) = ((b( ))(a)), (b 1 + + b n ) = b n b 1, where b, b, b 1,..., b n are certain combinations of operations on the elements of B, i.e. the elements of the type b i1 1 n 1 b in, n > 1. Denote by B (A) the set of all functions (Ω 2 Maps(A A)) obtaine by performing all kin of operations efine above on the elements of B an on the new obtaine elements as the results of operations. In what follows we will write for simplicity b a or b a instea of (b( ))(a) or (b(+))(a), respectively, where b B (A), a A. Note that it may happen that b a = b a, for any a A an any Ω 2, but we o not have the equality ω(b) a = ω(b ) a, for any a A, any Ω 2 an any unary operation ω (i.e. for any ω which is a finite combination of elements of Ω 1). We efine the following relation: we will write b b, for b, b B (A), if an only if b a = b a an ω(b) a = ω(b ) a, for any a A, any Ω 2 an any unary operation ω. This relation is a congruence relation on B (A), i.e. it is compatible with the operations we have efine in B (A). We efine B(A) = B (A)/. The operations efine on B (A) efine the corresponing operations on B(A). For simplicity we will enote the elements of B(A) by the same letters b, b etc. instea of the classes clb, clb etc. 3.32. Theorem. Let A C, then we have: 1. B(A) is an object of C G ; 2. The set of actions of B(A) on A, efine as in [Casas, Datuashvili an Lara, 2010] for categories of interest, is a set of strict erive actions in C G ; 3. B(A) is a universal strict general actor for A. Proof. The proofs of the statements 1., 2., an 3. are similar to the ones of Proposition 4.1, Proposition 4.2 an Theorem 4.3, respectively, for categories of interest [Casas, Datuashvili an Lara, 2010].

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 897 3.33. Corollary. Let C be a moifie category of interest. For any universal strict general actor C of an object A, we have an isomorphism θ : C B(A) with θ(c) a = c a, for all c C, a A. The unique morphism η given in Definition 3.29 satisfies the analogous conition η(x) a = x a, for all x USGA(A), a A. Proof. See the proof of Corollary 4.4 in [Casas, Datuashvili an Lara, 2010]. Define a map : A B(A) by (a) = a, where a = {a, a : Ω 2}. By efinition we have (a) a = a + a a, an (a) a = a a, for all a, a A, Ω 2. By irect checking we have also ker = Z(A). 3.34. Proposition. For any object A in C we have: 1. : A B(A) is a crosse moule in C G. 2. For any crosse moule γ : A B in C there exists a unique crosse moule morphism A: (A, B, γ) (A, B(A), ) in C G uner the object A. 3. : A B(A) is the unique crosse moule (up to isomorphism) with the property that the operator object on the right sie is a universal strict general actor of A. Proof. 1. In the proof we will only show that the new equalities, which occur in the moifie category of interest case, are satisfie. The others can be foun in [Casas, Datuashvili an Lara, 2010]. First we will prove that is a homomorphism in C G. For this we will show that (ω (a)) = ω((a)), for any ω Ω 1 an a A. Therefore, we nee to prove for all a, a A, ω, ω Ω 1, Ω 2, that (ω(a)) a = ω((a)) a, ω ((ω(a))) a = ω (ω((a))) a, (ω(a)) a = ω((a)) a, ω ((ω(a))) a = ω (ω((a))) a, We will prove the fourth equality. Since we have (ω(a)) a = ω(a) a, ω((a)) a = ω(a) a = ω(a) a, ω ((ω(a))) a = ω (ω(a)) a = ω (ω((a))) a, as require. On the other han, for ω = we have ( a) = (a) an (a 1 + a 2 ) = (a 1 ) + (a 2 ).

