Extending Structures for Dendriform Algebras

TL;DR

This paper develops a comprehensive framework for extending and factorizing dendriform algebras, introducing new structures like unified products, matched pairs, and deformation maps, with theoretical solutions and practical applications.

Contribution

It introduces extending datums, unified products, and deformation maps for dendriform algebras, providing new methods to solve extension and factorization problems.

Findings

Defined extending datums and unified products for dendriform algebras.

Solved the extending structure problem theoretically.

Introduced matched pairs, bicrossed products, and deformation maps.

Abstract

In this paper, we devote to extending structures for dendriform algebras. First, we define extending datums and unified products of dendriform algebras, and theoretically solve the extending structure problem. As an application, we consider flag datums as a special case of extending structures, and give an example of the extending structure problem. Second, we introduce matched pairs and bicrossed products of dendriform algebras and theoretically solve the factorization problem for dendriform algebras. Moreover, we also introduce cocycle semidirect products and nonabelian semidirect products as special cases of unified products. Finally, we define the deformation map on a dendriform extending structure (more general case), not necessary a matched pair, which is more practical in the classifying complements problem.