Polynomial Identities of RA2 Loop Algebras

Transcription

1 Ž. Journal of Algebra 213, Artcle ID jabr , avalable onlne at on Polynomal Identtes of RA2 Loop Algebras S. O. Juraans and L. A. Peres Departamento de Matematca, Unersdade de Sao Paulo, Caxa Postal 66281, Sao Paulo, SP, Brazl, E-mal: and Communcated by E. Klenfeld Receved January 15, 1998 In ths paper we study the polynomal denttes of the loop algebra of some RA2 loops of order Academc Press 1. INTRODUCTION A loop s a nonempty set L havng a bnary operaton wth a two-sded dentty element such that, for all a, b L, each one of the equatons ax b and xa b has a unque soluton. Let L be a loop and R be a rng. The loop rng RL s the free R-module wth bass L and multplcaton nduced by the bnary operaton of L. When R s a feld RL s called a loop algebra. Loop rngs have been the object of study for many years. In a classcal paper Chen and Goodare 5 mposed the alternatve denttes on the loop algebra and were able to gve a descrpton of the loops whose loop algebra satsfes these denttes. They called these loops RA loops. It turned out that these loops have very nce propertes and that ther descrpton nearly does not depend on the characterstc of the rng nvolved. Ths s qute remarkable because smlar results n group rngs have been obtaned only by requrng that the unt group satsfes some dentty. On the other hand, f G s a fnte group then by Wedderburn s Theorem CG s a drect sum of matrx algebras over the complex feld C and thus, by the AmtsurLevtzk Theorem, CG satsfes a standard * Ths paper was wrtten whle the second-named author held a grant from CNPq of Brazl $30.00 Copyrght 1999 by Academc Press All rghts of reproducton n any form reserved.

2 558 JURIAANS AND PERESI polynomal dentty. A result of Bruck 3 tells us that f L s a fnte loop then K L s sem-smple f K s a feld and charž K. 0. As far as we know, t s not known that every smple drect summand of K L satsfes a polynomal dentty and f they do t s not obvous that K L should also satsfy a polynomal dentty. There are 3 RA2 loops of order 16 whch we label L 3, L 4, L5 and whch are not RA loops. In 2 de Barros and Watson showed that f F2 s the feld wth two elements then F2 L s not somorphc to F2 Lj for j, and n 1 de Barros and Juraans proved that f Z s the rng of ntegers then Z L determnes L. In ths paper we study the polynomal denttes of the loop algebras of these 3 non-alternatve RA2 loops of order 16. It turns out that all these loop algebras satsfy many denttes whch are far from beng trval. Moreover the set of degree 4 denttes s the same for these loop algebras. In Secton 2 we gve the prelmnary results. In Secton 3 we present the polynomal denttes. All denttes and central denttes of degree 4 are determned. Some of degree 5 are gven. 2. SOME PRELIMINARIES In 6 Chen and Goodare defned a RA2 loop as a loop whose loop algebra over a feld of characterstc 2 s alternatve but not assocatve. Formerly they have defned RA loops as those loops whose loop algebra over a feld of characterstc p 2 s alternatve but not assocatve. There are fve RA2 loops whose order are 16. All of them can be constructed from a subgroup usng the method of duplcaton ntroduced by Chen 4. It s known that two of them are RA loops and the other three are not. We shall denote these three loops by L 3, L 4, and L 5. We gve a bref descrpton of these notatons. Let G be a group and 2 wrte L G Gu where u g ZŽ G. 0. Let : G G be an nvoluton such that g 0 g 0. Defne the multplcaton n L by the rules g hu hg u, gu h gh u, and gu hu g h 0 g for all g, h G. Then L s a loop and ths multplcaton s extended n the obvous way to the loop algebra K L. If we wrte K L KG Ž KG. uthen the multplcaton s gven by Ž x yu.ž z wu. xz gw y Žwx yz. 0 u and s the obvous extenson of : G G to the group algebra KG. The loop L s Ž. ² 4 2 denoted by L G,, g. Let D a, b : a b 1, bab a 3 : o 4 be the 2 dhedral group of order 8 and s a. Then ZŽ D. D 1, s Wth these notatons the loops L are obtaned as

