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Quantum Segre maps via cocycle twists

Yuri Bazlov, Runyang Chen

TL;DR

The paper develops a general framework for cocycle twist quantization of monoid-graded algebras and extends it to morphisms and twisted tensor products, enabling systematic quantization of projective-space constructions. It shows that quantum projective spaces arise as cocycle twists of classical coordinate rings, and that Segre embeddings admit quantized analogues via factorizable cocycles, yielding a quantum Segre map that recovers Arici–Galuppi–Gateva-Ivanova’s construction in a concrete setting. The key contribution is combining second-cohomology calculations for monoids, Yamazaki factorization, and twisted tensor products to produce explicit quantizations of the classical Segre map, including a precise description in terms of Kronecker products of antisymmetric matrices. This framework provides a unifying, algebraic approach to constructing and relating quantum products of projective spaces, with potential implications for noncommutative algebraic geometry and multiparameter quantum groups. The results establish that quantum Segre maps can be realized by factoring cocycles and Kronecker-product deformations, linking classical and quantum geometries in a concrete, computable way.

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.

Quantum Segre maps via cocycle twists

TL;DR

The paper develops a general framework for cocycle twist quantization of monoid-graded algebras and extends it to morphisms and twisted tensor products, enabling systematic quantization of projective-space constructions. It shows that quantum projective spaces arise as cocycle twists of classical coordinate rings, and that Segre embeddings admit quantized analogues via factorizable cocycles, yielding a quantum Segre map that recovers Arici–Galuppi–Gateva-Ivanova’s construction in a concrete setting. The key contribution is combining second-cohomology calculations for monoids, Yamazaki factorization, and twisted tensor products to produce explicit quantizations of the classical Segre map, including a precise description in terms of Kronecker products of antisymmetric matrices. This framework provides a unifying, algebraic approach to constructing and relating quantum products of projective spaces, with potential implications for noncommutative algebraic geometry and multiparameter quantum groups. The results establish that quantum Segre maps can be realized by factoring cocycles and Kronecker-product deformations, linking classical and quantum geometries in a concrete, computable way.

Abstract

A well-known noncommutative deformation of the polynomial algebra can be obtained as a twist of by a cocycle on the grading semigroup. Of particular interest to us is an interpretation of 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 , 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.
Paper Structure (15 sections, 16 theorems, 37 equations)

This paper contains 15 sections, 16 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. The second cohomology of the free commutative monoid $\mathbb{N}^n$
  3. The second cohomology of a monoid
  4. The Yamazaki factorization of a cocycle
  5. Abelian cocycles on a commutative monoid
  6. The second cohomology of $\mathbb{N}^n$ and multiplicatively antisymmetric matrices
  7. Cocycle twists of graded algebras
  8. A cocycle twist of a graded associative algebra
  9. Morphisms between twists of algebras graded by different monoids
  10. Twisted tensor product of graded algebras
  11. Quantum Segre maps via cocycle quantization
  12. Classical Segre embeddings
  13. Gradings on the homogeneous coordinate rings
  14. Quantum projective spaces as cocycle twists
  15. Quantizations of the classical Segre map. The factorizable cocycle case

Key Result

Proposition 2.3

View $S$ and $T$ as submonoids of $S\times T$. The Yamazaki factorization map where $\alpha_\mu(s,t) = \dfrac{\mu(s,t)}{\mu(t,s)}$ for $s\in S$ and $t\in T$, is a surjective homomorphism of groups.

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 29 more