Classification of 3-Dimensional Complex Diassociative Algebras

Transcription

1 Malayian Journal of Mahemaical Science 4 () (010) Claificaion of -Dimenional Complex Diaociaive Algebra 1 Irom M. Rihiboev, Iamiddin S. Rahimov and Wiriany Bari 1,, Iniue for Mahemaical Reearch,, Deparmen of Mahemaic, Faculy of Science, Univerii Pura Malayia, 4400 UPM Serdang, Selangor, Malayia 1 iromr@gmail.com, riamiddin@gmail.com and wiri@cience.upm.edu.my ABSTRACT The paper deal wih he claificaion problem of a ubcla of finie-dimenional algebra. One conider a cla of algebra having wo algebraic operaion wih five ideniie. They have been called diaociaive algebra by Loday. In hi paper we decribe all diaociaive algebra rucure in complex vecor pace of dimenion a mo hree. Keyword ociaive algebra, diaociaive algebra, iomorphim. INTRODUCTION In 199 Loday (Loday, (199)) inroduced everal clae of algebra. Thee clae of algebra iniially have arien from ome problem of algebraic K-heory. However, laer on i urned ou hey have ome geomerical and phyical applicaion a well. Le u ae one of hem o moivae he reearch problem of hi paper. I i well nown ha any aociaive algebra give rie o a Lie algebra, wih brace [ a, b] = ab ba. Le D be an algebraquipped wih wo binary operaion, - he lef produc and he righ produc, aifying he following five axiom a, b, c D. (a b) c=a (b c), (a b) c=a (b c), (a b) c=a (b c), (a b) c=a (b c), (a b) c=a (b c),

2 Irom M. Rihiboev, Iamiddin S. Rahimov & Wiriany Bari Then D, according o Loday (Loday e al., (001)), i aid o be an aociaive dialgebra (or a diaociaive algebra). In fac, hee axiom are variaion of he aociaive law. Therefore aociaive algebra are dialgebra for which he wo produc coincide. The peculiar poin i ha he brace [a, b] = a b - b a define a rucure in D, called Leibniz algebra rucure, wih ideniy [[ a, b], c] = [[ a, c], b] + [ a,[ b, c]]. The operaion [, ] in D i no aniymmeric, unle he lef and righ produc coincide. If we require he aniymmericiy of [, ] hen ( D,[, ]) become a Lie algebra. The main moivaion of Loday o inroduce he cla of Leibniz algebra wa he earch of an obrucion o he periodiciy in algebraic K-heory. Beide hi purely algebraic moivaion ome relaionhip wih claical geomery, non-commuaive geomery and phyic have been recenly dicovered. We will briefly dicu he caegorie of Loday' algebra and inerrelaion beween hem in Secion (Loday Diagram). The goal of hi paper i o give a complee claificaion of complex diaociaive algebra in dimenion a mo hree. The ouline of he paper i a follow. Secion (Loday Diagram) and Secion (On Algebraic Variey Dia) deal wih he definiion and imple properie of he Loday algebra. The main reul of he paper i in Secion 4 (Claificaion of Low-Dimenional Complex Diaociaive Algebra), where we decribe all diaociaive algebra rucure on wo and hree dimenional complex vecor pace. Furher all algebra are aumed o be over complex number. LODAY DIAGRAM Definiion.1 Leibniz algebra L i an algebra wih a binary operaion [, ]L L L aifying he following Leibniz ideniy [[ a, b], c] = [[ a, c], b] + [ a,[ b, c]], a, b, c L. When he brace happen o be ew-ymmeric, we ge a Lie algebra ince he Leibniz ideniy become he Jacobi ideniy. 4 Malayian Journal of Mahemaical Science

3 Claificaion of -Dimenional Complex Diaociaive Algebra Any aociaive algebra give rie o a Lie algebra by [ a, b] = ab ba. In 199 Loday propoed o inroduce a new noion of algebra which give, by a imilar procedure, a Leibniz algebra. He ared wih wo diinc operaion for he produc ab and he produc ba, o ha he brace i no necearily ew-ymmeric. Explicily, he defined an aociaive dialgebra (or a diaociaive algebra) a a vecor pace D equipped wih wo aociaive operaion and called repecively lef and righ produc, aifying hree more axiom a, b, c D. (a b) c=a (b c), (a b) c=a (b c), (a b) c=a (b c), I i immediae o chec ha [a,b]=a b-b a define a Leibniz brace. Hence any diaociaive algebra give rie o a Leibniz algebra. Definiion. Le (D 1, 1, 1 ) and (D,, ) be diaociaive algebra. Then a homomorphim of dialgebra D 1 and D i a linear mapping f D D uch ha 1 a, b D 1. f(a 1 b)=f(a) f(b), f(a 1 b)=f(a) f(b), Becive homomorphim i aid o be iomorphim. Loday and hi colleague have conruced and udied a (co)homology heory for diaociaive algebra (Loday e al., (001)). Since an aociaive algebra i a paricular cae of diaociaive algebra, we ge a new (co)homology heory for aociaive algebra a well. Moreover, Loday inroduced anoher cla of algebra, called dendriform algebra, which are cloely relaed o he above defined clae of algebra in (co)homological manner. Malayian Journal of Mahemaical Science 4

