A New Type of Nonassociative AlgebrasAuthor(s): Robert B. BrownSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 50, No. 5 (Nov. 15, 1963), pp. 947-949Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/71948 .

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A NEW TYPE OF NONASSOCIATIVE ALGEBRAS*

BY ROBERT B. BROWN

THE UNIVERSITY OF CHICAGO

Communicated by A. A. Albert, October 7, 1963

A problem of interest in nonassociative algebras is the study of generalized Cayley algebras and exceptional simple Jordan algebras which are closely related to the exceptional simple Lie algebras. Thus, the derivations of a generalized Cayley algebra form an exceptional Lie algebra of type G, the derivations of an exceptional simple Jordan algebra form an exceptional Lie algebra of type F, and the derivations and right multiplications of elements of trace zero of an exceptional simple Jordan algebra generate an exceptional Lie algebra of type E6.1 Further- more, all of these representations involve the representations of lowest possible dimension for the respective Lie algebras. The lowest possible dimension for a representation space for the exceptional Lie algebra Q7 is 56. Starting from the class of exceptional simple Jordan algebras, we will define a new class of simple non- associative algebras of dimension 56 over their centers and possessing nondegenerate trace forms, such that the derivations and left multiplications of elements of trace zero generate Lie algebras of type E7. Some basic properties of these algebras will be given without proof (most of the proofs consist of direct computations).

DEFINITION. Let 3 be an exceptional simple Jordan algebra over a ground field W

of characteristic not two or three. We define the algebra 91 to consist of all quadruples (a, f, a, b) with a, f in ~ and a, b in a. Setfi = (1, 0, 0, 0), f2 = (0, 1, 0, 0), a12 =

(0, O, a, 0), b2l = (0, 0, O, b). Furthermore, let a X b denote the cross product on 3 and (a, b) = tr (ab) the trace form on a.2 Then the product in the algebra 91 is defined as follows: fi2 = fil, fif2 = fAfi = , f2 = f2, fial2 = Ma12, f2a12 =( (1 - /)a12, fib12 =

(1 - A)b21, f221 = b2l, al2f1 = (1 - v)al2, al2f. = va12, b2lfi = vb2l, b21f2 = (1 - )b2l, a12c12 = wl(a X c)21, b21d21 = c2(b X d)12, al2b2l- = l(a, b)fi, b2la12 = 62(a, b)f2. The

parameters wi, W2, 51, 2 in o are assumed to be nonzero, while the possibility that u or v in ~ is zero is allowed.

The algebra 9 is central simple and is not associative, flexible, or power-associa- tive. The element f = fi + f2 is a unity element, and the left, middle, and right nuclei of 9 all consist of scalar multiples of f only. A trace may be defined on 9 as

follows. If x = afi + 3f2 + a12 + b2l, we set tr (x) = a + F. Then (x, y) = 1/2

tr(xy + yx) is a nondegenerate symmetric bilinear trace form on 9. (If 81 = 62, we

may define (x, y) as simply tr(xy).) We have (xy, z) = (x, yz) for all x, y, z in 9 if

and only if 61 = 62 andu = v = 1. However, this particular special case is not of too

much interest to us, because the derivations and left multiplications of elements of

trace zero do not generate a Lie algebra of type E7. On the other hand, without any restrictions on the parameters oi, , W 6, ,U, Y, v other than that the first four be non-

zero, the derivation algebra of 9 is a Lie algebra of type E6, as is seen by the follow-

ing theorem.

THEOREM 1. If oW, W2, 81, 62 are not zero and the characteristic of the ground field is not two or three, then the derivations D of the algebra 9 are given by D: fi -- O, f2 0, al12 (aDl2)i2, b2l -- (- bDl2*)21, where D12 is a linear transformation on 3 contained

in the Lie algebra generated by the derivations of a and the multiplications of elements

947

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948 MATHEMA TICS: R. B. BROWN PROC. N. A. S.

of trace zero in ,. Further, D12* is the transpose of D12 with respect to the trace form on

In general, the 133-dimensional space ?(91) spanned by the derivations and left multiplications of elements of trace zero in 9 is not closed under the operation of commutation. In fact, we have closure in ?(9) if and only if 81 = 62, g = 1/3, v = 1, and co1i2 = 482. In this case we will write 9 as T (3; 8, c), where 6 = 81 = 62 and o = '/2W1. If 9 = 91(a; 8, c), then (91) is a central simple Lie algebra, 9 is an irreducible representation space for e(9w), and we have the following theorem.

