# The Z3-graded extension of the Poincar\'e algebra

## TL;DR

This paper constructs a Z3-graded extension of the Poincaré algebra, involving an extended 12-dimensional space-time with complex-valued metrics, and explores its representations and symmetry properties.

## Contribution

It introduces a novel Z3-graded extension of the Poincaré algebra requiring a 12-dimensional space-time with complex metrics, expanding the algebraic framework of relativistic symmetries.

## Key findings

- Extended 12D Minkowski space-time with complex metrics
- Representation via differential operators and Casimir operators
- Discussion of symmetry properties of the generalized algebra

## Abstract

A Z3 symmetric generalization of the Dirac equation was proposed in recent series of papers, where its properties and solutions discussed. The generalized Dirac operator acts on "coloured spinors" composed out of six Pauli spinors, describing three colours and particle-antiparticle degrees of freedom characterizing a single quark state, thus combining Z2 x Z_2 x Z_3 symmetries of 12-component generalized wave functions. Spinorial representation of the Z3-graded generalized Lorentz algebra was introduced, leading to the appearance of extra Z2 x Z2 x Z3 symmetries, probably englobing the symmetries of isospin, flavors and families. The present article proposes a construction of Z3-graded extension of the Poincar\'e algebra. It turns out that such a generalization requires introduction of extended 12-dimensional Minkowskian space-time containing the usual 4-dimensional space-time as a subspace, and two other mutually conjugate "replicas" with complex-valued vectors and metric tensors. Representation in terms of differential operators and generalized Casimir operators are introduced and their symmetry properties are briefly discussed.