Barros, Luiz G.X. de

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Resenhas IME-USP 1996 , Vol. 2, No.3, 3'1, - "o~. The Isomorphism Problem for Loop Rings 1 Luiz G~ X. de Barros Abstract: We present some positive answers to the ques-tion: "For which rings R and loops L and M the ring isomor-phism RL ~ RM implies the loop isomorphism L ~ M?'. Key words: loop, loop ring, nonassociative algebra . 1 Loops In this section we introduce some fundamental concepts about loops. The basic references are the books by R.H. Bruck [6] and H.O. Pflugfelder [18]. A loop is a set L together with a binary operation· such that i) There exists an element 1 E L such that 1 . x = x . 1 = x for all x E L. ii) For all element a E L, the maps Ra and La defined by Ra(x) = x . a and La (x) = a . x for all x E L are bijections. As a consequence, it follows that in a loop L (a) The equations a . X = b and X . a = b have unique solutions. (b) Every element a E L has a unique left inverse aA and a unique right inverse aP defined as the solutions of the equations X . a = 1 and a . X = 1 respectively. Loops of order 2,3 or 4 are groups, and, up to isomorphism, there are 6 loops of order 5; 109 loops of order 6 and 23 , 750 loops of order 7. A diassociative loop is a loop in which , for all elements x and y, the subloop < x, y > generated by x and y is a group . In a diassociative loop it holds that a A = a P = a-I for all a. The commutator of two elements x and y of a loop L is the element in L , denoted by (x,y),-such that xy = (yx)· (x,y); and, the commutator subloop L' is the subloop generated by all commutators of L. The associator of three elements x, y and z of a loop L is the element in L , denoted by (x, y, z), such that (xy)z = (x(yz)) . (x, y , z); and, the associator subloop A( L) is the sub loop generated by all associators of L. The nucleus N(L) of a loop L is the subset {x ELI (x , a, b) = (a, x, b) (a,b,x) = 1 , Va , b E L}. If we denote by C(L) the subset {x ELI (x,y) 1 Research partially supported by CNPq (Proc. 300411/94). 347 348 Luiz G. X. de Barros 1 , Vy E L}, then the centre of L is the set Z(L) = C(L) n N(L). A M oufang loop is a loop in which, for all x, y and z, the following M oufang identities hold: (xy)(zx) = (x(yz))x ((zx)y)x = z(x(yx)) ((xy)x)z = x(y(xz)) In 1 974 Orin Chein [7] gave a method to construct nonassociative Moufang loops from nonabelian groups: Theorem 1.1 Let G be a nonabelian group with an involution 9 --+ g* such that gg* is in the center of G for all 9 E G and let go E G be a central element such that go = g; . Let u be an indeterminate, set L = G U Gu and define g(hu) = (hg)u (gu)h = (gh*)u (gu)(hu) = goh*g for all g, hE G. Then L is a Moufang loop which is not a group . We write L = L(G,*,go) to indicate that L is constructed in this way from G . It's easy to see that L is not associative. To see this it is enough to take two elements 9 and h in G such that 9 . h # h . g. Then: (u . h*) . g* = (h· u) . g* = hg . u and u· (h* . g*) = u · (gh)* = gh . u . The smallest nonassociative Moufang loop is L(S3 , 0- 1,1) of order 12, where S3 denotes the symmetric group of order 6. Given an associative and commutative ring R with unity and a loop L, we can mimic the construction of a group ring to form the loop ring RL. A ring A is said to be alternative if for all x and y in A the following equalities hold: x.xy = x2.y and xy.y = x .y2 . The study of alternative loop rings begun in 1983 with Edgar G . Goodaire who published the article [14] about those loop rings . In 1986 himself and Orin Chein [9] defined R.A. loops as the loops whose loop algebra over some ring with char-acteristic different from 2 is alternative but not associative and gave a complete description of those loops. There they proved the following Theorem 1.2 Let L be an R.A . loop . Then, there exists a nonabelian group G c L and an element u E L such that L = GUG · u, G' = L' = {I, s} ~ Z (G) = Z(L) The Isomorphism Problem for Loop Rings and L = L(G, *,90) with the involution * : G -+ G given by * {9 9 = sg and u 2 = 90 is an element in Z(L): if 9 E Z(G) if 9 ¢ Z(G) 349 Denoting by Q the quaternion group of order 8, the smallest R.A. loops are two loops of order 16, denoted, as in [7], by M 16 (Q) = L(Q, *, s), the so called Cayley loop, and M 16 (Q,2) = L(Q, *,1), where * is as in the above theorem and s is the nonidentity commutator in Q. Given an R .A. loop L, since the Moufang identities hold in the alternative ring RL, in particular, the elements of L also verify those identities and, thus , L must be a Moufang loop. 2 The Isomorphism Problem The isomorphism problem for group rings asks for which rings R and groups G and H the isomorphism of group algebras RG ~ RH implies that the groups G and H are isomorphic. Or, more compactly, under what conditions is a group "determined" by its group ring? Of course, this problem admits a version for loop rings. We will describe some results about the isomorphism problem for some types of loops over Z, the ring of integers, and Q, the rational field. In 1988, Edgar G. Goodaire and Cesar Polcino Milies [15] proved the following result : TheoreIll 2.1 Let Land M be R.A. loops such that ZL ~ ZM . Then L ~ M. A subloop N of a loop L is said to be normal if for all x and y in L we have that x . (yN) = (xy) . N , (Nx) . y = N . (xy) and xN = Nx . If N is a normal subloop of a loop L we can define the quotient loop LIN. The natural epimorphism L -+ LIN extends to an algebra epimorphism RL -+ R[LIN], and we will denote by tl(L : N) the kernel of this epimorphism. In 1993, Guilherme Leal and Cesar Polcino Milies [17] extended to loop rings of R.A. loops a result of Donald B. Coleman [13] for group rings. TheoreIll 2.2 Let Land M be R.A. loops. Then QL ~ QM if and only if LI L' ~ MIM' and tl(L : L') ~ tl(M : M'). 350 Luiz G. X. de Barros In the same article they proved the following Theorem 2.3 Let L be an R.A. loop with L' = {I, s}. Assume there exists an element Q' E Z(L) such that Q'2 = s. Let M be another loop . Then QL ~ QM if and only if L/L' ~ M/M' and Z(QL) ~ Z(QM). In 1993, Luiz G.X. de Barros [1] completed that result proving Theorem 2.4 Let L be an R.A. loop with L' = {I, s} and assume that there exists no element Q' E Z(L) such that Q'2 = s. Let M be another loop. Then, QL ~ QM if and only if L ~ M . In 1990 , Orin Chein and Edgar G. Goodaire [10] defined (RA2) loops as being those loops whose loop algebra over a ring with characteristic 2 is alternative but not associative and proved that R.A. loops are also (RA2) loops. Then, the modular case, that is, the case where the characteristic of the ring divides the order of the loop, could be studied. Theorem 2.5 (L.G.X. de Barros and C. Polcino Milies [4]) Let Z2 denote the field with two elements. Let Land M be R.A. 2-loops such that Z2L ~ Z2M. Then L ~ M . Code loops were introduced by R .L. Griess Jr . in [16] and classified by O. Chein and E .G. Goodaire in [11] and [12] who showed that nonassociative code loops are (RA2) loops (and thus , Moufang loops) with a unique nonidentity commutator, a unique nonindentity associator and a unique