898 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU The next equality that we must prove is (a 1 a 2 ) = (a 1 ) (a 2 ). For this we nee to show the following four equalities: (a 1 a 2 ) a = ((a 1 ) (a 2 )) (a), ω((a 1 a 2 )) a = ω((a 1 ) (a 2 )) a, (a 1 a 2 ) a = ((a 1 ) (a 2 )) (a), ω((a 1 a 2 )) a = ω((a 1 ) (a 2 )) a. We will only show the fourth equality. For this, we have ω((a 1 a 2 )) a = (ω(a 1 a 2 )) a = (ω(a 1 ) ω(a 2 )) a = W (ω (a 1 ), ω (a 2 ), a;, ) = W ((ω (a 1 )), (ω (a 2 )), a;, ) = ((ω(a 1 )) (ω(a 2 ))) a = (ω((a 1 )) ω((a 2 ))) a = ω((a 1 ) (a 2 )) a as require. Now we will show that is a crosse moule. We have to check conitions a)-) from the efinition of a crosse moule given in Definition 3.13. We will check conition b). Other conitions are checke similarly, see also [Casas, Datuashvili an Lara, 2010]. The conition b) states (b a) = b (a), for any b B(A), a A, Ω 2. Thus we have to show (b a) a = (b (a)) a, ω((b a)) a = ω(b a) a, where each equality represents two equalities corresponing to the ot an star actions. We only show the secon equlity of the secon one, others are prove analogously. Consier the case b = b i, i J. Recall that by efinition ω (b i ) a = ω (b i ) a. We have ω ((b i a)) a = (ω(b i a)) a = (ω(b i a)) a = (ω(b i ) ω(a)) a = (ω(b i ) ω(a)) a = W (ω(b i ), ω(a), a ;, ) = W (ω(b i ), ω(a), a ;, ) = W (ω(b i ), (ω(a)), a ;, ) = (ω(b i ) (ω(a))) a = (ω(b i ) ω((a))) a = ω(b i (a)) a. The case b = b i1 1 n 1 b in, or b is the sum of elements of the type b i1 1 n 1 b in is prove in a similar way as in the case of categories of interest [Casas, Datuashvili an

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 899 Lara, 2010], i.e. by application of the result in the case b = b i an Axiom 2. The proofs of 2. an 3. are the same as the ones of Proposition 4.6 for categories of interest [Casas, Datuashvili an Lara, 2010]. 4. Actor of an object in Cat 1 -Ass In this section we will construct an object (A(A), ω 0, ω 1 ) for an object (A, w 0, w 1 ) in Cat 1 - Ass an prove that uner certain conitions it is an actor of (A, w 0, w 1 ). The construction is euce from the general construction of a universal strict general actor in moifie categories of interest given in Section 3 an its interpretation for the case C = Cat 1 -Ass, i.e. from the construction of (B(A),w B(A) 0, w B(A) Consier the triples (f l,r, f 0 l,r, f 1 l,r M1. f i l,r ω i = ω i f l,r, for i = 0, 1, 1 ), for (A, w 0, w 1 ) Cat 1 -Ass. ), which consist of bimultipliers of A such that M2. f j l,r ω i = ω i f j l,r, for i = 0, 1, j = 0, 1, an ω if j l,r = ω jf l,r ω i, for i = 0, 1, j = 0, 1, i j, M3. f l,r (x) = f 1 l,r (x), for all x ker ω 0, M4. f l,r (x) = f 0 l,r (x), for all x ker ω 1. Denote the set of all this kin of triples by A(A). This set is not empty, we will show now that the elements of (B(A),w B(A) 0, w B(A) 1 ) are elements of A(A). We have to show that the elements of (B(A),w B(A) 0, w B(A) 1 ) satisfy conitions M1-M4. The proofs of M1 an M2 are obvious. We will emonstrate M3; M4 is prove in an analogous way. First we will show that if (B i, w B i 0, w B i 1 ) has a erive action on (A, w0 A, w1 A ) in Cat 1 -Ass, then for any b i B i the triple (b i(l,r), ω B i 0 (b i(l,r) ), ω B i 1 (b i(l,r) ) ) (i.e. cl(b