Cohomology of Linear Cycle Sets when the adjoint group is finite abelian

TL;DR

This work investigates the second cohomology group $\mathrm{H}^2_{\blackdiamond,\Yleft}(H,I)$ for linear cycle sets $H$ with a finite abelian additive structure and a finite abelian adjoint group, focusing on commutative adjoint. The authors develop explicit cocycle construction methods and establish that $\mathrm{H}^2_{\blackdiamond,\Yleft}(H,I)$ is computable as a subquotient of $I^{(s+1)n}$, where $n$ is the rank of the additive group of $H$ and $s$ is the rank of the adjoint group; they also describe how extensions of $H$ by a trivial cycle set $I$ correspond to these cocycles. A key contribution is a detailed construction of all cocycles via free magmas and a concrete diagrammatic complex, plus a procedure to determine the cohomology in terms of explicit data $(\gamma_i)$ and $(\mathfrak{f}_{ji})$. The paper concludes with a thorough treatment of the trivial-case, illustrative examples, and multiple applications to cyclic $p$-groups, including explicit decompositions of $\mathrm{H}^2_{\blackdiamond}(H,I)$ in several parameter regimes, thereby enabling classification of corresponding extensions. This framework advances the understanding of extensions and cohomology for braces/linear cycle sets and has potential implications for set-theoretic Yang-Baxter solutions and related algebraic structures.

Abstract

This paper analyzes the second cohomology group of a linear cycle set with coefficients in an abelian group I, for linear cycle sets with commutative adjoint operation, focusing on the finite abelian case. It aims to classify extensions of such structures through cohomological methods. Techniques are developed to systematically construct explicitly 2-cocycles. Finally, some illustrative examples are explored to validate the theoretical framework.

Theorems & Definitions (78)

• Remark 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Remark 2.5
• Remark 2.6