COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

302 Lowell J. Paige
this problem are of the order of 10’ years and hence it seems appropriate to
make rather severe restrictions on the type of identities desired. I would like
to illustrate the possible use of the computer with an example from the area
of Jordan algebras.
Let us recall a few facts concerning Jordan algebras. An abstract Jordan
algebra 8 over a field @ is a non-associative algebra satisfying the identities
ab = ba,
(a2b)a = a2(ba)
for all a, b E 5X.
The simplest examples of Jordan algebras arise from associative algebras.
Thus, let % be an associative algebra over a field @ of characteristic not
two. In terms of the associative multiplication of elements in 9X, written
a* b, define a new multiplication
ab = +(a*b+b.a).
If we retain the vector space structure of 9l and replace the associative multiplication,
a*b, by the new multiplication, ab, we obtain a Jordan algebra
which we denote by a+.
If a Jordan algebra 3 is isomorphic to a subalgebra of an algebra Ql+
(5?l associative), then 3 is called a special Jordan algebra. One of the fascinating
aspects of Jordan algebras is that there exist Jordan algebras which
are not special; these Jordan algebras are called exceptional.
The best known example of an exceptional Jordan algebra is constructed
as follows : Let C be an eight-dimensional generalized Cayley algebra over
the field Cp. Denote the involution in C by x + X, where x+X E @. Consider
the set H(C) of 3 X 3 matrices
ra c 61
where a, ,B, y E @ and a, b, c E C; i.e. the hermitian 3 X 3 matrices. Multiplication
for the elements of H(C) is the usual Jordan product
XY = gx* Ys- Y-X]
and it is not difficult to see that H(C) is a Jordan algebra. Professor A. A.
Albert and I [ll] have shown that H(C) is not the homomorphic image of
any special Jordan algebra, and this implies that there are identities satisfied
by special Jordan algebras which are not valid for all Jordan algebras.
A search for these identities presents interesting possibilities for a computer.
In order that I might sketch a possible attack on the problem of identities
in special Jordan algebras, we shall need a few more results about special
Jordan algebras which are due to Professor P. M. Cohn.
Non-associative algebras 303
Let A@) be the free associative algebra on the set of generators
{Xl, x2, * * ., xn} and denote by J$‘) the Jordan subalgebra of A(“)+ generated
by {xi, x2, . . . , x,}. On A(“) define a linear mapping x-+x*, the
reversal operator, by the equation
(XjlXj2 . . . XiJ* = XikXikml . . . Xi,Xil
for monomials consisting of products of the generators x1, x2, . . . , x,.
Since the monomials form a basis for A(“), the reversal operator * is uniquely
determined and
(xy)* = y*x*, x** = x
for all x, y E A(“). An element x of A(“) is said to be reversible (symmetric) if
x* = x. The set of all reversible elements H(*) of A@) is easily seen to be a
Jordan subalgebra of A(“)+ ; furthermore,
fib) 2 J$’
for all n. Cohn has shown that
H(2) 2 JP) and H(3) 2 56”)
and otherwise J$‘) is properly contained in H(“).
The exceptional Jordan algebra H(C) described earlier is generated by
three elements. Hence, it is the homomorphic image of the free Jordan
algebra Jc3) on three generators. On the other hand, H(C) is not the homomorphic
image of Jh3) (the free special Jordan algebra on three generators).
Thus, we know that the natural homomorphism v from
,, : J’3’ -t J;3’
has a non-zero kernel Kc3). A basis for Kc31 has not been found. Professor
Glennie [12] has shown that there are no elements of degree less than 7 in
Kc31 and that there are elements of degree 8 in Kc3J. It should be clear that
any non-zero element of Kc3) will provide an identity for special Jordan
algebras which is not valid for all Jordan algebras.
The importance of Cohn’s relationship, H c3) 2 Ji3), lies in the fact that we
have an explicit way to write the elements of the free special Jordan algebra
Jh3) in terms of reversible elements. Hence, treating Ji3) as a graded algebra
(by degree), we can compute the number of basis elements of a fixed degree.
For example, if we let the generators of Ji3) be a, b and c, then the number
of basis elements for the vector subspace spanned by all elements of total
degree 8 and degrees 3,3 and 2 in a, b and c respectively is 280.
A computer attack for the determination of the elements in Kc31 for the
natural mapping
y : J(3) + J$3’
may now be described.