i(l,r), ω B i 0 (b i(l,r) ), ω B i 1 (b i(l,r) ) )) is an element of A(A). We will show this fact for the left multipliers an omit the corresponing inex l. Let a ker ω0 A, then since (b i w B i 1 (b i )) ker ω B i 1, we have (b i w B i 1 (b i )) a = 0 (see Example 3.12 for the erive action conitions in Cat 1 - Ass), from which follows that conition M3 is satisfie. Now let b = b i b j, where b i B i, b j B j, an B i an B j have erive actions on A. Then by efinition of action of (B(A),w B(A) 0, w B(A) 1 ) on A we have (b i b j ) a ω B(A) 1 (b i b j ) a = (b i b j ) a (ω B i 1 (b i ) ω B j 1 (b j )) a = b i (b j a) ω B i 1 (b i ) (ω B j 1 (b j ) a) = b i (ω B j 1 (b j ) a) ω B i 1 (b i ) (ω B j 1 (b j ) a) = 0, since ω B j 1 (b j ) a ker ω0 A. The case where b is any type of element in (B(A),w B(A) 0, w B(A) 1 ) is prove by inuction on the length of b, applying the istributive property of the action an the fact that ω 0 an ω 1 are homomorphisms for the aition. The fact that a b = a ω B(A) 1 (b) is prove in an analogous way. An easy checking shows that if (f l,r, f 0 l,r, f 1 l,r triples (f 0 l,r, f 0 l,r, f 0 l,r ) an (f 1 l,r, f 1 l,r, f 1 l,r ) satisfies conitions M1-M4, then the ) also satisfy the same conitions. We will emon- 1 ); the proof for the case (fl,r, f l,r 1, f l,r 1 ) is analogous. strate this fact for the triple (f 0 l,r, f 0 l,r, f 0 l,r

900 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU Checking of conition M1. We have to show that f 0 l,rω 0 = ω 0 f 0 l,r anf 0 l,rω 1 = ω 1 f 0 l,r. The first equality follows from M2, for the triple (f l,r, fl,r 0, f l,r 1 ), for i = j = 0, an the secon one follows again from M2, for i = 1 an j = 0. Checking of conition M2, the case i=j=0. We have to prove f 0 l,rω 0 = ω 0 f 0 l,r. This equality was prove above as conition M1. The case i = 0, j = 1. We have to prove the following equalities f 0 l,rω 0 = ω 0 f 0 l,r = ω 1 f 0 l,rω 0. The first equality has been alreay prove in the previous case. For the secon equality we apply the first equality of this case, the fact that ω 1 ω 0 = ω 0 an obtain ω 0 fl,r 0 = ω 1 ω 0 fl,r 0 1fl,r 0 0. The case i = 1, j = 0. We have to prove f 0 l,rω 1 = ω 1 f 0 l,r = ω 0 f 0 l,rω 1. The first equality follows from M2 for the triple (f l,r, fl,r 0, f l,r 1 ), for i = 1, j = 0. For the secon equality we apply the fact that ω 1 ω 1 = ω 1, the first equality of this case an obtain ω 1 fl,r 0 = ω 0ω 1 fl,r 0 = ω 0fl,r 0 ω 1. The case i = 1, j = 1. We have to prove f 0 l,rω 1 = ω 1 f 0 l,r. This equality was prove alreay in the previous case. It is obvious that conitions M3 an M4 are satisfie for the triple (f 0 l,r, f 0 l,r, f 0 l,r ). A(A) has an associative algebra structure with componentwise aition, scalar multiplication an multiplication of the corresponing bimultipliers. The zero element is the triple (0, 0, 0), where 0 is the zero map. (A(A), ω 0, ω 1 ) is a cat 1 -algebra with unary operations, ω 0, ω 1 : A(A) A(A) efine by ω 0 (f, f 0, f 1 ) = (f 0, f 0, f 0 ) an ω 1 (f, f 0, f 1 ) = (f 1, f 1, f 1 ), respectively. Define an action of A(A) on A by the maps A(A) A A, ((f, f 0, f 1 ), a) = f l (a), an A A(A) A, (a, (f, f 0, f 1 )) = f r (a), for all a A an (f, f 0, f 1 ) A(A). We have an injective homomorphism ψ : (B(A),w B(A) 0, w B(A) 1 ) (A(A), ω 0, ω 1 ). It is worth to recall that we write left or right bimultipliers as maps on the left sie of an element (i.e. f l (a) an f r (a), see Section 2). Therefore for the prouct of two right bimultipliers b an b, from the triples in A(A), which is a composition of bimultipliers

