Representability of actions in the category of (Pre)crossed modules in Leibniz algebras

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1 Communications in Algebra ISSN: (Print) (Online) Journal homepage: Representability of actions in the category of (Pre)crosse moules in Leibniz algebras M. Atık, A. Aytekın & E.Ö. Uslu To cite this article: M. Atık, A. Aytekın & E.Ö. Uslu (2017) Representability of actions in the category of (Pre)crosse moules in Leibniz algebras, Communications in Algebra, 45:5, , DOI: / To link to this article: Accepte author version poste online: 07 Oct Publishe online: 07 Oct Submit your article to this journal Article views: 45 View relate articles View Crossmark ata Full Terms & Conitions of access an use can be foun at Downloa by: [The UC Irvine Libraries] Date: 29 September 2017, At: 11:57

2 COMMUNICATIONS IN ALGEBRA 2017, VOL. 45, NO. 5, Representability of actions in the category of(pre)crosse moules in Leibniz algebras M.Atık a,a.aytekın b,ane.ö.uslu c a DepartmentofRiskManagementanInsurance,FacultyofEconomicsanAministrativeSciences,ÇankırıKaratekin University,Çankırı,Turkey; b DepartmentofMathematics,FacultyofArtanScience,AksarayUniversity,Aksaray,Turkey; c DepartmentofMathematicsanComputerSciences,ArtanScienceFaculty,OsmangaziUniversity,Eskisehir,Turkey Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 ABSTRACT We investigate the representability of actions in the category of (pre)crosse moules in Leibniz algebras. For this, we construct an actor of a (pre)cat 1 - Leibniz algebra an then by using the natural equivalence of the categories of (pre)cat 1 -Leibniz algebras an that of (pre)crosse moules, we construct the split extension classifier of the corresponing(pre)crosse moule. 1. Introuction ARTICLE HISTORY Receive 15 July 2014 Revise 1 April 2016 Communicate by I. Shestakov KEYWORDS Action; actor; precrosse moule; split extension classifier 2010 MATHEMATICS SUBJECT CLASSIFICATION 17A32; 16W25; 17A36; 17B40 In an algebraic category, many homological properties such as obstruction theory of the objects epen on the representability of actions in the category. Representability of actions in semi-abelian categories wasinvestigatein[3].aifferentstuyofthisprobleminthecategoriesofinterestwasgivenin[7]with a combinatorial approach. The same was given for moifie categories of interest in[8]. The efinition of split extension classifier(object which represents actions), is formulate in[2] for semi-abelian categories in terms of categorical notions of internal object action an semiirect prouct. Categories of interest are semi-abelian categories. As an application of[4], in this special case these notions coincie with the ones given in[12]. Analogous situation exists in the case of moifie category of interest an categories equivalent to them. Split extension classifiers (which calle actors in [8, 10]) play an important role by analogy with automorphism group of a group in the category of groups or erivation algebra of a Lie algebra in the category of Lie algebras. Manywellknowncategoriesofalgebraicstructuressuchas(pre)cat 1 -Leibnizalgebras(Liealgebras, associative algebras, associative commutative algebras), commutative von Neumann rings etc. are not categories of interest. These kins of notions are unifie uner the name of moifie categories of interest, which satisfy all axioms of a category of groups with operations in [13] except one, which is replace by a new axiom; these categories satisfy as well two aitional axioms introuce in[12]. The category of(pre)crosse moules in Leibniz algebras(which is equivalent to a moifie category ofinterest,namely,thecategoryof(pre)cat 1 -Leibnizalgebras)wasintroucein[6].In[9],itisshown that the thir imensional cohomology of Leibniz n-algebras classifies crosse moules of Leibniz n- algebras. The notion of crosse moules in Leibniz algebras can be thought as a generalization of Leibniz algebras. For any Leibniz algebra M, we have the crosse moule M i M an so the category of Leibniz algebras is a full subcategory of crosse moules in Leibniz algebras. The same thing is true for precrosse moules, namely, M 0 is a precrosse moule. Naturally, it will be important to CONTACT E.Ö. Uslu enveruslu@ogu.eu.tr Department of Mathematics an Computer Sciences, Art an Science Faculty, Osmangazi University, Eskisehir, Turkey. 2017Taylor&Francis

3 1826 M.ATİKETAL. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 investigate the representability of actions, in other wors, investigate the existence an construction of split extension classifiers in the category of(pre)crosse moules. At this vein, for a given (pre)crosse moule M : M 1 M 0, we foun a conition uner which we construct an actor of the corresponing (pre)cat 1 -Leibniz algebra (M 1 M 0,ω 1,ω 0 ) by using the generalconstructionofuniversalstrictgeneralactorof (M 1 M 0,ω 1,ω 0 )givenin[5].thenapplying the equivalence of the categories (Pre)Cat 1 -Lbnz (Pre)XLbnz of (pre)cat 1 -Leibniz algebras an (pre)crosse moules, we carry the construction of an actor of (M 1 M 0,ω 1,ω 0 ) to the category of (pre)crossemoules,whichisasplitextensionclassifierofthe(pre)crossemoulem : M 1 M 0 uner the appropriate conition on it. Therefore we foun a new example of a category an iniviual objects there with representable actions. This problem is state in[3](problem 2). Theoutlineofthepaperisasfollows:inSection2,wegivesomeneeenotionsfromliteraturean introuce the notions such as bierivations an (generalize) crosse bierivations of a (pre)crosse moule. In Section 3, we construct an actor of a precat 1 -Leibniz algebra an consequently in Section 4, we construct the split extension classifier of a precrosse moule by using the natural equivalence between the categories of precat 1 -Leibniz algebras an precrosse moules. In Section 5, we give the same constructions for cat 1 -Leibniz algebras an crosse moules with aitional moifications. In Section 6, we give a comparison between the split extension classifiers of precrosse moules an crosse moules in the categories of Lie algebras an that of Leibniz algebras. 2. Preliminaries In this section we will recall some basic efinitions an properties about (pre)crosse moules in Leibniz algebras which neee in the rest of the paper. Aitionally, we efine new notions such as bierivations an (generalize) crosse bierivations of a (pre)crosse moule an some relate results. Also we will recall the notion of moifie category of interest, some relate efinitions an resultsfrom[5].inaition,wewillgivetheconstructionofuniversalstrictgeneralactorofaprecat 1 - Leibnizalgebra (M,w M 1,wM 0 )byusingthegeneralconstructiongivenformoifiecategoriesofinterest in[5] (Pre)crosse moules in Leibniz algebras Leibniz algebras, which are a non-antisymmetric generalization of Lie algebras, were introuce in 1965 bybloh[1],whocallethemd-algebrasanin1993loay[11]maethempopularanstuietheir (co)homology. Let k be a commutative ring with unit. In the rest of the paper all Leibniz algebras an Lie algebras willbeoverk. Definition2.1. LetLbeak-moulean [, ] : L L Lbeabilinearmap.IftheLeibnizientity [ x,[y,z] ] = [[ x,y ],z ] [ [x,z],y ] issatisfie,forallx,y,z L,thenLiscalleaLeibnizalgebrawiththebracket [, ]. Definition2.2. LetL,L beleibnizalgebras.ak-linearmaph : L L iscalleahomomorphismof Leibnizalgebrasifh [ l,l ] = [ h(l),h(l ) ],foralll,l L. Consequently, we have the category of Leibniz algebras which will be enote here by Lbnz. Now, we recall the construction of the Leibniz algebra Bier(L) of bierivations of a Leibniz algebra LintroucebyLoay[11].Here,weusethenotationgivenin[7].

