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Extensions of a family of Linear Cycle Sets

Jorge Guccione, Juan José Guccione, Christian Valqui

TL;DR

The paper develops a cohomological framework for extensions of a specific linear cycle set H by a trivial abelian group I, focusing on the second cohomology H^2_{⊲,⊳}(H,I) and providing explicit formulas and conditions that classify all extensions. It treats a concrete H with underlying group $\mathds{Z}_{p^{\eta}}$ and operation $i\cdot j=(1-p^{\nu}i)j$, deriving an explicit description of the associated brace and constructing concrete endomorphisms $A,B$ to realize the actions $h\blackdiamond y$ and $y\Yleft h$. The core results show that every 2-cocycle is equivalent to a pair $(\alpha_1(\gamma),f)$ and that H^2 can be computed as ker(F1)∩ker(F2)/Im(G); in the general case with $\Yleft\neq 0$, this is governed by polynomial data P(A),Q(A),R(A),S and the pair (f0,γ) subject to explicit relations. The paper then provides a broad collection of detailed examples across many regimes of primes, exponents, and decomposition types for H and I, yielding complete parameterizations of 2-cocycles and corresponding extensions, thereby offering a systematic method to construct and classify extensions of linear cycle sets in this setting.

Abstract

This paper explores the cohomology of linear cycle sets, focusing on extensions of a specific linear cycle set H by an abelian group I. We derive explicit formulas for the second cohomology group, which classifies these extensions, and establish conditions under which the extensions are fully determined. Key results include a characterization of extensions when I lies in the socle of the extended structure and H is trivial, and the construction of explicit examples for both trivial and non-trivial cases. The paper provides a systematic approach to understanding the structure of these extensions, with applications to various families of abelian groups.

Extensions of a family of Linear Cycle Sets

TL;DR

The paper develops a cohomological framework for extensions of a specific linear cycle set H by a trivial abelian group I, focusing on the second cohomology H^2_{⊲,⊳}(H,I) and providing explicit formulas and conditions that classify all extensions. It treats a concrete H with underlying group and operation , deriving an explicit description of the associated brace and constructing concrete endomorphisms to realize the actions and . The core results show that every 2-cocycle is equivalent to a pair and that H^2 can be computed as ker(F1)∩ker(F2)/Im(G); in the general case with , this is governed by polynomial data P(A),Q(A),R(A),S and the pair (f0,γ) subject to explicit relations. The paper then provides a broad collection of detailed examples across many regimes of primes, exponents, and decomposition types for H and I, yielding complete parameterizations of 2-cocycles and corresponding extensions, thereby offering a systematic method to construct and classify extensions of linear cycle sets in this setting.

Abstract

This paper explores the cohomology of linear cycle sets, focusing on extensions of a specific linear cycle set H by an abelian group I. We derive explicit formulas for the second cohomology group, which classifies these extensions, and establish conditions under which the extensions are fully determined. Key results include a characterization of extensions when I lies in the socle of the extended structure and H is trivial, and the construction of explicit examples for both trivial and non-trivial cases. The paper provides a systematic approach to understanding the structure of these extensions, with applications to various families of abelian groups.
Paper Structure (20 sections, 46 theorems, 336 equations, 1 figure)

This paper contains 20 sections, 46 theorems, 336 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Braces and linear cycle sets
  3. Extensions of linear cycle sets
  4. Building extensions of linear cycle sets
  5. Cohomology of linear cycle sets
  6. The actions * and -< for a specific linear cycle set
  7. Case p odd or p=2 and (v,n)=/(1,2)
  8. Case p=2, v=1 and n=2
  9. Computing the cohomology in some cases
  10. First properties
  11. Cocycles when -<=0
  12. Cocycles in the general case
  13. Examples
  14. Case H=Zpn and I=Zpr with r<=n, if p is odd, and r<= min(2,n), if p=2
  15. Case H=Zpn and I=Zpr with r>n, if p is odd, and (n,r)=(1,2), if p=2
  16. ...and 5 more sections

Key Result

Proposition 1.4

Let $\triangleleft\colon I \times H\to I$ be the map defined by $y\triangleleft h \coloneqq h\blackdiamond (y - y\Yleft h)$. For each $y\in I$ and $h\in H$, we set $y^h\coloneqq h^{\times -1}\blackdiamond (y \triangleleft h)$. By GGV*Theorem 5.6 and Remark 5.12 we know that $(\iota, I\times_{\beta, for all $h,h',h"\in H$ and $y,y'\in I$.

Figures (1)

  • Figure 1: $\bigl(\widehat{C}_N^{**}(H,I),\partial_{\mathrm{h}},\partial_{\mathrm{v}}, D\bigr)$

Theorems & Definitions (118)

  • Definition 1.2
  • Proposition 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10: GGV*Theorem 6.1
  • Proposition 1.12: GGV*Proposition 7.9
  • Corollary 1.13: GGV*Corollary 7.10
  • ...and 108 more