4 Irom M. Rihiboev, Iamiddin S. Rahimov & Wiriany Bari Definiion. Dendriform algebra E i an algebra wih wo binary operaion E E E, E E E aifying he following axiom ( a b) c = ( a c) b + a ( b c), ( a b) c = a ( b c), a, b, c E. ( a b) c + ( a b) c = a ( b c), The reul inerwining diaociaive and dendriform algebra can be expreed in he framewor of algebraic operad. In order o illurae i, Loday define a cla of Zinbiel algebra, which i Kozul dual o he caegory of Leibniz algebra. Definiion.4 Zinbiel algebra R i an algebra wih a binary operaion R R R, aifying he condiion ( a b) c = a ( b c) + a ( c b), a, b, c R. Each one of hee ype of algebra define a binary quadraic operad. For hee operad, here i a well-defined noion of Kozul dualiy heory! devied by Ginzburg and Kapranov. Le P be he dual of he operad P (noe ha P!! = P ). The noion of diaociaive algebra define an algebraic operad Dia, which i binary and quadraic. By he heory of Ginzburg and Kapranov, here i a well-defined dual operad Dia!. Loday ha hown ha hi i preciely he operad Dend of he dendriform algebra. In oher word a dual diaociaive algebra i nohing bu a dendriform algebra. The caegorie of algebra over hee operad aemble ino a commuaive diagram of funcor below which reflec he Kozul dualiy (Ginzburg and Kapranov, (1994)). 44 Malayian Journal of Mahemaical Science

5 Claificaion of -Dimenional Complex Diaociaive Algebra Dend Dia Zinb Leib Com Lie Figure 1 Loday diagram. In he diagram Zinb, Com,, Lie, Leib and for Zinbiel, Commuaive, ociaive, Lie and Leibniz algebra caegorie, repecively, and he Kozul dualiy in i correpond o ymmery around he verical axi paing hrough!!!! =, Com = Lie, Zinb = Leib, Dend = Dia. Oberve ha claificaion of complex low dimenional Lie, Leibniz and Zinbiel algebra ha been given in (Jacobon, (196)), (Ayupov and Omirov, (1999)) and (Dzhumadildaev and Tulenbaev, (005)) repecively. ON ALGEBRAIC VARIETY DIAS In hi ecion we recall ome elemenary fac on diaociaive algebra ha will be ued laer on. Le V be an n-dimenional vecor pace and e 1,... n be a bai of V. Then a diaociaive rucure on V can be defined a wo bilinear mapping λ V V V repreening he lef produc and µ V V V repreening he righ produc, conened via diaociaive algebra axiom. Malayian Journal of Mahemaical Science 45

6 Irom M. Rihiboev, Iamiddin S. Rahimov & Wiriany Bari Hence, an n-dimenional diaociaive algebra D can be een a a riple D = ( V, λ, µ ) where λ and µ are aociaive law on V. We will denoe by Dia he e of diaociaive algebra law on V. Le u denoe by q γ and δ, where i, j,,,, q = 1,,,..., n, he rucure conan of a diaociaive algebra wih repec o he bai e 1,..., e n of V, where e i e j = γ e and e i e j = δ e for i, j, = 1,,,..., n. Then Dia can be conidered a a cloed ube of n -dimenional affine pace pecified by he following yem of polynomial equaion wih repec o he rucure conan γ and δ i q j γ γ = γ γ, i j γ γ = γ δ, i j δ γ = δ γ, i j γ δ = δ δ, δ δ = δ i δ j. Thu Dia can be conidered a a ubvariey of n -dimenional affine pace. On Dia he linear marix group GL ac by changing of bai. Thi acion can be expreed a follow if r D = { γ, δ }, hen n j g = [ ] GL n and g i r p q 1 p q r 1 i j l γ pq l γ pq {( g D),( g D) } = { g g ( g ), g g ( g ) }. CLASSIFICATION OF LOW-DIMENSIONAL COMPLEX DIASSOCIATIVE ALGEBRAS In hi ecion we dicu a claificaion of low-dimenional diaociaive algebra. Our raegy coni of he following wo ep. For he fir ep we conider aociaive algebra wih he operaion. Geing aociaive algebra claificaion, we fix one of hem and inead of he 46 Malayian Journal of Mahemaical Science