4 COMMUNICATIONS IN ALGEBRA 1827 Definition2.3. Anelement ϕ ofbier(l)isapair ϕ := (ϕ l,ϕ r )consistsofk-linearmaps ϕ l : L L, ϕ r : L Lwith forallx,x L. ϕ r [x,x ] = [x,ϕ r (x )] + [ϕ r (x),x ], ϕ l [x,x ] = [ϕ l (x),x ] [ϕ l (x ),x], [x,ϕ r (x )] = [x,ϕ l (x )], Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Bier(L) is a Leibniz algebra with the bracket [ϕ,ϕ ] = ([ϕ,ϕ ] l,[ϕ,ϕ ] r ) where [ϕ,ϕ ] l = ϕ l ϕ l + ϕ r ϕ l, [ϕ,ϕ ] r = ϕ r ϕ r ϕ r ϕ r,forall ϕ,ϕ Bier(L). Let M,N Lbnz. Recall from [7] that an action(i.e. erive action)of N on M is a pair of bilinear mapsn M M, (n,m) [n,m]anm N M, (m,n) [m,n]suchthat i) [m,[m,n]] = [[m,m ],n] [[m,n],m ], ii) [m,[n,m ]] = [[m,n],m ] [[m,m ],n], iii) [n,[m,m ]] = [[n,m],m ] [[n,m ],m], iv) [m,[n,n ]] = [[m,n],n ] [[m,n ],n], v) [n,[m,n ]] = [[n,m],n ] [[n,n ],m], vi) [n,[n,m]] = [[n,n ],m] [[n,m],n ], forallm,m M,n,n N. It is well known that, for a Lie algebra g, the Lie algebra Der(g) of all erivations has an action on g, which was calle as ajoint representation. A similar action exists for Leibniz algebras uner certain conitions. Remark2.4. LetM Lbnz.Consierthemaps an Bier(M) M M (ϕ,m) ϕ l (m) M Bier(M) M (m,ϕ) ϕ r (m). Ingeneral,thesemapsonotefineanactionofBier(M)onM.Nevertheless,ifMsatisfiesAnn(M) = 0 or[m,m] = M,thenthesemapsefineanactionofBier(M)onM (See[7],foretails). Definition 2.5 ([6]). A precrosse moule M : M 1 M 0 in Leibniz algebras consists of a Leibniz homomorphism : M 1 M 0,callebounarymap,togetherwithanactionofM 0 onm 1 satisfying the ientities forallm 1 M 1,m 0 M 0.Inaition,if ([m 0,m 1 ]) = [m 0,(m 1 )],([m 1,m 0 ]) = [(m 1 ),m 0 ], [(m 1 ),m 1] = [m 1,m 1], [m 1,(m 1 )] = [m 1,m 1 ], forallm 1,m 1 M 1 thenm : M 1 M 0 iscalleacrossemoule. Let M : M 1 M 0 an M : M 1 M 0 be (pre)crosse moules. A homomorphism from M to M is a pair (µ 1,µ 0 ) of Leibniz algebra homomorphisms µ 1 : M 1 M 1, µ 0 : M 0 M 0 such

5 1828 M.ATİKETAL. that µ 1 = µ 0, µ 1 ([m 0,m 1 ]) = [µ 0 (m 0 ),µ 1 (m 1 )] an µ 1 ([m 1,m 0 ]) = [µ 1 (m 1 ),µ 0 (m 0 )], for all m 1 M 1, m 0 M 0. Consequently, we have the category of precrosse moules an the category of crosse moules in the category of Leibniz algebras, which will be enote here by PXLbnz, XLbnz, respectively. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Example 2.6. (i) LetN bea(two-sie)iealofaleibnizalgebram,thenn inc. Misacrossemoule,wherethe action is given by the ajoint representation. Consequently, M i M an 0 inc. M are crosse moules. (ii) LetL beanon-abelianleibnizalgebrawithanactionoflonl.thenl 0 L is a precrosse moule which is not a crosse moule. Consequently, L 0 L an L 0 0 are precrosse moules. (iii) Let M be a Leibniz algebra. Then M : M M π 1 M is a precrosse moule with a componentwise action of M on M M where π 1 is the projection. M M π 1 M is not a crosse moule. (iv) LetM bealeibnizalgebrasatisfyingann(m) = 0or[M,M] = M.Then : M Bier(M) m ϕ m = ([m, ],[,m]) isacrossemoulewiththeactionefineinremark2.4. A(pre)crossemouleM : M 1 M 0 isa(pre)crossesubmoule(orasubobjectin (P)XLbnz) of M : M 1 M 0 if M 1, M 0 are Leibniz subalgebras of M 1 an M 0, respectively; = M 1 an the actionofm 0 onm 1 isinucebytheactionofm 0 onm 1.ThissituationwillbeenotebyM M. In aition, if M 1 an M 0 are ieals of M 1 an M 0, respectively, [m 0,m 1 ],[m 1,m 0] M 1, for all m 0 M 0,m 1 M 1 an [m 0,m 1],[m 1,m 0 ] M 1,forallm 0 M 0,m 1 M 1,thenM iscalleanieal ofm. Let M : M 1 M 0 be an ieal of a (pre)crosse moule M : M 1 M 0. Then the quotient(pre)crossemoulem/m isthe(pre)crossemoulem 1 /M 1 M 0/M 0 withtheinuce bounary map an action. Definition 2.7. Let M : M 1 M 0 be a (pre)crosse moule. A bierivation of M is a pair (f,g) consistsofbierivationsf,g ofm 1 anm 0 respectively,suchthatf l = g l,f r = g r an a) f l ([m 1,m 0 ]) = f l ([m 0,m 1 ]) = [f l (m 1 ),m 0 ] [g l (m 0 ),m 1 ], b) f r ([m 0,m 1 ]) = [g r (m 0 ),m 1 ] + [m 0,f r (m 1 )], c) f r ([m 1,m 0 ]) = [m 1,g r (m 0 )] + [f r (m 1 ),m 0 ], ) [m 0,f r (m 1 )] = [m 0,f l (m 1 )], e) [m 1,g r (m 0 )] = [m 1,g l (m 0 )], forallm 0 M 0, m 1 M 1. Notation2.8. Intheefinition,insteaofwritingα l = β l,α r = β r,wemaywritethesetwoequalities inoneasα l,r = β l,r.intherestofthepaperwewillusethisnotation,forshortness. Thesetofallbierivationsofa(pre)crossemouleMwillbeenotebyBier(M).Itcanbeeasily checke that Bier(M) is a Leibniz algebra with usual scalar multiplication, aition an the bracket efine by [ (f,g),(f,g ) ] = ( [ f,f ], [ g,g ] ), forall (f,g),(f,g ) Bier(M).