THEOREM 2. If the characteristic of the ground field is not two or three, then the derivations and the left multiplications of elements of trace zero in 9(,; 8, co) generate an exceptional simple Lie algebra V(9) of type E7. (This resembles the representation of the Lie algebra C7 given by Freudenthal.3)

We have also computed the automorphism groups of the algebras 918(; 8, c). This is not difficult, because fl, f2, and f are the only idempotents which are anni- hilated by all derivations of 9(3; 8, co); consequently, if p is an automorphism, then either fi = fi and f'2 = f2, or fiq = f2 and f2 = fi. Let L(3) be the group of all nonsingular linear transformations S on 3 such that N(aS) = N(a) for all a in 3, where N is the norm on 3.

THEOREM 3. If there is a nonsingular linear transformation P on a such that N(aP) = 'o28-1N(a) for all a in a, then the automorphisms of 9(3,; 8, o) are the linear transformations of the form S': fi -- fi, f2 -' f2, al2 -- (aS)i2 for S in L(3), b2l --

(bS*-1)2i, and 'rS', where r: fi - f2, f2 -+ fi, a12 --- (aP)21, b2 - (bP*l)12. If P does not exist, then the automorphisms of 91(3; , c) are just the group { S'ISeL() }. However, such a transformation P always exists if 3 is reduced.

The structure of the group L(3) for 3 reduced has been found by Jacobson.4 He has shown that L(3) modulo its center is simple and that the center consists of the identity transformation I alone or else of the three transformations I, el, e2I, where e is a primitive cube root of unity. This result is essential for the following theorem on the structure of the automorphism groups of the algebras 9(3.; 8, o).

THEOREM 4. Let 3 be a reduced exceptional simple Jordan algebra over a field of characteristic not two or three. The automorphism group Aut (S1) of 9(3; 8, co) has trivial center, and the subgroup L(3)' = {S'ISeL(3) } is a normal subgroup. The center 3 of L(3) ' is a normal subgroup of Aut (9) also and consists of either the identity transformation I or of the three transformations I, el, e2I. Furthermore, L(3,)'/3 is simple. Finally, if ~ is any normal subgroup of Aut (91) containing an element not in L() ', then ' = Aut (9).

Our final result is a classification of the algebras 91(3; 8, o) for 3 reduced. THEOREM 5. Let 31 be a reduced exceptional simple Jordan algebra and 32 be an

arbitrary exceptional simple Jordan algebra over a field of characteristic not two or three. Then 9(31; 61, co,) and 9(32; 62, o2) are isomorphic if and only if 32 is reduced and the coefficient algebras of 31 and 32 are isomorphic.

Research supported by a National Science Foundation Summer Fellowship. Jacobson, N., "Exceptional Lie algebras," Yale mimeographed notes (1957).

2 Springer, T. A., "On a class of Jordan algebras," Amsterdam Academy, Series A, 62, 254-264 (1959), and "The projective octave plane. I," Amsterdam Academy, Series A, 63, 74-88 (1960).

3Freudenthal, H., "Beziehungen der E7 und Es zur Oktavenebene. I," Amsterdam Academy, Series A, 57, 218-230 (1954).

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VOL. 50, 1963 MATHEMATICS: CLIFTON AND SMITH 949

4Jacobson, N., "Some groups of transformations defined by Jordan algebras. III," J. reine angew. Math., 207, 61-85 (1961).

