Positivity 8: 369–378, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Jordan Algebras and Dual Afﬁne Connections on
KEIKO UOHASHI AND ATSUMI OHARA
Osaka Prefectural College of Technology and Osaka University, Osaka, Japan
August 28, 2002
Abstract. We study dually ﬂat structures on symmetric cones associated with Jordan algebras. We
give an interpretation of connections, a geometrical concept, in terms of Jordan algebras and show a
relation between doubly autoparallel submanifolds and Jordan subalgebras.
Mathematics Subject Classiﬁcation (2000): 17C36, 53A15
Key words: symmetric cone, Jordan algebra, dual afﬁne connection, doubly autoparallel submani-
We describe dual afﬁne connections on symmetric cones in terms of the Jordan
algebra. Shima gave correspondence between Jordan algebras and afﬁne connec-
tions on symmetric spaces with Hessian structure [10,11]. Ohara presented a relation
between a Jordan algebra and the dual connection especially on a cone of positive
deﬁnite symmetric matrices . The references [10,11] make use of vector ﬁelds
induced by Lie algebras, and  treats of vector ﬁelds along the canonical afﬁne
coordinates. In this paper, we shall generalize the result in  to other symmet-
ric cones. In addition, we deﬁne a new property called doubly autoparallelism
 of submanifolds on symmetric cones by means of dual connections. Doubly
autoparallel submanifolds are useful tools in information geometric analysis or ap-
plications such as interior-point method for convex programming and conditional
independence model and so on [5,7–9]. For example, in optimization problems
called Semideﬁnite Programming (SDP), the result of  shows that if feasible
regions of the problems are doubly autoparallel in the cone of positive deﬁnite
matrices, then the optimal solution can be obtained without numerical iterations
and expressed by an exiplicit formula.
First we give notation and formulas on Jordan algebras, and introduce dually ﬂat
structures on symmetric cones associated with simple Euclidean Jordan algebras in
Section 2. We derive a relation among the dual connections, Jordan algebras and
mutations of Jordan algebras in Section 3. We ﬁnally give a characterization of