7 Claificaion of -Dimenional Complex Diaociaive Algebra acion of GL n, conider he ame acion of he choen aociaive algebra abilizer. In he econd ep we ae rucure conan wih repec o algebraic operaion and elemen of abilizer group a variable and hen examine diaociaive algebra axiom. Alhough hi procedure mae i lighly eaier o olve he claificaion problem in low dimenional cae, he problem of claificaion in general, however, remain a big problem. q From now and wha follow we ue he following noaion n - q and Dian - and for q-h aociaive and q-h diaociaive algebra rucure in n-dimenional vecor pace, repecively. Two Dimenional Diaociaive Algebra Theorem 4.1 Le A be a -dimenional complex aociaive algebra. Then i i iomorphic o one of he following pairwie non-iomorphic aociaive algebra Abelian 1 e e = e e e = e e e = e 1e = e e e = e e1 = e e e = e 1e = e e = e e e = e e = e. For -dimenional complex diaociaive algebra he following rucural reul hold. Theorem 4. Any -dimenional complex diaociaive algebra eiher i aociaive or iomorphic o one of he following pairwie non-iomorphic diaociaive algebra 1 Dia e 1 e 1 =e 1 1 e 1 =e 1 e 1 =e Dia e 1 e 1 =e 1 1 e =e 1 e 1 =e 1 Dia e 1 e 1 =e 1 e 1 =αe 4 Dia e 1 e 1 =e 1 1 e =e 1 e 1 =e 1 e 1 =e. Malayian Journal of Mahemaical Science 47

8 Irom M. Rihiboev, Iamiddin S. Rahimov & Wiriany Bari Proof. Le D be a wo-dimenional vecor pace. To deermine a diaociaive algebra rucure on D, we conider D wih repec o one aociaive operaion. I i one of algebra from he li of Theorem 4.1. Le A 1 =(D, ) be he algebra e 1 e 1 =e 1. The econd muliplicaion operaion in D, we define a follow e 1 e 1 =α 1 e 1 +α e, e 1 e =α e 1 +α 4 e, e e 1 =α 5 e 1 +α 6 e, e e =α 7 e 1 +α 8 e. Now verifying diaociaive algebra axiom, we ge everal conrain for he coefficien α i where i = 1,,...,8. Applying (e 1 e 1 ) e 1 = e 1 (e 1 e 1 ), we ge (α 1 e 1 +α e ) e 1 = e 1 e 1 and hen α 1 e 1 = e 1. Therefore α 1 = 1. The verificaion of (e 1 e 1 ) e 1 = e 1 (e 1 e 1 ) lead o e 1 e 1 = e 1 (e 1 +α e ) and from hi we ge e 1 +α e = e 1. Hence we obain α = 0. Conider (e 1 e 1 ) e = e 1 (e 1 e ). I implie ha e 1 e = 0, herefore α = 0 and α = 0. 4 The nex relaion o conider i (e 1 e ) e 1 = e 1 (e e 1 ). I implie ha 0 = e 1 (α 5 e 1 +α 6 e ) and we ge 5 = 0 α. By he following obervaion we find α6( α6 1) = 0. Therefore α 6 i eiher equal o 0 or 1. Indeed, (e e 1 ) e 1 = e (e 1 e 1 ) α 6 (e e 1 ) = e e 1. To find α 7 and α 8, we noe ha (e e ) e 1 = e (e e 1 ) (α 7 e 1 +α 8 e ) e 1 =0 α 7 e 1 +α 6 α 8 e = 0. Hence we have α7 = 0, α6α8 = 0. Finally, we apply (e e ) e = e (e e ) α 8 (e e ) = 0, and ge α = Malayian Journal of Mahemaical Science