3 RA2 LOOP ALGEBRAS 559 Ž. 1 L3 LD, 4, s, where g g for all g D 4, L LD, Ž, s., where the nvoluton s gven by Žab n m. 4 4 a nž1. m b m, Ž. 1 L5 LD, 4, 1, where g g for all g D 4. It s easy to see, and well known, that K D 4K M Ž K. 4 2 f K s a feld of characterstc 2, and from ths we see easly that f 3, 4, 5 then K L 8K A, where A s an algebra of dmenson eght over K. Note that the algebra A s obtaned duplcatng M Ž K. 2 usng a process whch s smlar to that used to obtan CayleyDckson algebras Žsee. 9. In fact A M Ž K. M Ž K. 2 2 u where the multplcaton s gven by the rules above. The central dempotent assocated to A s e 1 Ž 1 s. 2. Havng ths t s now easy to obtan a natural bass for these algebras. Let : KL A be gven by Ž x. xe. Then ths bass s a subset of Ž L.. Usng the notatons of de Barros and Watson 2, Tables 35 we have that s corresponds to 3 and that Ž The bass elements 1, e,...,e of A correspond to Ž 1., Ž 2., Ž 5., Ž 6., Ž 9., Ž 10., Ž 13., Ž The multplcatons tables are: LEMMA 1. algebra. Assume that charž K. 2. Then, for 3, 4, 5, A s a smple

4 560 JURIAANS AND PERESI Proof. We have that e 1 Ž 1 s. 2 s a prmtve dempotent of K D 4.As n group rngs we have that K L Ž L, L. ž / L K KŽ C C C. 8K. L 2 2 2

5 RA2 LOOP ALGEBRAS 561 Hence K L K L Ž 1 e. K Le 8K K Le. Thus A K Le. Therefore to prove that A s smple we have just to prove that e s a prmtve dempotent of K L. In fact, let f x yu be a central element. Then gf fg for all g D.So gx yg u xg yg 4 u. Ths mples that yg yg and thus yg g 1 y. If 3, 5 take g a and f take g ab to obtan ya y and thus that yž 1 a ,.e., ye 0. Hence f e e1 e2 s the sum of two central dempotents then, lettng e x yuwe have e eež x yu. e xe K D 4. But e s prm- tve n K D 4. Thus e1 0 or e2 0. It follows that e s a prmtve dempotent. 3. THE POLYNOMIAL IDENTITIES In 8 Hentzel and Peres lsted all polynomal denttes and polynomal central denttes of degree 6 for the CayleyDckson algebras CŽ,,..A polynomal central dentty s a polynomal whch evaluates nto the feld for all substtutons of the varables by all choces of elements of the algebra, but s not tself an dentty. The CayleyDckson algebras are alternatve. Thus the process to obtan these polynomal denttes was descrbed usng alternatve reductons. However, wth some changes the process can be appled to obtan the polynomal denttes of small degree of any nonassocatve algebra of small dmenson. We apply ths process to the algebras A 3, A 4, and A 5. We denote by a, b the commutator ab ba and by Ž a, b, c. the assocator Ž ab. c abc. Ž. THEOREM 1. Let K be a feld of characterstc 2, 3. Then for 3, 4, 5 we hae: Ž. The algebras A do not satsfy any polynomal dentty or central dentty of degree 3. Ž. The degree 4 polynomal denttes of A are consequences of the denttes 2 2 x, y, x y, x, x xy, xx xx, yx 0, Ž 1. Ž x, x, y., y x, Ž y, x, y. Ž x, y, y, x. Ž y, y, x., x y, Ž x, y, x. Ž y, x, x, y. 0, Ž 2. Ž x, y, y., x x, Ž y, x, y. Ž x, y, x, y. Ž y, x, x., y y, Ž x, y, x. Ž y, x, y, x. 0. Ž 3.

6 562 JURIAANS AND PERESI Ž Ž. Ž.. In 4 8 we denote by Ý the alternatng sum oer the arables y and z. Ý Ž x, x, z. y zž x, x, y. Ž x, y, z. x Ž z, x, y. x Ž y, z, x. x yž x, z, x. Ž y, x, xz. Ž x, yx, z. Ž x, zy, x. Ž xz, x, y. Ž xz, y, x. Ž z, x, yx. 2Ž x, xz, y. 2Ž yx, x, z. 2Ž zy, x, x. 4 0, Ž 4. Ý Ž x, x, y. z Ž z, y, x. x xž y, z, x. Ž xz, x, y. Ž x, z, x. y yž x, x, z. Ž yx, x, z. xž z, x, y. Ž xy, z, x. Ž y, xz, x. Ž z, x, yx. 2Ž x, x, zy. 2Ž x, zy, x. 2Ž y, x, xz. 2 zž x, y, x. 2Ž z, yx, x. 4 0, Ž 5. Ý 2Ž x, xy, z. Ž x, x, y. z 2Ž xy, z, x. 2Ž x, yz, x. xž y, z, x. Ž x, y, zx. Ž x, zx, y. Ž xz, x, y. 2Ž x, z, xy. Ž y, x, xz. yž x, x, z. 2Ž yz, x, x. Ž zx, y, x. Ž z, xy, x. 2 zž x, y, x. Ž z, y, x. x4 0, Ž 6. Ý 2Ž x, x, zy. xž x, z, y. Ž x, yx, z. 2 xž y, x, z. 3Ž x, y, xz. 3Ž x, z, yx. 3Ž x, zy, x. xž z, y, x. 3Ž y, xz, x. Ž y, x, zx. yž z, x, x. 2Ž z, yx, x. Ž z, x 2, y. zž x, x, y. Ž z, x, xy.4 0, Ž 7. Ý xž x, y, z. 2Ž x, x, zy. Ž x, y, x. z Ž xz, x, y. xž z, x, y. Ž xy, z, x. Ž x, z, y. x 2Ž x, zy, x. Ž z, x, x. y Ž yx, x, z. 2Ž y, x, xz. Ž y, xz, x. Ž z, x, yx. 2Ž z, yx, x. zž y, x, x. 4 0, Ž 8. Ž 1. Ž x x, x, x. Ž x, x, x x. 0, Ž 9. 4 Ý Ž1. Ž2. Ž3. Ž4. Ž1. Ž2. Ž3. Ž4. S 4 Ý Ž1. Ž2. Ž3. Ž4. Ž1. Ž2. Ž3. Ž4. 4 S 4 Ž 1. Ž x, x x, x. Ž x, x, x x. 0. Ž 10.

