proceedings of the american mathematical society Volume 104, 4, December 1988

FINDING IN ASSOCIATIVE ALGEBRAS JOHN R. FAULKNER (Communicated by Donald S. Passman)

ABSTRACT. An embedding of octonion algebras in central simple degree three associative algebras is given.

Using standard operations on trace 0 elements of a simple 9-dimensional associa- tive (such as 3 x 3 matrices) over a of not 3 containing a primitive cube root of 1, we obtain in the Theorem the somewhat surprising embedding of an in an . One clue to this em- bedding is that the Tits construction in [4] applied to F © cf (cf split octonion) or to F3 (3 x 3 matrices) gives the same 27-dimensional exceptional Jordan algebra, suggesting that F © (f is related to F3. A similar embedding of the 27-dimensional exceptional Jordan algebra JF as trace 0 elements in the degree 4 special Jordan algebra J4 of 4 x 4 hermitian quaternionic matrices was previously known (see [1 and 2]). In this case, a relation between F ® fí and ^4 was suggested by the Cayley-Dickson process in [1] which applied to F © f or to %\ gives the same 56-dimensional structurable algebra. Our approach is to show that the set An of trace 0 elements of a separable degree 3 A carries a nondegenerate S permitting composition S(x * y) — S(x)S(y), from which it follows that an isotope of (An,* ) is a unital . If A has dimension 9, then Aq has dimension 8, and the composition algebra must be octonion. We begin by establishing some basic facts about degree 3 alternative algebras. LEMMA. Let A be a finite-dimensional alternative algebra of degree 3, with generic minimum polynomial px(X) = A3 - T(x)A2 + S(x)X —N(x)l. Setting x# = x2-T(x)x+S(x)l,S(x,y) = S(x+y)-S(x)-S(y), andxxy = (x+y)*-x*-y*, we have (a) N(xy) = N(x)N(y), (b)S(l)=T(l) = Z,N(l) = l, (c)S(x) = T(x*), (d) S{x,y)=T{xxy), (e)S(x,l) = 2T(x), (î)T(x)T(y)=T(xy)+T(xxy), (g)2S(x) = T(x)2-T(x2), (h) T(xy) = T(yx), (i)xxy = xy + yx- T(x)y - T(y)x + (T(x)T(y) - T(xy))l, (j) x** = N(x)x,

Received by the editors March 15, 1988. 1980 Subject Classification (1985 Revision). Primary 17A75; Secondary 17C40.

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(k) (x x y)* + x* x y* = T(x*y)y + T(xy*)x, (1) (xy)* =2/#x#, (m)S(xy) = S(yx). PROOF. We may assume the field is infinite and use differential calculus as in [5] where (a), (b) and px(X) = N(X1 — x) are proved. We have x_1 = N(x)~1x# if N(x) ^ 0, since xx# - 7V(x)l = x#x - N(x)l = px(x) = 0. Since N(x + Xy) = X3N(y)+X2dxN[y+XdyN[x+N(x), we see T(x) = dxN[i and S(x) = diN\x. Since N(x + Xy) = N(xy~l + Xl)N(y), we also see c^TV^ = T(xy~1)N(y) = T(xy*), if N(x) ^ 0. By density, dxN\y = T(xy*). Now x = 1 gives (c), and (d) follows by linearization. We see x x y = dx#\y so T(x(y x z)) = dydxN\z is symmetric in x,y, and z. In particular, S(x, 1) — dxS[i = dxdiN\i = T(x(l x 1)) = 2T(x) showing (e), since 1x1 = 2(1#) = 2 by the definition of X, #, and (b). We also compute x x 1 = dx#\i = xl + lx - T(x)l - T(l)x + S(x, 1)1 = T(x)l - x. Hence, T(x x y) = T((x x l)y) = T(T(x)y - xy) = T(x)T(y) - T(xy) showing (f). Clearly, (g) and (h) follow from (f). Moreover, (i) follows from (d) and (f). Since x** = (N(x)x~1)# = N(x)2N(x~1)(x-1)-1 and N(x)N(x~1) = N(l) = 1, we see (j) holds by density. Moreover, (k) is the coefficient of A2 in (j) with x replaced by x + Xy, using dxN\y = T(xy*). To get (1), note (xy)* = N(xy)(xy)_1 = N(x)N(y)y_1x~l = y*x* and use density. Finally, (m) follows from (c), (1), and (h). Before looking at the general situation, we motivate the definition of x * y in the following Theorem by investigating the relationship between An and ¿f in the particular degree 3 alternative algebra A = F ©