6 COMMUNICATIONS IN ALGEBRA 1829 Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Definition 2.9. Let M : M 1 M 0 be a precrosse moule, α, α 1 Bier(M 1 ) an the pair := ( l, r )consistsofk-linearmaps l, r : M 0 M 1 suchthat 1) a) α l ([m 1,m 0 ]) = α l ([m 0,m 1 ]) = [α l (m 1 ),m 0 ] [ l (m 0 ),m 1 ], b) α r ([m 0,m 1 ]) = [ r (m 0 ),m 1 ] + [m 0,α r (m 1 )], c) α r ([m 1,m 0 ]) = [m 1, r (m 0 )] + [α r (m 1 ),m 0 ], ) l [m 0,m 0 ] = [ l(m 0 ),m 0 ] [ l(m 0 ),m 0], e) r [m 0,m 0 ] = [ r(m 0 ),m 0 ] + [m 0, r (m 0 )], f) [m 0,α r (m 1 )] = [m 0,α l (m 1 )], g) [m 1, r (m 0 )] = [m 1, l (m 0 )], 2) (α 1,β 1 )isabierivationofm, 3) β 1 l,r (m 1) = α l,r (m 1 ), for all m 0 M 0, m 1 M 1 where l,r = β 1 l,r. Then the triples (α,,α1 ) will be calle a generalize crosse bierivations of M. The set of all generalize bierivations of M will be enote bygbier(m 0,M 1 ). Proposition2.10. LetM : M 1 M 0 beaprecrossemoule.gbier(m 0,M 1 )isnon-empty. Proof. Choosea 1 M 1.Define α l : M 1 M 1, m 1 [a 1,m 1 ],α r : M 1 M 0, m 1 [m 1,a 1 ], l : M 0 M 1, m 0 [a 1,m 0 ], r : M 0 M 1, m 0 [m 0,a 1 ], α 1 l : M 1 M 1, m 1 [(a 1 ),m 1 ],α 1 r : M 1 M 1, m 1 [m 1,(a 1 )], β 1 l : M 0 M 0, m 0 [(a 1 ),m 0 ],β 1 r : M 0 M 0, m 0 [m 0,(a 1 )], forallm 0 M 0,m 1 M 1.Denote α := (α l,α r ), := ( l, r ), α 1 := (α 1 l,α1 r ).Wehavethat (α,,α1 ) Gbier(M 0,M 1 ).Inee α,α 1 Bier(M 1 )an α,α 1, satisfytheientitiesgivenindefinition2.9. Gbier(M 0,M 1 )isaleibnizalgebrawithusualscalarmultiplication,aitionanthebracketefine by [(α,,α 1 ),(γ,λ,γ 1 )] = ( [α,γ],[,λ], [α 1,γ 1 ] ) where [α,γ], [α 1,γ 1 ]areusualbracketsofbierivationsan forall (α,,α 1 ),(γ,λ,γ 1 ) Bier(M 0,M 1 ). [,λ] l = α l λ l + γ r l, [,λ] r = γ r r α r λ r, Remark Let M : M 1 M 0 be a precrosse moule which is not a crosse moule. So there exist some m 1,m 1 M 1 such that [m 1,(m 1 )] = [m 1,m 1 ] or [(m 1 ),m 1] = [m 1,m 1]. Consier the triple (α,,α 1 )efinebym 1 asinproposition2.10.thenwehave α = α 1. Definition2.12. LetK : K 1 K 0 beacrossemoule.bier(k 0,K 1 )isefineasthesetofallpairs := ( l, r )suchthat l, r : K 0 K 1 arek-linearmapsan l [k 0,k 0 ] = [ l(k 0 ),k 0 ] [ l(k 0 ),k 0], r [k 0,k 0 ] = [ r(k 0 ),k 0 ] + [k 0, r (k 0 )], [k 0, r (k 0 )] = [k 0, l (k 0 )], forallk 0,k 0 K 0.Thistime,wewillcalltheelementsofBier(K 0,K 1 )ascrossebierivations.

7 1830 M.ATİKETAL. ByasimilarconstructiongiveninProposition2.10,Bier(K 0,K 1 )isnon-empty.alsobier(k 0,K 1 )is aleibnizalgebrawithusualscalarmultiplication,aitionanthebracket efineby [,λ] l = l λ l + λ r l, [,λ] r = λ r r r λ r,forall,λ Bier(K 0,K 1 ). Remark Definitions 2.7, 2.9 an 2.12 are euce from the construction of bierivations of a semiirect prouct in Section 4. One can see that bierivations an(generalize) crosse bierivations of a (pre)crosse moule M : M 1 M 0 can be easily obtaine from certain type of bierivations ofthesemiirectprouctm 1 M 0.InthecaseofLiealgebrastheseefinitionscoinciewiththeones givenin[8,10]. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Now we recall the categories of (pre)cat 1 -Leibniz algebras an their functorial relation between the categories PXLbnz an XLbnz. Details can be foun in[9]. Aprecat 1 -LeibnizalgebraisaLeibnizalgebrawithtwoaitionalunaryoperations ω 0,ω 1 : M M suchthat ω 0 ω 1 = ω 1, ω 1 ω 0 = ω 0 where ω 0 an ω 1 are Leibniz algebra homomorphisms. We will enote such a precat 1 -Leibniz algebra by (M,ω 0,ω 1 ). A homomorphism of precat 1 -Leibniz algebras is a Leibniz algebra homomorphism compatiblewithunaryoperations.theresultingcategorywillbeenotebyprecat 1 -Lbnz. LetCat 1 -LbnzbethefullsubcategoryofPrecat 1 -Lbnz,consistsofthoseobjectssatisfying; [kerw 0,kerw 1 ] = 0, [kerw 1,kerw 0 ] = 0. TheobjectsofCat 1 -Lbnzarecalleascat 1 -Leibnizalgebras. DefineafunctorC : PXLbnz Precat 1 -Lbnzasfollows;foranyprecrossemouleM : M 1 M 0,C(M) := (M 1 M 0,w 0,w 1 )wherem 1 M 0 isasemi-irectprouctan w 0 (m 1,m 0 ) = (0,m 0 ), w 1 (m 1,m 0 ) = (0,(m 1 ) +m 0 ), for all m 1 M 1, m 0 M 0. Note that to construct the semi-irect prouct we use the action of M 0 on M 1 fromtheprecrossemoulem. Conversely, efine a functor P : Precat 1 -Lbnz PXLbnz as follows; for any object (M,w 1,w 0 ) in Precat 1 -Lbnz, P((M,w 1,w 0 )) is the precrosse moule M : M 1 M 0 where M 1 = kerw 0, M 0 = Imw 0 an = w 1 kerw0. These two functors give rise to a natural equivalence between the categories PXLbnz an Precat 1 - Lbnz. On the other han, the obvious restrictions of the functors give a natural equivalence between the categoriesxlbnzancat 1 -Lbnz Moifie category of interest Moifie categories of interest were introuce in [5]. The conition () in the efinition of moifie categoriesofinterestwas Foreachω 1 an 2,Eincluestheientitiesω(x+y) = ω(x)+ω(y) an ω(x y) = ω(x) ω(y).butthisconitionoesnotcontainthescalarmultiplication.theauthors of [5] change this conition. After this change all constructions an proofs in [5] are true. Accoring to this correcte efinition every category of interest is a moifie category of interest as well. Now we will recall the efinition of moifie category of interest with its revise form.

