This preprint was accepted July 30, 1997.
Contact: S.A. Evdokimov, I.N. Ponomarenko, A. M. Vershik
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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