Quantum Segre maps via cocycle twists

TL;DR

This paper develops a general cocycle twist method to quantize tensor products and morphisms, applying it to construct noncommutative Segre embeddings of projective spaces, linking to quantum projective spaces.

Contribution

It introduces a general framework for cocycle twist quantization of graded algebras and applies it to create noncommutative Segre maps, extending previous proposals.

Findings

Constructed deformations of classical Segre embeddings using cocycle twists.

Connected noncommutative Segre maps to factorizable cocycles.

Provided a unified approach to quantum projective space deformations.

Abstract

A well-known noncommutative deformation $\mathcal A^N_{\mathbf{q}}$ of the polynomial algebra $\mathcal A^N$ can be obtained as a twist of $\mathcal A^N$ by a cocycle on the grading semigroup. Of particular interest to us is an interpretation of $A^N_{\mathbf{q}}$ as a quantum projective space. We outline a general method of cocycle twist quantization of tensor products and morphisms between algebras graded by monoids and use it to construct deformations of the classical Segre embeddings of projective spaces. The noncommutative Segre maps $s_{n,m}$, proposed by Arici, Galuppi and Gateva-Ivanova, arise as a particular case of our construction which corresponds to factorizable cocycles in the sense of Yamazaki.