THE EULER CLASS AS AN OBSTRUCTION IN THE THEORY OF FOLIATIONS*

BY YEATON H. CLIFTON AND J. WOLFGANG SMITH

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALlFORNIA, LOS ANGELES

Communicated by S. S. Chern, September 13, 1963

Let X denote an oriented differentiable manifold of dimension n and ? an oriented foliation' of dimension p < n. We ask whether there exists an oriented foliation ?' of X by leaves of dimension p + 1 which is concordant with ? in the sense that each p-leaf of ? lies on a single (p + 1)-leaf of ?'. This is clearly a basic question in the theory of foliations, and we shall refer to it as the first extension problem. Let us suppose, for the moment, that ? constitutes an actual fibering of X, so that there exists a base manifold M of dimension n - p and a projection map r: X -- M with ? = { ~-l(x) :x e M}. It is apparent that an oriented foliation ?' of X by (p + 1)-leaves concordant to ? is now equivalent to a direction field on M, in which case the topological obstruction to the first extension problem is none other than the Euler class of the tangent bundle to M. For an arbitrary foliation ?, on the other hand, the leaf-space A (which results from the identification of all points lying on a single leaf of ?) is usually not a manifold and quite unsuitable to give us an Euler class, since it carries no natural vector bundle. Let us recall at this point that in a previous paper2 we have introduced a category e which extends the category C of topological spaces and continuous maps, and we have shown that a foliation S canonically determines an object 9c in e which generalizes the notion of a base manifold. In particular, whenever ? does admit a classical base manifold M, C is naturally isomorphic to M in C. In general, S occupies an intermediate position between X and A (there exist natural projections X - 9C -- A), and appears to be a more suitable candidate for the role of a base manifold than the classical leaf-space A. This raises the question whether by a suitable development of these ideas it may not be possible to define a notion of object tangent bundle to 9C and a corresponding Euler class which will constitute an obstruction for the first extension problem. It is the purpose of this note to indicate in outline how this can be done and to state the essential result constituting a general solution to the given prob- lem. For a detailed development of the requisite theory of object fiber bundles and for all proofs we refer to a forthcoming paper. Some geometrical applications of the theory will be considered in the last section of this note.

1. Objest Bundles.-We recall that an object C in the category e was defined' to be a pair (X,I), X being a topological space and I a set of local maps u: X -- X (i.e., maps whose domain D(u) is an open subset of X) subject to three axioms: (i) I contains the identity map of X; (ii) given u,v I and x e D(u) n D(v), there shall exist u,v e I such that uou = vrv near x (extension axiom); (iii) given a local map u: X -- X such that for every x e D(u) there exist v,U,v e I such that u o u = V ? v near x, then u e I (closure axiom). X will be referred to as the fundamental space and

VOL. 50, 1963 MATHEMATICS: CLIFTON AND SMITH 949

4Jacobson, N., "Some groups of transformations defined by Jordan algebras. III," J. reine angew. Math., 207, 61-85 (1961).

THE EULER CLASS AS AN OBSTRUCTION IN THE THEORY OF FOLIATIONS*

BY YEATON H. CLIFTON AND J. WOLFGANG SMITH

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALlFORNIA, LOS ANGELES

Communicated by S. S. Chern, September 13, 1963

Let X denote an oriented differentiable manifold of dimension n and ? an oriented foliation' of dimension p < n. We ask whether there exists an oriented foliation ?' of X by leaves of dimension p + 1 which is concordant with ? in the sense that each p-leaf of ? lies on a single (p + 1)-leaf of ?'. This is clearly a basic question in the theory of foliations, and we shall refer to it as the first extension problem. Let us suppose, for the moment, that ? constitutes an actual fibering of X, so that there exists a base manifold M of dimension n - p and a projection map r: X -- M with ? = { ~-l(x) :x e M}. It is apparent that an oriented foliation ?' of X by (p + 1)-leaves concordant to ? is now equivalent to a direction field on M, in which case the topological obstruction to the first extension problem is none other than the Euler class of the tangent bundle to M. For an arbitrary foliation ?, on the other hand, the leaf-space A (which results from the identification of all points lying on a single leaf of ?) is usually not a manifold and quite unsuitable to give us an Euler class, since it carries no natural vector bundle. Let us recall at this point that in a previous paper2 we have introduced a category e which extends the category C of topological spaces and continuous maps, and we have shown that a foliation S canonically determines an object 9c in e which generalizes the notion of a base manifold. In particular, whenever ? does admit a classical base manifold M, C is naturally isomorphic to M in C. In general, S occupies an intermediate position between X and A (there exist natural projections X - 9C -- A), and appears to be a more suitable candidate for the role of a base manifold than the classical leaf-space A. This raises the question whether by a suitable development of these ideas it may not be possible to define a notion of object tangent bundle to 9C and a corresponding Euler class which will constitute an obstruction for the first extension problem. It is the purpose of this note to indicate in outline how this can be done and to state the essential result constituting a general solution to the given prob- lem. For a detailed development of the requisite theory of object fiber bundles and for all proofs we refer to a forthcoming paper. Some geometrical applications of the theory will be considered in the last section of this note.

1. Objest Bundles.-We recall that an object C in the category e was defined' to be a pair (X,I), X being a topological space and I a set of local maps u: X -- X (i.e., maps whose domain D(u) is an open subset of X) subject to three axioms: (i) I contains the identity map of X; (ii) given u,v I and x e D(u) n D(v), there shall exist u,v e I such that uou = vrv near x (extension axiom); (iii) given a local map u: X -- X such that for every x e D(u) there exist v,U,v e I such that u o u = V ? v near x, then u e I (closure axiom). X will be referred to as the fundamental space and

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