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Finding Octonion Algebras in Associative Algebras John R proceedings of the american mathematical society Volume 104, Number 4, December 1988 FINDING OCTONION ALGEBRAS 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 algebra (such as 3 x 3 matrices) over a field of characteristic not 3 containing a primitive cube root of 1, we obtain in the Theorem the somewhat surprising embedding of an octonion algebra in an associative algebra. 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 alternative algebra A carries a nondegenerate quadratic form S permitting composition S(x * y) — S(x)S(y), from which it follows that an isotope of (An,* ) is a unital composition algebra. 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 Mathematics Subject Classification (1985 Revision). Primary 17A75; Secondary 17C40. ©1988 American Mathematical Society 0002-9939/88 $1 00 + $.25 per page 1027 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1028 J. R. FAULKNER (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 © <f. If x = (a, a), then T(x) = a + t(a), S(x) = at(a) + n(a), and N(x) — an(a), in terms of the trace t and the norm n on if. The projection x —►a restricted to An is an isometry of (Ao,5) with (cf,n — t2) since a — —t(a) on A0- Also, if w + u>2+ 1 = 0 in F, then a —y a + u>t(a) is an isometry of (tf, n — t2) with (¿f, n). Combining, we get an isometry 9: (—t(a),a) —y a + u)t(a) of (An,i>) with (cf,n). Since n admits composition relative to the product in &, S automatically admits composition 5(x *y) = S(x)S(y) relative to the product (1) x*y = 9í\92(x)93(y)) which has been pulled back from cf by any isometries 9i of (A0,5) with (¿f, n). In particular, taking all #, = 9 above gives a product x\3y on An isomorphic to that on tf. We would like to express xDy in terms of the original product on A. For y = (/?, b) e Aq, a direct computation (using a + a = t(a)) shows (2) 9(x)9(y) = -cj2ab + ujba - -y, where 7 = ut(ab) + t(a)t(b). On the other hand, if x —»xn is the projection onto An, i.e. xo = x — |T(x)l, then we also have (3) 9((xy)o) = -bä + P1, where p = (u> — ui2)~l. Consequently, 9((uxy — uj2yx)o) = —u>ba+ ui2ab + 7 = (J9(x))(-j9(y)) where j(a) = ä. Taking 9i = 9,92 = j9, and 93 = -JÔ in (1), shows S admits composition relative to x*y = (uxy —oj2yx)o- Moreover, if u e An with 9(u) = l, then 9(x * u) = -j9(x) = 9(u * x) so 9((x * u) * ((-u) * y)) = 9(x)9(y) and xOy is the product in the (u, —u)-isotope of (An, *). This suggests the product * used in the Theorem. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OCTONION ALGEBRAS IN ASSOCIATIVE ALGEBRAS 1029 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 identity element 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.
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