9 Claificaion of -Dimenional Complex Diaociaive Algebra The verificaion of all oher cae lead o he obained conrain. Thu, in hi cae we come o he diaociaive algebra wih he muliplicaion able where α6( α6 1) = 0. e 1 e 1 = e 1 1 e 1 = e 1 e 1 =α 6 e If α = 0 6, hen he righ and lef produc coincide and we ge he aociaive algebra. If α = 1, one obain he diaociaive algebra 1 6 Dia. The oher algebra of he li of Theorem 4. can be obained by a minor modificaion of he obervaion above. The Claificaion of -Dimenional Complex ociaive Algebra menioned above, o claify he low-dimenional diaociaive algebra we need complee li of aociaive algebra in repecive dimenion. By he following heorem we give a reul from (Bari and Rihiboev, (007)) on claificaion of -dimenional complex aociaive algebra. Theorem 4. Any -dimenional non decompoable complex aociaive algebra A i iomorphic o one of he following pairwie non-iomorphic algebra 1 1e e1 1e e 1e e 4 e1 e 5 1e e1 6 e e 7 e e 8 1e e1 9 1e1 e 10 1e e 11 1e e 1 1e1 e e = e = e e1 e = e e = e = e e1 = e e = e e = e e = e = e e = e e = e = e e = e e = e = e e1 e = e e = e e = e1 e = e e = e e = e1 e = e e = e = e e1 e = e e = 1e e = e e1 e = e e = e e = e1 = e e = e1 = e, C \ {1} e = 1e = e e1 = e Malayian Journal of Mahemaical Science 49

10 Irom M. Rihiboev, Iamiddin S. Rahimov & Wiriany Bari Remar 4.1 There exi he following pairwie non-iomorphic decompoable aociaive algebra in dimenion hree e e = e e = e e = e e = e1 e = e e = e e = e = e e = e e = e = e e = e e = e = e e e1 15 1e e1 16 e1 e1 17 1e1 e The Claificaion of -Dimenional Complex Diaociaive Algebra Uing he reul from Theorem 4., we have he following Theorem 4.4 Any -dimenional complex diaociaive algebra D i eiher aociaive or iomorphic o one of he following pairwie non-iomorphic algebra. Dia 1 e 1 e =e 1 e =e e =e e =e e =e Dia e 1 e =e 1 e =e e =e e 1 =e 1 e =e e =e Dia e 1 e =e 1 e =e e =e e =e e 1 =e 1 Dia 4 e 1 e =e e =e e =e e =e Dia 5 e 1 e =e e =e e =e e 1 =e 1 - e e =e Dia 6 e 1 e =e e =e e =e e 1 =e 1 e =e e =e Dia 7 e 1 e =e e = e e =e e 1 =e e =e e =e Dia 8 e 1 e =e e =e e =e e 1 =e e =e e 1 =e 1 - e, e e = e Dia 9 e e 1 =e e =e e =e e 1 =e 1 e =e e =e Dia 10 e e 1 =e 1 e =e e =e e 1 =e 1 e =e Dia 11 e e 1 =e 1 e =e e =e e 1 =e 1 e =e e =e 1 Dia e 1 e =e 1 e =e e 1 =e 1 e =e 1 e =e 1 e 1 =e 1, e e =e 50 Malayian Journal of Mahemaical Science

11 Claificaion of -Dimenional Complex Diaociaive Algebra Dia 1 e 1 e =e 1 e =e e 1 =e 1 e =e 1 e =e 1 e 1 =e 1, e e =e e =e Dia 14 e 1 e =e 1 e =e e 1 =e 1 e =e 1 e =e 1 +e e 1 =e 1, e e =e e =e Dia 15 e 1 e 1 =e e =e e =e Dia 16 e 1 e =e e 1 =e 1 e 1 =me 1 e =ne e 1 =pe, e e =qe Dia 17 e 1 e =e 1 e =e e 1 =e 1 e 1 =e +e where, m, n, p, q C. Proof. We give he proof only for one cae. The oher cae can be carried ou by a minor changing. Suppoe ha aociaive algebra A 1 =(D, ) ha he following muliplicaion able (hi i he algebra from Theorem 4.) 7 e e =e e 1 =e 1 e =e. We define A =(D, ) by he muliplicaion able e 1 e 1 =α 1 e 1 +α e +α e 1 e =α 4 e 1 +α 5 e +α 6 e 1 e =α 7 e 1 +α 8 e +α 9 e, e e 1 =β 1 e 1 +β e +β e e =β 4 e 1 +β 5 e +β 6 e e =β 7 e 1 +β 8 e +β 9 e, e e 1 =γ 1 e 1 +γ e +γ e e =γ 4 e 1 +γ 5 e +γ 6 e e =γ 7 e 1 +γ 8 e +γ 9 e, where α i, β i and γ are unnown ( i, j, =1,,...,9 ). Verifying he diaociaive algebra axiom, we ge he following conrain for he rucure conan and α1 = α = α = α4 = α5 = α6 = α7 = α8 = α 9 = 0, β1 = β = β = β4 = β5 = β6 = β7 = β 9 = 0, γ = γ = γ = γ = γ = 0, γ = γ = β = β, β γ = 0, β γ = 0, γ γ = 0, γ = γ. 5 5 Malayian Journal of Mahemaical Science 51