8 COMMUNICATIONS IN ALGEBRA 1831 Let C be a category of groups with a set of operations an with a set of ientities E, such that E incluesthegroupientitiesanthefollowingconitionshol.if i isthesetofi-aryoperationsin, then: (a) = ; (b) the group operations (written aitively : 0,,+) are elements of 0, 1 an 2 respectively. Let 2 = 2\{+}, 1 = 1\{ }.Assumethatif 2,then 2 contains efinebyx y = y x anassume 0 = {0}; (c) foreach 2,Eincluestheientityx (y +z) = x y+x z; () foreach ω 1 an 2,Eincluestheientity ω(x+y) = ω(x)+ω(y)aneithertheientity ω(x y) = ω(x) ω(y)ortheientity ω(x y) = ω(x) y. LetCbeanobjectofCanx 1,x 2,x 3 C: Axiom1.x 1 + (x 2 x 3 ) = (x 2 x 3 ) +x 1,foreach 2. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Axiom2.Foreachorerepair (, ) 2 2 thereisaworw suchthat (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 ), whereeachjuxtapositionrepresentsanoperationin 2. Definition A category of groups with operations C satisfying conitions (a) (), Axioms 1 an 2, is calle a moifie category of interest. LetE G bethesubsetofientitiesofewhichincluesthegroupientitiesantheientities(c)an (). We enote by C G the corresponing category of groups with operations. Thus we have E G E, C = (,E),C G = (,E G )anthereisafullinclusionfunctorc C G.C G iscalleageneralcategory of groups with operations of a moifie category of interest C. Example2.15. ThecategoriesCat 1 -Lbnz,anPreCat 1 -Lbnzaremoifiecategoriesofinterest,which are not categories of interest. Further examples can be foun in[5]. Definition2.16. LetA,B C.AnextensionofBbyAisasequence 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 Esuchthatps = 1 B. Definition2.17. ForA,B C,itissaithatthereisasetofactionsofBonA,wheneverthereisamap f : A B A,foreach 2. AsplitextensionofBbyA,inucesanactionofBonAcorresponingtotheoperationsinC.Fora 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 2. Actionsefine by (3.2) an (3.3) will be calle erive actionsofb ona.thenotationb a,isusetoenoteboththeotanthestaractions.

9 1832 M.ATİKETAL. Definition Given an action of B on A, a semiirect prouct A B is a universal algebra, whose unerlyingsetisa Bantheoperationsareefineby foralla,a A,b,b B. 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), Theorem2.19([5]). AnactionofBonAisaeriveactionifanonlyifA BisanobjectofC. Now we will efine the actions in the categories Precat 1 -Lbnz an Cat 1 -Lbnz accoring to the efinition of action in a moifie category of interest. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Example2.20. (i) Let (M,ω M 1,ωM 0 ),(N,ωN 1,ωN 0 ) Precat1 -Lbnzanlet (N,ω N 1,ωN 0 )hasaerive actionon (M,ω M 1,ωM 0 ).ByDefinition2.17wehavetheientities(2.1)analsotheientities; [ ω N i (n),ω M i (m) ] = ω M i [n,m], [ ω M i (m),ω N i (n) ] = ω M i [m,n], [ωj N (n),ωi M (m)] = ωi M [ωj M (n),m], [ωi M (m),ωj N (n)] = ωi M [m,ωj N (n)], [ωj N (n),ωi M (m)] = ωj M [ n,ω M i (m) ], [ωi M (m),ωj N (n)] = ωj M [ ω M i (m),n ], i,j = 0,1,i = j,foranym (M,ω1 M,ωM 0 ),n (N,ωN 1,ωN 0 ). (ii) Let (M,ω1 M,ωM 0 ),(N,ωN 1,ωN 0 ) Cat1 -Lbnz an let (N,ω1 N,ωN 0 ) has a erive action on (M,ω1 M,ωM 0 ).Thenwehavetheientities(2.1),(3.4)aninaitionwehave[m,n] = [n,m] = 0, ifn ker ω0 N,m ker ωm 1 orn kerωn 1,m kerωm 0. The efinition of a split extension classifier in 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 the efinition, an object [A] is a split extension classifier for A ifthereexistsaeriveactionof [A]onA,suchthatthecorresponingextension 0 A A [A] is a terminal object in the above efine category. s [A] 0 (3.4) Proposition2.21([5]). LetCbeamoifiecategoryofinterestanAbeanobjectinC.AnobjectB C isasplitextensionclassifierforainthesenseof[2]ifanonlyifitsatisfiesthefollowingconition:bhas aeriveactiononasuchthatforallcincanaeriveactionofconathereisauniquemorphism ϕ : C B,withc a = ϕ(c) a,c a = ϕ(c) a,forall 2,a Aanc C. The object B in C satisfying the above state conition is calle an actor of A an enote by Act(A). The corresponing universal acting object, which represents actions in the sense of[2, 3], in the categories equivalent to moifie categories of interest is calle a split extension classifier an enote by [A], as it is in semi-abelian categories.