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THEOREM. If A is a finite-dimensional alternative algebra of degree 3 over a field F of characteristic not S containing an element ui with w2 + w + 1 = 0, then the coefficient S(x) of X in the generic minimum polynomial permits composition S (a * b) = S(a)S(b) on Aq = {x: T(x) = 0} relative to the product a* b = (wab — ui2ba)o where xo = x — |T(x)l. If A is separable over F, then the quadratic form S is nondegenerate, and (Ao,*) is a quasi-composition algebra with a unital isotope which is a composition algebra. If A also has dimension 9, then the isotope is an octonion algebra. PROOF. We have S(a)S(b) = T(a*)T(b*) (by (c)) = T(a*b*) + T(a* x b*) (by (f)) = T(a*b*) - T((a x b)*)) (by T applied to (k) using a,b G Ao) = S(ab) - S(a x b) by (c), (1), and (m)) = S(ab)-S(ab + ba-T(ab)l) (by (i)) = S(ab) - S(ab) - S(ba) - S(ab, ba) + T(ab)S(ab + ba, 1) -T(ab)2S(l) = -S(ab) - S(ab, ba) + T(ab)2 (by (b), (m), (e), and (h)). On the other hand, S(x0) = S(x - \T(x)l) = S(x) - \S(x, l)T(x) + ¿T(x)25(l) = S(x)-±T(x)2 by (e) and (b). Thus, S(a*b) = S(uab-u2ba)- ±T(wab-w2ba)2 = Lü2S(ab)+cü4S(ba)-uoj2S(ab,ba)-±(uj-u2)2T(ab)2 (by (h)) = -S(ab)-S(ab,ba)+ T(ab)2, since u3 = 1, u) + ui2 = —1, and (w — w2)2 = —3. If A is separable over F, then T(xy) is nondegenerate on A and on Ao- Since S(x,y) = —T(xy) on A0 by (d) and (f), S is also nondegenerate on Ao. The remainder now follows from Kaplansky [3]. Indeed, whenever S(u) ^ 0, we get a unital composition algebra from the isotope with product aOb = (R*_1a) * (L*_16) with unit u * u. Here i?* and L* denote the right and left multiplication in (Ao, *)■ EXAMPLE. If A = F3 and u = diag{2p, -p, -p} for p = (w - w3)-1, then (F3)0 is a split octonion algebra with u and norm form tr(adj a) relative to the product aOb = (u* a) * (b*u) where x * y = uxy — w2yx + ptr(xy) Id. PROOF. We use A = F3 in the Theorem, so T(x) = tr(x), N(x) = det(x), S(x) = tr(adjx), and x* — adjx, the matrix adjoint. Since p~2 = -3, we see —|(w — w2) = p2p~l = p and x* y = (wxy — u2yx)o — ojxy — ui2yx + ptr(xy) Id. Let ei = diag{l, w, w2}, e2 = diag{l,w2,cj}, so e¿ G A0 have e¿e_,= Id, iV(e¿) = 1, et = el — ej f°r z ^ 3- We see e¿ * e¿ = (u; - w2)(e2)0 = P~lCj, i ^ j, and e¿ * ej — (w — w2)(e¿ej)o =0. If u = pei + pe2 = diag{2p,—p,—p}, we see u * ei = a * u — ej and u * u = u. If Lxy = R*x = x * y on Ao, we note that L*u is the inverse of R*. on Fei + Fe2. In general, if Lxy = RyX = xy, and if d = diag{Ai, A2, A3}, then Ld and Rd act on matrices FEij(i ^ j) by multiplica- tion by Xi and A¿. On FEij, we have L*d— u>Ld — u)2Rd = (wA¿ - w2Xj)\d and R* = -u2Ld + tjRd = (-u2X% + uXj)Id, so L*dR*d= -(A2 + A2 + A,Aj)Id. Note that for d = w, Ai = 2p, A2 = -p,A3 = -p, this is -((2p)2 + (-p)2 + (~2p2)) or -((-p)2 + (-p)2 + (p)2) both of which are -3p2 = 1. Thus, R* = (L;)_1.

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Since S(u) = tr(adj(u)) = trdiag{p2, -2p2, -2p2} = 1, we get (F3)0 with product adb — (R*.~1a) * (L* _16) = (u * a) * (b * u) is an eight-dimensional unital composi- tion algebra from [3]. Since S(ei) = tr(adj(ei)) = tr(e2) = 0, it is a split octonion algebra. I wish to thank Kevin McCrimmon for suggestions improving the exposition of this paper.

REFERENCES

1. B. N. Allison and J. R. Faulkner, A Cayley-Dickson process ¡or a class of structurable algebras, Trans. Amer. Math. Soc. 283 (1984), 185-210. 2. J. R. Faulkner, A geometric construction of Moufang planes (To appear in Geometrice Dedi- cata). 3. I. Kaplansky, Infinite-dimensional quadratic forms admitting composition, Proc. Amer. Math. Soc. 4 (1953), 956-960. 4. K. McCrimmon, The Freudenthal-Springer-Tits constructions o¡ exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495-510. 5. _, Generically algebraic algebras, Trans. Amer. Math. Soc. 127 (1967), 527-551.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF VIRGINIA, CHARLOTTESVILLE, VIR- GINIA 22903

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