The geometry of a deformation of the standard addition on the integral lattice

TL;DR

This paper explores a new associative multiplication on the integer lattice that induces a group structure on a specific subset where standard addition fails to do so, using geometric and algebraic deformation techniques.

Contribution

It introduces a novel deformation of the group multiplication on the integer lattice to establish a group structure on a subset defined by modular conditions.

Findings

Constructed a new associative multiplication on $\\mathbb Z^n$

Realized the group geometrically in the enveloping space

Identified generators and relations of the deformed group

Abstract

Let $\mathfrak A_n$ be the subset of the standard integer lattice $\mathbb Z^n$, $\mathfrak A_n\subset\mathbb Z^n$ which is defined by the condition $\mathfrak A_n=((a_1,...,a_n)\in\mathbb Z^n | a_i\not\equiv a_j\mod n, \forall i,j\in {1,... n})$. It is clear that the standard addition on the lattice $\mathbb Z^n$ does not induce the group structure on the set $\mathfrak A_n$ since the componentwise sum of some two vectors may contain components which are equal modulo $n$. Our aim is to find a new associative multiplication on the lattice $\mathbb Z^n$ such that the induced multiplication on the set $\mathfrak A_n$ gives it the group structure. In this paper the group structure on the subset $\mathfrak A_n$ of the integer lattice $\mathbb Z^n$ is studied by means of the constructions of a deformation of a group multiplication. The geometric realization of this group in the enveloping space and its generators and relations between them are found. We begin with the main constructions and the results we need for them.