12 Irom M. Rihiboev, Iamiddin S. Rahimov & Wiriany Bari Afer ome imple compuaion we ge a li of algebra a follow Cae 1. If β8 0 hen β = 1 and 8 γ 5 = γ 8 = 0. We hen have I i an aociaive algebra. e e =e e 1 =e 1 e =e, e e =e e 1 =e 1 e =e. Cae. Suppoe ha β = 0 8 hen we have Now le u conider he following γ γ = 0, 5 8 γ 5 = γ 5. Cae.1. If γ 5 0 hen γ = 1 and γ = algebra Dia 11 e e =e e 1 =e 1 e =e,. In hi cae we obain he e e 1 =e 1 e =e e =e. Cae.. If γ = 0 5 hen i give he following muliplicaion able e e 1 =e 1 e =γ 8 e +e. Taing he following bae change e ' = γ 8e + e ' = e 1 ' = e 1 we find he able of muliplicaion a follow e e 1 = e 1 e = e e = e e 1 =e 1 e = e. 10 Thi i he algebra Dia. Similar obervaion can be applied for he oher cae. The final reul hen can be lied ou a follow ociaive algebra A 1 = 1, A =(D, ) A 1 =, A =(D, ) A 1 =, A =(D, ) Correponding diaociaive algebra Trivial algebra* Trivial algebra Trivial algebra, Dia, Dia, Dia, Dia, Dia 5 Malayian Journal of Mahemaical Science

13 Claificaion of -Dimenional Complex Diaociaive Algebra ociaive algebra A 1 = 4, A =(D, ) Correponding diaociaive algebra Trivial algebra, 9 Dia A 1 = 5, A =(D, ) A 1 = 6, A =(D, ) A 1 = 8, A =(D, ) A 1 = 9, A =(D, ) A 1 = 10, A =(D, ) A 1 = 11, A =(D, ) Trivial algebra Trivial algebra Trivial algebra, Dia, Dia, Dia Trivial algebra 16 Dia 16 Dia A 1 = 1, A =(D, ) Trivial algebra, A 1 = 1, A =(D, ) Trivial algebra A 1 = 14, A =(D, ) Trivial algebra A 1 = 15, A =(D, ) Trivial algebra, A 1 = 16, A =(D, ) Trivial algebra A 1 = 17, A =(D, ) Trivial algebra, 17 Dia Dia, Dia, Dia 15 Dia 1 Some ymbolic compuaion in hi paper have been done by uing Maple ofware. ACKNOWLEDGEMENTS Thi reearch wa uppored by FRGS/FASA1-007/Sain Tulen/ UPM/97. The auhor han Shuhra Rahimov for uppor in Maple calculaion. REFERENCES Ayupov, Sh. A. and Omirov, B.A On -dimenional Leibniz algebra, Uzbe Mah. Jour., (in Ruian) Bari, W. and Rihiboev, I.M On low dimenional diaociaive algebra, Proceeding of Third Conference on Reearch and Educaion in Mahemaic (ICREM), UPM, Malayia Malayian Journal of Mahemaical Science 5

14 Irom M. Rihiboev, Iamiddin S. Rahimov & Wiriany Bari Dzhumadildaev, A.S. and Tulenbaev, K.M Nilpoency of Zinbiel Algebra, J. Dyn. Conrol Sy., 11() Ginzburg, V. and Kapranov, M.M Kozul dualiy for operad, Due Mah. J., Jacobon, N Lie algebra. Inercience Trac on Pure and Applied Mahemaic, Loday, J.-L., Frabei, A., Chapoon, F. and Goicho, F Dialgebra and Relaed Operad, Lecure Noe in Mah. Berlin Springer. Loday, J.-L Une verion non commuaive de algèbre de Lie le algèbre de Leibniz, Eneign. Mah, 9, Rahimov, I.S., Rihiboev, I.M. and Bari, W Complee li of low dimenional complex aociaive algebra. arxiv v1 [mah.ra]. 54 Malayian Journal of Mahemaical Science