A Finite Quantum Symmetry of M(3,C)

TL;DR

This paper explores a 27-dimensional Hopf algebra as a finite quantum symmetry of the matrix algebra M(3,C), relevant to the color sector in Connes' Standard Model formulation, establishing duality and representations.

Contribution

It introduces and analyzes a specific finite quantum group symmetry of M(3,C), connecting it with dual Hopf algebras and their representations in the context of noncommutative geometry.

Findings

Established duality between Hopf algebras A(F) and H.

Defined representations of H on M(3,C) and A(F).

Described the role of A(F) as a quantum symmetry in the Standard Model context.

Abstract

The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups A(SL(2,C))->A(SL_q(2))->A(F), q^3=1, is studied as a finite quantum group symmetry of the matrix algebra M(3,C), describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra H,investigated in a recent work by Robert Coquereaux, is established and used to define a representation of H on M(3,C) and two commuting representations of H on A(F).