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Extending structures for anti-dendriform algebras and anti-dendriform bialgebras

Qinxiu Sun, Xingyu Zeng

TL;DR

This work develops an extending-structures framework for anti-dendriform algebras, solving local and global extension problems via unified and crossed products and establishing Wells-type sequences for automorphism data. It then builds a bialgebra theory in this setting through anti-dendriform D-bialgebras and their double-construction realizations with commutative Connes cocycles, linking them to matched pairs of associative algebras and to the anti-dendriform Yang-Baxter equation (AD-YBE). O-operators are introduced as a mechanism to generate skew-symmetric AD-YBE solutions, connecting coboundary anti-dendriform D-bialgebras to operator theory. Overall, the paper reveals deep interactions among extending structures, automorphism inducibility, matched-pair formalisms, and operator-based constructions in the anti-dendriform landscape, with implications for non-abelian extension theory and integrable systems.

Abstract

In this paper, we first explore the extending structures problem by the unified product for anti-dendriform algebras. In particular,the crossed product and non-abelian extension are studied. Furthermore, we explore the inducibility problem of pairs of automorphisms associated with a non-abelian extension of anti-dendriform algebras, and derive the fundamental sequences of Wells. Then we introduce the bicrossed products and matched pairs of anti-dendriform algebras to solve the factorization problem. Finally, we introduce the notion of anti-dendriform D-bialgebras as the bialgebra structures corresponding to double construction of associative algebras with respect to the commutative Cone cocycles. Both of them are interpreted in terms of certain matched pairs of associative algebras as well as the compatible anti-dendriform algebras. The study of coboundary cases leads to the introduction of the AD-YBE, whose skew-symmetric solutions give coboundary anti-dendriform D-bialgebras. The notion of O-operators of anti-dendriform algebras is introduced to construct skew-symmetric solutions of the AD-YBE. We also characterize the relationship between the skew-symmetric solutions of AD-YBE and O-operators.

Extending structures for anti-dendriform algebras and anti-dendriform bialgebras

TL;DR

This work develops an extending-structures framework for anti-dendriform algebras, solving local and global extension problems via unified and crossed products and establishing Wells-type sequences for automorphism data. It then builds a bialgebra theory in this setting through anti-dendriform D-bialgebras and their double-construction realizations with commutative Connes cocycles, linking them to matched pairs of associative algebras and to the anti-dendriform Yang-Baxter equation (AD-YBE). O-operators are introduced as a mechanism to generate skew-symmetric AD-YBE solutions, connecting coboundary anti-dendriform D-bialgebras to operator theory. Overall, the paper reveals deep interactions among extending structures, automorphism inducibility, matched-pair formalisms, and operator-based constructions in the anti-dendriform landscape, with implications for non-abelian extension theory and integrable systems.

Abstract

In this paper, we first explore the extending structures problem by the unified product for anti-dendriform algebras. In particular,the crossed product and non-abelian extension are studied. Furthermore, we explore the inducibility problem of pairs of automorphisms associated with a non-abelian extension of anti-dendriform algebras, and derive the fundamental sequences of Wells. Then we introduce the bicrossed products and matched pairs of anti-dendriform algebras to solve the factorization problem. Finally, we introduce the notion of anti-dendriform D-bialgebras as the bialgebra structures corresponding to double construction of associative algebras with respect to the commutative Cone cocycles. Both of them are interpreted in terms of certain matched pairs of associative algebras as well as the compatible anti-dendriform algebras. The study of coboundary cases leads to the introduction of the AD-YBE, whose skew-symmetric solutions give coboundary anti-dendriform D-bialgebras. The notion of O-operators of anti-dendriform algebras is introduced to construct skew-symmetric solutions of the AD-YBE. We also characterize the relationship between the skew-symmetric solutions of AD-YBE and O-operators.
Paper Structure (10 sections, 33 theorems, 183 equations)

This paper contains 10 sections, 33 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Representation of anti-dendriform algebras
  3. Extending structures and the ES-problem for anti-dendriform algebras
  4. Crossed products, the GE-problem and non-abelian extensions
  5. Inducibility of pairs of anti-dendriform algebra automorphisms
  6. Wells exact sequences for anti-dendriform algebras
  7. Bicrossed product and matched pairs
  8. Anti-dendriform D-bialgebras and Yang-Baxter equations
  9. Anti-dendriform D-bialgebras
  10. Coboundary anti-dendriform D-bialgebra

Key Result

Proposition 2.2

Let $(A,\succ,\prec)$ be an anti-dendriform algebra and $V$ a vector space. Assume that $l_{\succ}, r_{\succ},l_{\prec},r_{\prec} : A \longrightarrow \hbox{End} (V)$ are four linear maps. Then $(V,l_{\succ}, r_{\succ},l_{\prec},r_{\prec})$ is a representation of $(A,\succ,\prec)$ if and only if $(A\ for all $x,y\in A$ and $a,b\in V$. In this case, the anti-dendriform algebra $(A\oplus V,\succeq,\p

Theorems & Definitions (78)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 68 more