nonidentity square, which coincide. This element is central of order 2. Theorem 2.6 (L.G .X. de Barros and C. Polcino Milies [5]) Let Z2 denote the field with two elements. Let Land M be nonassociative code loops such that Z2L ~ Z2M. Then L ~ M. A similar result holds over the ring of integers. Theorem 2.7 (L .G.X. de Barros and O.S . Juriaans [3]) Let Land M be nonas-sociative code loops such that ZL ~ ZM. Then L ~ M . We recall that an algebra A is flexible if for all x, yEA it holds that x · (y. x) = (x· y) . x. In [2], Luiz G.X. de Barros and Orlando S. Juriaans defined R .F. loops as those loops whose loop algebra over a ring with characteristic different from 2 is flexible but not alternative. Using Chein's method (Theorem 1.1), a loop M = M(L, *, go) can be constructed from an R.A. loop L, with * and go as in Theorem 1.2. This loop M is a non-Moufang diassociative R.F. loop. Theorem 2.8 (L.G.X. de Barros and O.S. Juriaans [2]) Let M and N be R .F. loops constructed from R .A . loops. Then ZM ~ ZN if and only if M ~ N. Theorem 2.9 (L.G.X. de Barros and O.S. Juriaans [2]) Let M and N be R.F. loops constructed from R.A. loops. Then QM ~ QN if and only if M/M' ~ N/N' and 6..(M : M') ~ 6..(N : N'). The Isomorphism Problem for Loop Rings 351 References [1] L.G.X. de Barros, Isomorphisms of Rational Loop Algebras, COInIn. in Al-gebra, 21, 11 (1993),3977-3993 . [2] L.G.X. de Barros and O.S. Juriaans, Some Loops whose Loop Algebras are Flexible, preprint . [3] L.G.X. de Barros and O.S . Juriaans, Integral Loop Rings of Code Loops, Nova Journal of MatheInatics, GaIne Theory and Algebra, to appear. [4] L.G.X. de Barros and C. Polcino Milies, Modular Loop Algebras of R.A. Loops, J. Algebra 175 (1995) , 1027-1040. . [5] L.G.X. de Barros and C. Polcino Milies, Loop Algebras of Code Loops, COInIn. in Algebra, 23 (13) (1995), 4781-4790. [6] R.H. Bruck, A Survey of Bynary Systems, Springer-Verlag, Berlin , 1968. [7] O.Chein, Moufang Loops of Small Order I, Trans. AIner. Math. Soc. 188 (1974),31-51. [8] O. Chein, Moufang Loops of Small Order II, MeInoirs AIner. Math. Soc. 197 (13) (1978) . [9] O. Chein and E.G. Goodaire, Loops whose Loop Rings are Alternative, COInIn. in Algebra, 14 (1986), 293-310. [10] O. Chein and E.G . Goodaire, Loops whose Loop Rings in Characteristic 2 are Alternative, COInIn. in Algebra, 18(3) (1990),659-688. [11] O. Chein and E.G. Goodaire, Moufang Loops with a unique nonidentity com-mutator (associator, square), J . Algebra, 130 (1990), 369-384. [12] O. Chein and E.G. Goodaire, Code Loops are RA 2 Loops, J. Algebra, 130 (1990),385-387. [13] D. B. Coleman, Finit e Groups with Isomorphic Group Algebras, Trans. AIner. Math. Soc., 105 (1962) 1-8. [14] E.G . Goodaire, Alternative Loop Rings, Pub!. Math. Debrecen, 30 (1983), 31-38. [15] E.G. Goodaire and C. Polcino Milies, Isomorphisms of Integral Alternative Loop Rings, Rend. Circolo Mat. PalerIno, II 37 (1988), 126-135. [16] R.L. Griess, Jr ., Code loops, J. Algebra 100 (1986) , 224-234. [17] G. Leal and C. Polcino Milies, Isomorphic Group (and Loop) Algebras, J. Algebra, 155, 1 (1993) , 195-210. 352 Luiz G . X. de Barros [18] H.O. Pflugfelder, Quasigroups and loops: Introduction , Helderman Verlag , Berlin , 1990. Luiz G. X. de Barros Universidade de Sao Paulo Instituto de Matematica e Estatistica Caixa Postal 66281 - Sao Paulo - SP CEP 05389-970 19xb@ime.usp .br Brasil 

The Isomorphism Problem for Loop Rings

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The Isomorphism Problem for Loop Rings

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