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 901 by efinition of prouct in A(A) we have ( b)( b )(a) = (a b ) b, since ( b)( b ) enotes the composition an we apply first b an then b. Conition 2. Let A be an associative algebra such that Ann(A) = 0 or A 2 = A. Let A 0, A 1 Ass an A 0 has an action on A 1. It is easy to see that if A i, i = 0, 1, satisfy Conition 2, then the semiirect prouct A 1 A 0 also satisfies this conition. We will show first that if Ann(A i ) = 0, then Ann(A 1 A 0 ) = 0. Suppose (a 1, a 0) (a 1, a 0 ) = 0, for any a i A i, i = 0, 1. By the efinition of multiplication in A 1 A 0, it follows that a 0 a 0 = 0 an a 1 a 1 +a 1 a 0 +a 0 a 1 = 0, for any a i A i, i = 0, 1. From the first equality it follows that a 0 = 0. Taking in the secon equality a 0 = 0 we obtain that a 1 = 0, which proves that Ann(A 1 A 0 ) = 0. Now suppose that A 2 i = A i, i = 0, 1; we have to show that (A 1 A 0 ) 2 = A 1 A 0. Suppose (a 1, a 0 ) A 1 A 0, where a 0 = a 1 1 0 a 1 2 0 + + a i 1 0 a i 2 0 an a 1 = a 1 1 1 a 1 2 1 + + a j 1 1 a j 2 1. We have the following equalities (a 1, a 0 ) = (0, a 1 1 0 a 1 2 0 + + a i 1 0 a i 2 0 ) + (a 1 1 1 a 1 2 1 + + a j 1 1 a j 2 1, 0) = (0, a 1 1 0 ) (0, a 1 2 0 ) + + (0, a i 1 0 ) (0, a i 2 0 ) + (a 1 1 1, 0) (a 1 2 1, 0) + + (a j 1 1, 0) (a j 2 which proves that (A 1 A 0 ) 2 = A 1 A 0. 4.1. Proposition. If A satisfies Conition 2, then there is an isomorphism (A(A), ω 0, ω 1 ) (B(A),w B(A) 0, w B(A) 1 ). 1, 0), Proof. It is a well-known fact that uner Conition 2 the action of Bim(A) on A is a erive action in the category of associative algebras [Lavenhomme an Lucas, 1996, Casas, Datuashvili an Lara, 2010, Borceux, Janelize an Kelly, 2005 ]. In an analogous way, from the efinition of action of A(A) on A, one can easily see that this action is a erive action in Ass an in Cat 1 -Ass as well. Therefore, since (B(A),w B(A) 0, w B(A) 1 ) has general actor property, by efinition (see Definition 3.27) there exists a unique morphism ϕ : A(A) B(A) in Cat 1 -Ass G such that ϕ((f, f 0, f 1 )) a = (f, f 0, f 1 ) a an a ϕ((f, f 0, f 1 )) = a (f, f 0, f 1 ), for all a A an (f, f 0, f 1 ) A(A). We have shown above that there exists an injective homomorphism ψ : (B(A),w B(A) 0, w B(A) 1 ) (A(A), ω 0, ω 1 ). For any b (B(A),w B(A) 0, w B(A) 1 ) an a A we have ψ(b) a = b a an a ψ(b) = a b. By the construction of (A(A), ω 0, ω 1 ) its action on (A, ω 0, ω 1 ) is strict. By Theorem 3.32.2. the action of (B(A),w B(A) 0, w B(A) 1 ) on (A, ω 0, ω 1 ) is also strict. From these facts it follows that ϕψ = 1 an ψϕ = 1. Therefore we fin that ϕ is an isomorphism. 4.2. Corollary. If A satisfies Conition 2, then (A(A), ω 0, ω 1 ) is an actor of (A, ω 0, ω 1 ). Proof. Since (A(A), ω 0, ω 1 ) Cat 1 -Ass an its action on A is a erive action in this category, the semiirect prouct (A, ω 0, ω 1 ) (A(A), ω 0, ω 1 ) Cat 1 -Ass. Now the result follows from Proposition 4.1 an Theorems 3.31 an 3.32.3.