7 RA2 LOOP ALGEBRAS 563 Ž. The degree 4 polynomal central denttes of A are consequences of the denttes Ž x, x, y., y Ž x, y, y., x 2 x, Ž y, x, y. 2x, y y, x Ž y, y, x., x Ž y, x, x., y 2 y, Ž x, y, x. 2 y, xx, y, Ž 11. Ý Ž1. Ž2. Ž3. Ž4. S 4 Ž 1. Ž x x.ž x x.. Ž 12. Proof. To prove that Ž.Ž are denttes of A we lnearze them, replace the varables by all combnatons of bass elements, and verfy that each substtuton evaluates to zero. To prove that Ž 11. and Ž 12. are central denttes we do the same and verfy that f a substtuton does not evaluate to zero then t evaluates to a non-zero element of the feld. We now prove that we have lsted all denttes. Snce the argument n the proof s the same for each A, each degree, and each representaton Ž n the sense of. 8 of the group algebra of the symmetrc group, we gve here only the proof for A 3, degree 4, and the thrd representaton. There are 5 ways to assocate 4 varables. They are f ŽŽ x x. x x 4, f Ž x Ž x x.. x, f Ž x x.ž x x., f x ŽŽ x x. x., f x Ž x Ž x x We consder each f as a functon on the varables x 1, x 2, x 3, x 4. Let g1ž x 1, x 2, x 3, x4. x1x2x3x4 x3x2x1x4 x1x4x3x2 x3x4x1x2 x2x1x3x4 x2x3x1x4 x4x1x3x2 x4x3x1x2 x1x2x4x3 x3x2x4x1 x1x4x2x3 x3x4x2x1 x2x1x4x3 x2x3x4x1 x4x1x2x3 x4x3x2x1 and g2ž x 1, x 2, x 3, x4. x1x3x2x4 x3x1x2x4 x1x3x4x2 x3x1x4x2 x2x3x1x4 x2x1x3x4 x4x3x1x2 x4x1x3x2 x1x4x2x3 x3x4x2x1 x1x2x4x3 x3x2x4x1 x2x4x1x3 x2x4x3x1 x4x2x1x3 x4x2 x3x 1. We denote by f Ž x, x, x, x. j the functon obtaned from g by assocat- ng each term of g n the way f. Then a polynomal dentty Ž j or central polynomal dentty. of A has the form Žsee. 8 3 ž / 2 5 ÝÝ j j j1 x f Ž x, x, x, x.. Ž 13.

