Characters of central piques

Abstract

As a first step towards a duality theory for central quasigroups, the paper presents an explicit computation of the characters of a central pique (quasigroup with pointed idempotent) using Wigner's "little groups" method. The characters of a central pique's cloop (principally isotopic abelian group) form a dual pique. The conjugacy classes of the dual correspond to the characters of the primal; indeed the unitary character table of the dual is the inverse of the unitary character table of the primal. Together with its dual, a central pique forms a structure known as the double. The double satisfies identities indexed by loops of 2-power order. These identities project onto the unit circle to yield identities involving character values.

Semisymmetrizations of Abelian Group Isotopes

“…The group structure on ( A, +) is pointwise, so that a(ξ+η) = aξ+aη for a in A and ξ, η in A. The dual pique Q is the set A equipped with the product ξ • η = Rξ + Lη, the left action of R and L on A being given by the mixed associative laws a(Rξ) = (aR)ξ and a(Lξ) = (aL)ξ for a in A and ξ in A [6]. Duals of semisymmetrizations of finite abelian groups and their isotopes are specified as follows.…”

“…The character table of the semisymmetrization, under the standard quasigroup normalization [4] (extending the usual normalization for group character tables) is given by Table 1 according to [6,Th. 7.1].…”

“…For an abelian group isotope Q, the semisymmetrization Q ∆ is a central pique. The final two sections apply the results of the paper [6] to this central pique. Section 4 investigates the corresponding dual.…”