902 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU 4.1 Actor of a cat 1 -algebra corresponing to a given crosse moule Let A 1, A 0 be associative algebras with a erive action of A 0 on A 1. Let f = (f l, f r ) Bim(A 1 A 0 ). Then f l : A 1 A 0 A 1 A 0 can be represente by four k-linear maps such that α l : A 1 A 1, γ l : A 1 A 0, β l : A 0 A 0, l : A 0 A 1 f l (a 1, a 0 ) = (α l (a 1 ) + l (a 0 ), β l (a 0 ) + γ l (a 1 )) for all a 1 A 1, a 0 A 0. Also, f r : A 1 A 0 A 1 A 0 can be represente by four k-linear maps such that α r : A 1 A 1, γ r : A 1 A 0, β r : A 0 A 0, r : A 0 A 1 f r (a 1, a 0 ) = (α r (a 1 ) + r (a 0 ), β r (a 0 ) + γ r (a 1 )) for all a 1 A 1, a 0 A 0. Let f = (f l, f r ) Bim(A 1 A 0 ) an suppose f satisfies the conition M1, for i = 0 an certain f 0 Bim(A 1 A 0 ). Then we fin that γ r = γ l = 0 an so we can represent f by the triple (α,, β). One can easily see that α Bim(A 1 ), β Bim(A 0 ) an we have the following conitions A1. α l (a 0 a 1 ) = l (a 0 ) a 1 + β l (a 0 ) a 1, A2. α l (a 1 a 0 ) = α l (a 1 ) a 0, A3. a 0 α l (a 1 ) = r (a 0 ) a 1 + β r (a 0 ) a 1, A4. α r (a 1 a 0 ) = a 1 r (a 0 ) + a 1 β r (a 0 ), A5. α r (a 0 a 1 ) = a 0 α r (a 1 ), A6. α r (a 1 ) a 0 = a 1 l (a 0 ) + a 1 β l (a 0 ), A7. r (a 0 a 0) = a 0 r (a 0), A8. l (a 0 a 0) = l (a 0 ) a 0, A9. a 0 l (a 0) = r (a 0 ) a 0, for all a 1 A 1, a 0, a 0 A 0. We have an analogous result for the representations of bimultipliers f 0 an f 1 from any triple (f, f 0, f 1 ) A(A 1 A 0 ), since we have showe that (f 0, f 0, f 0 ) an (f 1, f 1, f 1 ) are elements in A(A 1 A 0 ), an therefore f 0 an f 1 satisfy conition M1 for i = 0. Note that in the case where the semiirect prouct A 1 A 0 correspons to a certain crosse moule A 1 A 0, Conitions A7-A9 are crosse bimultiplier conitions of the crosse moule (Section 2, Definition 2.6).