10 COMMUNICATIONS IN ALGEBRA 1833 Remark As a consequence of this proposition, an actor of an object is unique up to an isomorphism. Definition2.23. LetA,B C.AsetofactionsofBonAisstrictifforanytwoelementsb,b B,from theconitionsb a = b a, ω(b) a = ω(b ) a,b a = b aan ω(b) a = ω(b ) a,foralla A, ω 1 an 2,itfollowsthatb = b. Definition2.24. AgeneralactorGA(A)ofanobjectAinC,isanobjectofC G,havingasetofactions ona,whichisasetoferiveactionsinc G anforanyobjectc CanaeriveactionofConAin C, there exists in C G a unique morphism ϕ : C GA(A) such that c a = ϕ(c) a, for all c C, a Aan 2. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Definition2.25. IftheactionofageneralactorGA(A)onAisstrict,thenitissaithatGA(A)isastrict generalactorofaanenotebysga(a). Conition2.26. LetA Can {B j } j J enotethesetofallobjectsofcwhichhaveeriveactionsona. Let ϕ j : B j GA(A),j J,enotethecorresponinguniquemorphismsuchthatb j a = ϕj (b j ) a, forallb j B j,a A, 2.TheelementsofGA(A)satisfythefollowingequality: foranyb i B i,b j B j, 2 ani,j J. (ϕ i (b i ) ϕ j (b j )) a = W(ϕ i (b),ϕ j (b );a;, ) Definition A universal strict general actor of an object A, enote by USGA(A), is a strict general actor with Conition 2.26, such that for any strict general actor SGA(A) with Conition 2.26 there exists auniquemorphism η : USGA(A) SGA(A)inthecategoryC G,with ψ j η = ϕ j,foranyj J,where ϕ j : B j SGA(A) an ψ j : B j USGA(A) enote the corresponing unique morphisms with the appropriate properties from the efinition of a general actor. Proposition 2.28 ([5]). 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. Theorem2.29([5]). LetCbeamoifiecategoryofinterestanA C.Ahasanactorifanonlyifthe semiirectproucta USGA(A)isanobjectofC.Ifitisthecase,thenAct(A) = USGA(A). Let (M,w M 0,wM 1 )beaprecat1 -Leibniz.ConsierallsplitextensionsofM inprecat 1 Lbnz E j : 0 (M,w0 M,wM 1 ) (K j,w K j 0,wK j 1 ) (L j,w L j 0,wL j 1 ) 0, j J where (L j,w L j 0,wL j 1 ) = (L k,w L k 0,wL k 1 ) = (L,wL 0,wL 1 ),forj = kinthiscasethecorresponingextensions eriveifferentactionsof (L,w0 L,wL 1 )on (M,wM 0,wM 1 ).Let {l j,[l j,-],[-,l j ]}bethesetoffunctionsefine by the action of (L j,w L j 0,wL j 1 ) on (M,wM 0,wM 1 ). For any element l j L j enote l j = {l j,[l j,-],[-,l j ]}. Let L = {l j,l j L j,j J}. Thus each element l j L, j J is a special type of a function l j = {+,[,],[,] op } Maps, (M,w0 M,wM 1 ) (M,wM 0,wM 1 ) efine by l j([-,-]) = [l j,-] : M M, l j ([-,-] op ) = [-,l j ] : M M,l j (+) = (l j +(-)) : M M.ThebracketonLisefineby [[l i,l k ],m] = [l i,[l k,m]] + [[l i,m],l k ], [m,[l i,l k ]] = [[m,l i ],l k ] [[m,l k ],l i ], [l i,l k ] (m) = m.

11 1834 M.ATİKETAL. Aitionally, we efine (l i +l k ) (m) = l i (l k m), [(l i +l k ),m] = [l i,m] + [m,l k ], w i (l k ) (m) = w L k i (l k ) m, [w i (l k ),m] = [w L k i (l k ),m], [m,w i (l k )] = [m,w L k i (l k )], i = 0,1, w i [l i,l k ] = [w L i i (l i ),w L k i (l k )], i = 0,1, w 1 (l 1 +l 2 ) = w L 1 i (l 1 ) +w L 2 i (l 2 ), i = 0,1, ( l i ) (m) = ( l i ) m, ( l) m = m, [( l i ),m] = [l i,(m)], [m,( l i )] = [(m),l i ], [( l),(m)] = [l,(m)], [(m),( l)] = [(m),l], Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 (l 1 +l 2 ) = l 2 l 1 wherel,l 1,l 2 arecertaincombinationsofthebracketoftheelementsofl. DenotebyL (M)thesetofallfunctions( 2 Maps(M,w M 0,wM 1 ) (M,wM 0,wM 1 ))obtaineby performingallkinofoperationsefineaboveontheelementsoflanonthenewobtaineelements as the results of operations. Note that if may happen that [l,m] = [ l,m ] or [m,l] = [ m,l ], for any m M,butweonothavetheequalities[w i (l),m] = [ w i ( l ),m ] or[m,w i (l)] = [ m,w i ( l )],i = 0,1, foranym M,respectivelywherewisafinitecombinationsofw 0 anw 1.DefinearelationonL (M) by l l if an only if [l,m] = [ l,m ], [m,l] = [ m,l ], [w i (l),m] = [ w i ( l ),m ], [m,w i (l)] = [ m,wi ( l )] foranyl,l L (M),m M ani = 0,1.ThisisacongruencerelationonL (M).Denote L (M)/ byl(m).theoperationsefineonl (M)efinethecorresponingoperationsonL(M). Theorem2.30([5]). LetA C.ThenwehaveB(A) = USGA(A). Corollary Let (M,w 0,w 1 ) Precat 1 Lbnz. Then (L(M),w L(M) generalactorof (M,w0 M,wM 1 ). Proof. Follows from Theorem ActorofanobjectinPrecat 1 -Lbnz 0,w L(M) 1 ) is a universal strict In this section, accoring to a given object (M,w 1,w 0 ) in Precat 1 -Lbnz, we will construct an object (A(M),w 0,w 1 ) an prove that it is an actor of (M,w 1,w 0 ) uner certain conitions. The construction euce from the interpretation of L(M) constructe in Section 2. Let (M,w 1,w 0 ) be a precat 1 -Leibniz algebra. Consier the triples (ϕ,ϕ 0,ϕ 1 ) of bierivations of M such that M1) ϕ i l,r w i = w i ϕ l,r,fori = 0,1, M2) ϕ j l,r w i = w i ϕ j l,r,fori = 0,1,j = 0,1. WewillenotethesetofallthesekinsoftriplesbyA(M). A(M) is a Leibniz algebra with componentwise aition, scalar multiplication an the bracket efine by [(ϕ,ϕ 0,ϕ 1 ),(ψ,ψ 0,ψ 1 )] = ([ϕ,ψ],[ϕ 0,ψ 0 ],[ϕ 1,ψ 1 ]), for all (ϕ,ϕ 0,ϕ 1 ),(ψ,ψ 0,ψ 1 ) A(M). The zero element is the triple (0,0,0) of zero maps. Also, (A(M),ω 0,ω 1 ) is a precat 1 -Leibniz algebra with the unary operations ω 0 : A(M) A(M), ω 1 : A(M) A(M)efineby ω 0 (ϕ,ϕ 0,ϕ 1 ) = (ϕ 0,ϕ 0,ϕ 0 ), ω 1 (ϕ,ϕ 0,ϕ 1 ) = (ϕ 1,ϕ 1,ϕ 1 ),respectively.