8 564 JURIAANS AND PERESI We now replace the varables x 1, x 2, x 3, x4 by bass elements of A3 to fnd the constrants the x must satsfy for Ž 13. j to be an dentty or central dentty for A 3. To smplfy the notaton we let e0 1 and denote the substtuton e, e, e, e smply by Ž, j, k, l.. The substtutons Ž 0, 1, 1, 2. j k l, Ž 0, 1, 2, 4., Ž 0, 2, 3, 5., Ž 1, 1, 2, 5., Ž 1, 1, 3, 3., Ž 1, 2, 2, 7., Ž 1, 2, 3, 4., and Ž2, 3, 6, 2. gve equatons 2 x11 x12 x13 x14 x154 x21 x22 x23 x24 x2544e 2 ; 2 x11 x12 x14 2 x15 x21 x22 x24 x254e 7; x12 2 x13 x14 x234e 4; x11 x13 x15 2 x21 x23 2 x254e 7; x11 x12 x13 x14 x154 2 x21 x22 x23 x24 x2544e 0; x12 x14 2 x22 2 x244e 6; x21 x22 x24 x254e 4; 2 x x 2 x x 2 x x x x x x 44e If Ž 13. s an dentty of A3 then all the coeffcents of these equatons must be zero. Ths gves a system wth 8 equatons and 10 varables x j. Ths system has rank 8. The solutons are then lnear combnatons of solutons x11 1, x12 0, x13 2, x14 0, x15 1, x21 1, x22 3, x23 4, x24 3, x25 1; x11 0, x12 1, x13 1, x14 1, x15 1, x21 0, x22 2, x23 2, x24 1, x25 1. The frst of these two solutons gves the lnearzed form of the dentty obtaned by subtractng Ž. 3 from Ž. 2. The second gves the lnearzed form of Ž. 3. Now, f Ž 13. s a central dentty then the coeffcent of e0 need not be zero. We have now a system wth 7 equatons and rank 7. The solutons of ths system are lnear combnatons of 3 solutons: the two solutons above and x11 0, x12 0, x13 4, x14 0, x15 0, x21 3, x22 3, x23 8, x24 3, x25 3. Ž. Ths last soluton s the lnearzed form of 11.

9 RA2 LOOP ALGEBRAS 565 Remark. It seems that the algebras A Ž 3, 4, 5. have the same set of degree 5 denttes. As an example we menton that Ý Ž ab.cd. e až bc.de. 2 až b, cd, e. 2Ž a, b.cd, e. 2 ab. Ž c, d, e.4 0, where Ý means the alternatng sum over the arguments a, b, c, d, s an dentty of A. THEOREM 2. Let K be a feld of characterstc 2, 3. Let 3, 4, 5. Then f s a polynomal dentty Ž central dentty. of A f and only f f s a polynomal dentty Ž central dentty. of K L. In partcular we hae found all polynomal denttes and central denttes of degree 4 of K L. Proof. Let f be a polynomal dentty of A. We know that K L 8K A where A s a smple algebra Ž by Lemma 1. whose center s K. So f s a polynomal dentty of K. Hence f s a polynomal dentty for all the smple components of K L. An element of K L s an orthogonal sum of elements belongng to the varous smple components and only the one belongng to A s possbly non-central. Ths makes t clear that f we substtute elements of dfferent components, each monomal n f wll vansh. Ths and the fact that f s a polynomal dentty for each component proves that f s one for K L. On the other hand, f f s a polynomal dentty for K L then t s also one for A. The proof s analogous f f s a central dentty of A. The last statement follows now from Theorem 1. Remark. For the loop algebras KL Ž 3, 4, 5., we found nterestng denttes, for example, Ž.Ž. 1, 9, and Ž 10.. It would be nterestng to know f t s possble to classfy the loops whose loop algebra satsfes one of these denttes. ACKNOWLEDGMENTS The authors thank Dr. Luz Gonzaga Xaver de Barros of Unversty of Sao Paulo who was nvolved n the begnnng of ths research project. Behnd the proof of Theorem 1 there are extensve calculatons on matrces and substtutons by elements of the algebras A Ž 3, 4, 5.. These calculatons were done usng the computer facltes at IME ŽInsttuto de Matematca e Estatıstca.. We used a X-termnal granted by FAPESP ŽFundaçao de Amparo `a Pesqusa do Estado de Sao Paulo..

10 566 JURIAANS AND PERESI REFERENCES 1. L. G. X. de Barros and S. O. Juraans, Some loops whose loop algebras are flexble, Noa J. Math. Game Theory Algebra 5 Ž 1996., L. G. X. de Barros and B. Watson, On modular loop algebras of RA2 loops, Algebras Groups Geom. 14 Ž 1997., R. H. Bruck, Some results n the theory of lnear nonassocatve algebras, Trans. Amer. Math. Soc. 56 Ž 1944., O. Chen, Moufang loops of small order, I, Trans. Amer. Math. Soc. 188 Ž 1974., O. Chen and E. G. Goodare, Loops whose loop rngs are alternatve, Comm. Algebra 14 Ž 1986., O. Chen and E. G. Goodare, Loops whose loop rngs n characterstc 2 are alternatve, Comm. Algebra 18 Ž 1990., E. G. Goodare, E. Jespers, and C. P. Mles, Alternatve Loop Rngs, Math. Studes, Vol. 184, North-Holland, New York, I. R. Hentzel and L. A. Peres, Identtes of CayleyDckson algebras, J. Algebra 188 Ž 1997., K. A. Zhevlakov, A. M. Sln ko, I. P. Shestakov, and A. I. Shrshov, Rngs That Are Nearly Assocatve, Academc Press, New York, 1982.