ACTIONS IN MODIFIED CATEGORIES OF INTEREST 903 4.3. Notation. In the rest of the paper, A will enote a crosse moule A : (A 1 A 0 ), S(A) will enote the corresponing cat 1 -algebra (A 1 A 0, ω 1, ω 0 ). 4.4. Proposition. Let f := (α,, β), f 0 := (α 0, 0, β 0 ), f 1 := (α 1, 1, β 1 ) be bimultipliers of A 1 A 0. Then (f, f 0, f 1 ) A(A 1 A 0 ) if an only if (f, f 0, f 1 ) satisfies the following conitions. 1. β l,r (a 0 ) = β 0 l,r (a 0), 2. i l,r (a 0) = 0, for i = 0, 1, 3. β 1 l,r (a 0) = l,r (a 0 ) + β l,r (a 0 ), 4. β 1 l,r (a 1) = α l,r (a 1 ), 5. β i l,r (a 1) = α i l,r (a 1), for i = 0, 1, 6. α l,r (a 1 ) = α 1 l,r (a 1), 7. α l,r (a 1 ) = α 0 l,r (a 1) + l,r (a 1 ), for all a 0 A 0, a 1 A 1. Proof. Let (f, f 0, f 1 ) A(A 1 A 0 ). All the properties 1-7 follow from the conitions M1-M4 an the presentations of the bimultipliers f l,r, fl,r 0, an f l,r 1 in terms of the corresponing linear maps α, β,, given in this section. We will emonstrate property 6. We have f l,r (a 1, a 0 ) = (α l,r (a 1 ) + l,r (a 0 ), β l,r (a 0 )). For any a 1 A 1 we obtain Analogously, f l,r (a 1, 0) = (α l,r (a 1 ), 0). f 1 l,r(a 1, 0) = (α 1 l,r(a 1 ), 0). Now the result follows from M3. The converse statement of the proposition is prove by irect checking an is left to the reaer. 4.5. Corollary. ker ω 0 = {(f, 0, f 1 ) A(A 1 A 0 )}, an any element (f, 0, f 1 ) ker ω 0 can be represente by ((α,, 0), (0, 0, 0), (α, 0, β 1 )). Proof. Follows from the efinition of the map ω 0 an Proposition 4.4.

904 Y. BOYACI, J. M. CASAS, T. DATUASHVILI AND E. Ö. USLU 4.6. Proposition. For any ((α,, 0), (0, 0, 0), (α, 0, β 1 )) ker ω 0 we have: 1. (α, β 1 ) is a bimultiplier of A, 2. α l,r (a 1 ) = l,r (a 1 ), 3. β 1 l,r (a 0) = l,r (a 0 ), for all a 0 A 0, a 1 A 1. Proof. 1. By Proposition 4.4, conition 6, we have α = α 1 an by A1-A6 we obtain that (α 1, β 1 ) satisfies conitions of Definition 2.4. Property 2. Follows from Proposition 4.4, conition 7 Property 3. Follows from Proposition 4.4, conitions 1, 2 an 3 an from the fact that β 0 l,r = 0 5. Split extension classifier of a crosse moule Accoring to the efinition of action in semi-abelian categories [Borceux, Janelize an Kelly, 2005 ], it is natural to efine an action (i.e. a erive action) in XAss in an analogous way as it is efine in a moifie category of interest, thus as an action erive from a split extension in this category. In this section we will construct a crosse moule for a given crosse moule A : A 1 A 0 an prove, that if A 0 an A 1 satisfy Conition 2, then is isomorphic to P (Act(S (A))), which means that is the split extension classifier of A. 5.1. Lemma. The bilinear maps given by Bim(A) BM(A) BM(A), BM(A) Bim(A) BM(A), ((α, β ), ), (, (α, β )), efine a erive action of Bim(A) on BM(A) in the category of associative algebras, where l = α l l, r = r β r an l = l β l, r = α r r, for all (α, β ) Bim(A), Bim(A 0, A 1 ). Proof. The proof is a irect consequence of the efinitions.