12 COMMUNICATIONS IN ALGEBRA 1835 Thereisanactionof (A(M),ω 0,ω 1 )on (M,w 1,w 0 )efinebythemaps (A(M),w 0,w 1 ) (M,w 1,w 0 ) (M,w 1,w 0 ) ((ϕ,ϕ 0,ϕ 1 ),m) ϕ l (m) an (M,w 1,w 0 ) (A(M),w 0,w 1 ) (M,w 1,w 0 ) (m,(ϕ,ϕ 0,ϕ 1 )) ϕ r (m), forallm M an (ϕ,ϕ 0,ϕ 1 ) A(M). ConitionAForaLeibnizalgebraM,Ann(M) = 0or [M,M] = M. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Proposition 3.1. If M satisfies Conition A, then (A(M),ω 0,ω 1 ) an (L(M), w L(M) isomorphic. 0,w L(M) 1 ) are Proof. UnerConitionAtheactionof (A(M),w 0,w 1 )on (M,w 1,w 0 )efinebyusisaeriveaction in Precat 1 -Lbnz. Therefore, from Definition 2.27, we have the unique morphism η : (A(M)) (L(M)) such that η( [ (f,f 0,f 1 ),m ] ) = [ (f,f 0,f 1 ),m ], η ([ m,(f,f 0,f 1 ) ]) = [ m,(f,f 0,f 1 ) ], for all m M an (f,f 0,f 1 ) A(M). By the constructions of (A(M),ω 0,ω 1 ) an (L(M),w L(M) 0,w L(M) 1 ), wefinthat ηisanisomorphism. Corollary3.2. IfM satisfiesconitiona,then (A(M),ω 0,ω 1 )isanactorof (M,ω 0,ω 1 ). Proof. Since (A(M),ω 0,ω 1 ) Precat 1 -Lbnz an its action on M is a erive action, then the result follows from Theorem 2.29, Corollary 2.31 an Proposition Actorofaprecat 1 -Leibnizalgebracorresponingtoagivenprecrossemoule LetM 1,M 0 beleibnizalgebraswithanactionofm 0 onm 1.Let ψ = (ψ l,ψ r ) Bier(M 1 M 0 ).Then ψ l : M 1 M 0 M 1 M 0 canberepresentebyfourk-linearmaps such that α l : M 1 M 1,δ l : M 1 M 0,β l : M 0 M 0 an l : M 0 M 1 ψ l (m 1,m 0 ) = (α l (m 1 ) + l (m 0 ),β l (m 0 ) + δ l (m 1 )), forallm 1 M 1,m 0 M 0.Similarly, ψ r : M 1 M 0 M 1 M 0 canberepresentebyfourk-linear maps such that α r : M 1 M 1,δ r : M 1 M 0,β r : M 0 M 0 an r : M 0 M 1 ψ r (m 1,m 0 ) = (α r (m 1 ) + r (m 0 ),β r (m 0 ) + δ r (m 1 )), for all m 1 M 1, m 0 M 0. Let M : M 0 M 1 be a precrosse moule an (M 1 M 0,ω 0,ω 1 ) be the corresponing precat 1 -Leibniz algebra. Suppose ψ = (ψ l,ψ r ) satisfy the Conition M2, for j = 0 or j = 1. Then by a irect checking, we fin that δ r = δ l = 0. So, any bierivation ψ = (ψ l,ψ r ) Bier(M 1 M 0 )canberepresentebythetriple (α,,β). Letm := (m 1,m 0 ),m := (m 1,m 0 ) M 1 M 0.Since ψ = (ψ m,ψ r )isabierivation,weobtain ψ r ([m,m ]) = ψ r ( [m1,m 1 ] + [m 0,m 1 ] + [m 1,m 0 ],[m 0,m 0 ]) = ( α r [m 1,m 1 ] + α r[m 0,m 1 ] + α r[m 1,m 0 ] + r[m 0,m 0 ],β r[m 0,m 0 ])

13 1836 M.ATİKETAL. an ψ r ([m,m ]) = [m,ψ r (m )] + [ψ r (m),m ] = [(m 1,m 0 ),ψ r (m 1,m 0 )] + [ψ r(m 1,m 0 ),(m 1,m 0 )] = [(m 1,m 0 ), ( α r (m 1 ) + r(m 0 ),β r(m 0 )) ] + [(α r (m 1 ) + r (m 0 ),β r (m 0 )),(m 1,m 0 )] := I 1 +I 0 where I 1 = ( [m 1,α r (m 1 ) + r(m 0 )] + [m 1,β r (m 0 )] + [m 0,α r (m 1 ) + r(m 0 )],[m 0,β r (m 0 )]) an I 0 = ( [α r (m 1 ) + r (m 0 ),m 1 ] + [α r(m 1 ) + r (m 0 ),m 0 ] + [β r(m 0 ),m 1 ],[β r(m 0 ),m 0 ]). Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Soweget: (1) β r [m 0,m 0 ] = [m 0,β r (m 0 )] + [β r(m 0 ),m 0 ] (2) Ifwetakem 0 = m 0 = 0,then (3) Ifwetakem 0 = 0,m 1 = 0,then (4) Ifwetakem 1 = 0,m 0 = 0,then (5) Ifwetakem 1 = 0 = m 1,then α r [m 1,m 1 ] = [m 1,α r (m 1 )] + [α r(m 1 ),m 1 ]. α r [m 0,m 1 ] = [m 0,α r (m 1 )] + [ r(m 0 ),m 1 ] + [β r(m 0 ),m 1 ]. α r [m 1,m 0 ] = [m 1, r (m 0 )] + [m 1,β r (m 0 )] + [α r(m 1 ),m 0 ]. r [m 0,m 0 ] = [m 0, r (m 0 )] + [ r(m 0 ),m 0 ]. Bysimilarcalculationswehave α Bier(M 1 ), β Bier(M 0 )an (L1.) l [m 0,m 0 ] = [ l(m 0 ),m 0 ] [ l(m 0 ),m 0], (L2.) r [m 0,m 0 ] = [ r(m 0 ),m 0 ] + [m 0, r (m 0 )], (L3.) [m 0, r (m 0 )] = [m 0, l (m 0 )], (L4.) α l [m 1,m 0 ] = α l [m 0,m 1 ] = [α l (m 1 ),m 0 ] [ l (m 0 ),m 1 ] [β l (m 0 ),m 1 ], (L5.) α r ([m 0,m 1 ]) = [ r (m 0 ),m 1 ] + [β r (m 0 ),m 1 ] + [m 0,α r (m 1 )], (L6.) α r ([m 1,m 0 ]) = [α r (m 1 ),m 0 ] + [m 1, r (m 0 )] + [m 1,β r (m 0 )], (L7.) [m 0,α r (m 1 )] = [m 0,α l (m 1 )], (L8.) [m 1,β r (m 0 )] = [m 1,β l (m 0 )], (L9.) [m 1, l (m 0 )] = [m 1,β l (m 0 )] [m 1, r (m 0 )] [m 1,β r (m 0 )], forallm 1 M 1,m 0,m 0 M 0. Proposition3.3. LetM : M 0 M 1 beaprecrossemoulean(m 1 M 0,ω 0,ω 1 )bethecorresponing precat 1 -Leibniz algebra. Let ϕ,ϕ 0,ϕ 1 Bier(M 1 M 0 ) an enote ϕ,ϕ 0,ϕ 1 by the triples (α,,β), (α 0, 0,β 0 ), (α 1, 1,β 1 ),respectively.then (ϕ,ϕ 0,ϕ 1 ) A(M 1 M 0 )ifanonlyif (ϕ,ϕ 0,ϕ 1 )satisfythe following ientities: (1.) β l (m 0 ) = β 0 l (m 0), β r (m 0 ) = β 0 r (m 0), (2.) i l (m 0) = 0, i r (m 0) = 0,i = 0,1, (3.) β 1 l (m 0) = β l (m 0 ) + l (m 0 ), β 1 r (m 0) = β r (m 0 ) + r (m 0 ),

