This preprint was accepted July 30, 1997.
Contact:
S.A. Evdokimov,
I.N. Ponomarenko,
A. M. Vershik
Abstract:
In this paper we discuss algebras in Plancherel duality, i.e.
a special class of pairs of semisimple finite-dimensional
algebras with involution being in a nondegenerate duality as
vector spaces. This class arised more than twenty years ago as
a generalization of the Krein-Tanaka duality and Hopf algebras.
We present new axiomatics of algebras in Plancherel duality according
to the properties of the corresponding pairing. It is proved that the
pairing is a Plancherel one iff it is positive, homogeneous and isometric.
It turns out that the above class provides a natural framework
for the algebraic approach to combinatorics connected with the
notion of C-algebra.
For an arbitrary C-algebra (possibly non-commutative)
a positivity
condition generalizing the Krein condition in commutative case,
is defined.
We show that the class of positive C-algebras includes those
arising in algebraic combinatorics from
association schemes (possibly non-commutative). It is proved
that the category of positive C-algebras is equivalent to
the category of pairs of algebras in Plancherel duality one
of which being commutative.
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