2
M . L. RACINE
inner and an outer ideal of $. For any b #, ^U is an inner ideal, the
principal inner ideal generated by b . A Jordan algebra is said to be simple if
it has no non-trivial ideals . For B an ideal of P, define B = B,
B = y u . , , j e ^ , j 0, the Penico serie s of B and B - B,
B = BU r. . , , j €^ , j 0, the derived series of B. We have B ^ z B^ ,
B [ J " J
B(j* D B(J and B D B *; B and
(
B'J-' are ideals . B is solvable (Penico
solvable) if B ^ = {0} (B = {0}) for some j . McCrimmon [33] has shown
that for finitely generated algebras solvability and Penico solvability are
equivalent.
Let G be an associativ e algebra over $. Then G = (G, U, 1), where
bU = aba, a, b c G, is a Jordan algebra. A Jordan algebra $ is special if $
a
is a subalgebra of G for some associativ e algebra G, otherwise, ^ is said
to be exceptional. If (G, *) is an associativ e algebra with involution then
M(G, *) = {a G | a* = a } is a subalgebra of G . Note that if we let
G = B 0 B where B is the opposite algebra of B and * the exchange
involution of G, then B is isomorphic to #(G, *).
Powers are defined inductively: a = 1, a = a and a = a U ,
a p, n ? Z n_ 0. An element c e # is invertible if there exists a b e $ such
2
that bU = c and b U = 1. Such a b can be shown to be unique and is
c c
-1 (c) (c) (c)
denoted c . If c is an invertible element of $ then (^ , U , 1 ),
(c) (c) (c) -1
where Q = J, U = U U and 1 = c , can be shown to be a Jordan
* *9 a c a ' J
algebra and is called the c-isotop e of p. If every non-zero element of ~$ is
invertible then ^ is said to be a Jordan division algebra. An element z e #,
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