14 COMMUNICATIONS IN ALGEBRA 1837 (4.) β 1 l (m 1) = α l (m 1 ), β 1 r (m 1) = α r (m 1 ), (5.) β i l (m 1) = α i l (m 1), β i r (m 1) = α i r (m 1),i = 0,1, forallm 0 M 0,m 1 M 1. Proof. Follows from relate efinitions an equalities L1 L9. Proposition3.4. LetM : M 1 M 0 beaprecrossemoule,m 0,M 1 satisfyconitionaan (A(M 1 M 0 ),w 1,w 0 )betheactorof (M 1 M 0,w 0,w 1 ).Then,kerw 0 = {(ϕ,0,ϕ 1 ) A(M 1 M 0 )}. Proof. Followsfromtheefinitionofw 0 anproposition3.3. Proposition3.5. kerw 0 = Gbier(M0,M 1 ) Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Proof. Let (ϕ,0,ϕ 1 ) kerw 0. It follows from Propositions 3.3 an 3.4 that (ϕ,0,ϕ 1 ) = ((α,,0),(0,0,0),(α 1,0,β 1 )) an the resulting triple (α,,α 1 ) is a generalize crosse bierivation. Conversely,forany (γ,λ,γ 1 ) Bier(M 0,M 1 )wehavethetriples θ = (γ,λ,0), θ 1 = ( γ 1,0,λ ) such that (θ,0,θ 1 ) kerw 0. Proposition3.6. Im(w 0 ) = Bier(M). Proof. Directchecking. 4. Split extension classifier of a precrosse moule Accoring to the efinition of action in semi-abelian categories [3], it is natural to efine an action in PXLbnz in analogous way as it is efine in a moifie category of interest. In this section we will construct a precrosse moule for a given precrosse moule M : M 1 M 0 an prove, that if M 0 an M 1 satisfy Conition A, then is isomorphic to P(Act(C (M))). Consequently, is the split extension classifier of M. Let, M : M 1 M 0 be a precrosse moule, (M 1 M 0,w 0,w 1 ) be the corresponing precat 1 - algebraan (A(M 1 M 0 ),w 0,w 1 )beitsactorinprecat 1 -Lbnz. Proposition 4.1. The bilinear maps an Bier(M) Gbier(M 0,M 1 ) Gbier(M 0,M 1 ) ((f,g),(α,,α 1 )) ([f,α],,[f,α 1 ]) Gbier(M 0,M 1 ) Bier(M) Gbier(M 0,M 1 ) ((α,,α 1 ),(f,g)) ([α,f],, [ α 1,f ] ) efine an action of Bier(M) on Gbier(M 0,M 1 ) where [α,f], [f,α],[α 1,f], [f,α 1 ], are brackets of bierivations, an l = f l l + r g l r = f r r + r g r l = f r l + l g l r = f r r r g r

15 1838 M.ATİKETAL. Proof. Direct checking by using the efinitions. Defineamap : Gbier(M 0,M 1 ) Bier(M)by (α,,α 1 ) (α 1,β 1 )where β 1 l,r = l,r. Proposition4.2. : Gbier(M 0,M 1 ) Bier(M)is aprecrosse moulewiththeaction efinein Proposition 4.1. Proof. The proof is irect consequence of the efinitions. Proposition4.3. = P(A(C (M))). Proof. FollowsfromPropositions3.5,3.6an4.2,since isisomorphictotherestrictionof ω 1. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Theorem 4.4. If M 1 an M 0 satisfy Conition A, then the precrosse moule : Gbier(M 0,M 1 ) Bier(M) efine in Proposition 4.2 is the split extension classifier of M. Proof. If M 0 an M 1 satisfy Conition A, then the semiirect prouct M 1 M 0 also satisfies this conition. So the result irect consequence of Corollary 3.2, Proposition 4.3 an the fact that P an SefineanequivalencebetweenthecategoriesPXLbnzanPrecat 1 -Lbnz. The split extension classifier of a precrosse moule M : M 1 [M] PXLbnz. M 0 will be enote here by Example 4.5. Let M be a Leibniz algebra. Consier the precrosse moule M : M i M. Then the actor of M is the precrosse moule (Bier(M), Bier(M), i). 5. Split extension classifier of a crosse moule Inthissection,byasimilariscussiongiveninSections3an4,wewillefineanactorofanobjectin Cat 1 -LbnzanthesplitextensionclassifierofanobjectinXLbnz.Weomittheproofssincetheresults are similar to those given in Sections 3 an 4 with aitional moifications ActorofanobjectinCat 1 -Lbnz Let (M,w 1,w 0 )beacat 1 -Leibnizalgebra.Consierthesubset (A(M),w 0,w 1 )of (A(M),w 0,w 1 )whose elements satisfy the following aitional conitions: M3) ϕ l,r (x) = ϕ 1 l,r (x),forallx kerw 0, M4) ϕ l,r = ϕ 0 l,r (x),forallx kerw 1. Proposition5.1. (A(M),w 0,w 1 )isasubobjectof (A(M),w 0,w 1 ). Proof. Directchecking. ByasimilariscussiongiveninSection3,ifMsatisfiesConitionA,then (A(M),w 0,w 1 )isanactor of (M,w 1,w 0 )incat 1 -Lbnz. LetM : M 1 M 0 beacrossemoule.anybierivation ϕ = (ϕ l,ϕ r ) Bier(M 1 M 0 )canbe representebyatriple (α,,β)asitwasfortheprecrossemoulecase.

