Groups with involution, and quasigroups with cracovian representations

TL;DR

This paper introduces a novel nonassociative product in groups with involution, leading to a new quasigroup representation via cracovian matrices, differing from traditional group matrix representations.

Contribution

It defines a new class of quasigroups with cracovian representations and explores their properties, extending the algebraic framework beyond classical group representations.

Findings

Defined a nonassociative product in groups with involution.

Introduced cracovian matrices and the column-by-column product.

Connected quasigroup properties with groups like Clifford and orthogonal groups.

Abstract

In groups with involution a nonassociative product of elements is defined, which leads to the definition of a certain type of quasigroups. These quasigroups are represented by square tables of complex numbers, with inverses, which differ from the matrix representations of groups in the rule of performing the product of two tables. The row-by-column product of two matrices in representations of groups is replaced by the column-by-column product, which is called the cracovian product, in representations of the defined type of quasigroups. The matrices undergoing the column-by-column product are called cracovians. The basic properties of the quasigroups connected with groups with involution are determined while only a summary of the properties of cracovian algebra is presented, as the basis of cracovian representation theory for the quasigroups connected with groups with involution. Clifford groups are groups with involution and the quasigroups connected with them are determined. The orthogonal and pseudo-orthogonal rotation groups belong to groups with involution. An analogy is drawn between Weyl's "hidden" symmetry group of an object, and the quasigroup connected with the group with involution.