16 COMMUNICATIONS IN ALGEBRA 1839 Proposition 5.2. Let M : M 1 M 0 be a crosse moule an (M 1 M 0,w 1,w 0 ) is the corresponing cat 1 -Leibniz algebra. Let ϕ,ϕ 0,ϕ 1 Bier(M 1 M 0 ) an enote ϕ,ϕ 0,ϕ 1 by the triples (α,,β),(α 0, 0,β 0 ),(α 1, 1,β 1 ), respectively. (ϕ,ϕ 0,ϕ 1 ) A(M 1 M 0 ) if an only if the triple (ϕ,ϕ 0,ϕ 1 )satisfythefollowingientities: 1. β l (m 0 ) = βl 0(m 0), β r (m 0 ) = βr 0(m 0), 2. l i(m 0) = 0, r i(m 0) = 0,i = 0,1, 3. β l (m 0 ) + l (m 0 ) = βl 1(m 0), β r (m 0 ) + r (m 0 ) = βr 1(m 0), 4. βl 1(m 1) = α l (m 1 ), βr 1(m 1) = α r (m 1 ), 5. βl i(m 1) = αl i(m 1), βr i(m 1) = αr i(m 1),i = 0,1, 6. α l,r (m 1 ) = αl,r 1 (m 1), 7. α l,r (m 1 ) = αl,r 0 (m 1) + l,r (m 1 ), forallm 0 M 0,m 1 M 1. Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 Proof. Let (ϕ,ϕ 0,ϕ 1 ) A(M 1 M 0 ). All equalities 1 7 are irect consequences of M1 M4. We will emonstrate the proofs of properties 6 an 7. By the efinition of w 0, we have (m 1,0) kerw 0, for all m 1 M 1. From M3, we have ϕ (l,r) (a 1,0) = ϕ 1 (l,r) (a 1,0), which means α l,r (m 1 ) = α 1 l,r (m 1), for all m 1 M 1. Also, by the efinition of w 1, we have (m 1,-(m 1 )) kerw 1, for all m 1 M 1. Since i l,r (m 0) = 0, i = 0,1,wehave α l,r (m 1 ) = α 0 l,r (m 1) + l,r (m 1 ),forallm 1 M 1. The converse statement can be prove by a irect checking. Remark5.3. LetM : M 1 M 0 beacrossemouleanlet (M 1 M 0,w 1,w 0 )bethecorresponing cat 1 -Leibnizalgebra.Fromtheefinitionofw 0 wehavekerw 0 = { (ϕ,0,ϕ 1 ) A(M 1 M 0 ) },anfrom Proposition 5.2, any element (ϕ,0,ϕ 1 ) kerw 0 can be represente by ((α,,0),(0,0,0),(α,0,β 1 )). Also,fromDefinition2.7,forany ((α,,0),(0,0,0), (α,0,β 1 ))wehavethat, (α,β 1 )isabierivationof Man l,r = β 1 l,r.consequently,wegetkerw 0 = Bier(M 0,M 1 ),Imw 0 = Bier(M) Split extension classifier of a crosse moule Proposition 5.4. The bilinear maps Bier(M) Bier(M 0,M 1 ) Bier(M 0,M 1 ), ((f,g), ), an Bier(M 0,M 1 ) Bier(M) Bier(M 0,M 1 ), (,(f,g)), efine a erive action of Bier(M)onBier(M 0,M 1 )where l = f l l + r g l, r = f r r + r g r, Proof. Direct consequence of the efinitions. l = f r l + l g l, r = f r r r g r. Proposition 5.5. Define the map : Bier(M 0,M 1 ) Bier(M) by (, ), for all Bier(M 0,M 1 ). Then Bier(M 0,M 1 ) Bier(M) is a crosse moule with the action efine in Proposition 5.4. Proof. Direct checking by using the efinitions.

17 1840 M.ATİKETAL. Proposition5.6. LetM : M 1 M 0 beacrossemoule.then = P(A(C (M))). Proof. Follows from Remark 5.3 an Proposition 5.5. Theorem 5.7. If M 1 an M 0 satisfy Conition A, then : Bier(M 0,M 1 ) Bier(M) is the split extensionclassifierofthecrossemoulem : M 1 M 0. ThesplitextensionclassifierofacrossemouleM : M 1 M 0 willbeenoteby[m] XLbnz. Downloae by [The UC Irvine Libraries] at 11:57 29 September Comparison In this section, we will give the relation between the split extension classifiers in the categories PXLbnz, XLbnz, PXLie of precrosse moules of Lie algebras an XLie of crosse moules of Lie algebras. The constructionofsplitextensionclassifiersinpxlieanxliecanbefounin[8,10].in[8]an[10],split extension classifiers were calle actors. LetM 1,M 0 beleibnizalgebrassatisfyingconitiona. Proposition 6.1. Let M : M 1 M 0 be a crosse moule of Leibniz algebras. Then [M] XLbnz is a subobjectof[m] PXLbnz,inPXLbnz. Proof. Everycrossebierivation Bier(M 0,M 1 )givesrisetoatriple (α,,α 1 ) Gbier(M 0,M 1 ) where α = α 1 =.ThenBier(M 0,M 1 )isaleibnizsubalgebraofgbier(m 0,M 1 ).Consequently,we get [M] XLbnz [M] PXLbnz. For a given precrosse moule L : L 1 L 0 in Lie algebras, enote its split extension classifier by [L] PXLie in PXLie an for a given crosse moule L : L 1 L 0 of Lie algebras, enote its split extensionclassifierby [ L ] XLie inxlie.asinicatein[8], [ L ] XLie isasubobjectof [ L ] PXLie. Proposition6.2. LetL : L 1 L 0 beaprecrossemouleinthecategoryofliealgebras.then[l] PXLie isasubobjectof[l] PXLbnz inpxlbnz. Proof. Directchecking. Proposition6.3. LetL : L 1 L 0 beacrossemouleinthecategoryofliealgebras.then. [ L ] XLie is asubobjectof [ L ] XLbnz inxlbnz. Proof. Directchecking. Corollary 6.4. Let L : L 1 following: L 0 be a crosse moule in the category of Lie algebras. We have the [ L ] XLie [ L ] XLbnz [ L ] PXLbnz an [ L ] XLie [ L ] PXLie [ L ] PXLbnz in PXLbnz.

18 COMMUNICATIONS IN ALGEBRA 1841 Proof. Follows from Propositions 6.2 an 6.3. Acknowlegments We woul like to thank the referee an G. Janelize for their suggestions espacially about the efinition of Moifie categories of interest which improve the work. Also, we woul like to thank T. Datuashvili for valuable comments an suggestions while her visit to Eskisehir Osmangazi University supporte by TÜBİTAK grant 2221 Konuk veya Akaemik İzinli Bilim İnsanı Destekleme Programı. References Downloae by [The UC Irvine Libraries] at 11:57 29 September 2017 [1] Bloh, A.(1965). A generalization of the concept of a Lie algebra. Sov. Math. Dokl. 6: [2] Borceux, F., Janelize, G., Kelly, G. M.(2005). Internal object actions. Comment. Math. Univ. Carolin. 46: [3] Borceux, F., Janelize, G., Kelly, G. M. (2005). On the representability of actions in a semi-abelian category. Theory Appl. Categ. 14: [4] Bourn, D., Janelize, G.(1998). Protomoularity, escent an semiirect proucts. Theory Appl. Categ. 4(2): [5] Boyacı, Y., Casas, J. M., Datuashvili, T., Uslu, E. Ö. Actions in moifie categories of interest with application to crosse moules. Theory Appl. Categ. 30(25): [6] Casas, J. M.(1999). Crosse extensions of Leibniz algebras. Commun. Algebra 27(12): [7] Casas, J. M., Datuashvili, T., Lara, M.(2010). Universal strict general actors an actors in categories of interest. Appl. Categ. Struct. 18: [8] Casas, J. M., Datuashvili, T., Lara, M., Uslu, E. Ö. (2012). Actions in the category of precrosse moules in Lie algebras. Commun. Algebra 40: [9] Casas, J. M., Khmalaze, K., Lara. M.(2008). Crosse moules for Leibniz n-algebras. Forum Math. 20: [10] Casas, J. M., Lara, M.(1998). The actor of a crosse moule in Lie algebras. Commun. Algebra 26(7): [11] Loay, J.-L. (1993). Une version non commutative es algèbres e Lie: les algèbres e Leibniz. Enseign. Math. 39(3 4): [12] Orzech, G.(1972). Obstruction theory in algebraic categories I an II. J. Pure Appl. Algebra 2: , [13] Porter, T.(1987). Extensions, crosse moules an internal categories in categories of groups with operations. Proc. Einburgh Math. Soc. 30: