Extending structures and classifying complements for left-symmetric algebras
Yanyong Hong

TL;DR
This paper introduces a unified framework for classifying extensions and complements of left-symmetric and Novikov algebras using cohomological methods, providing a theoretical foundation and concrete examples.
Contribution
It defines the unified product for these algebras and constructs cohomological objects to classify all possible algebra structures and complements.
Findings
Existence of a left-symmetric algebra structure on E characterized by a unified product.
Development of cohomological objects for classification of extensions and complements.
Application of the theory to several detailed examples.
Abstract
Let be a left-symmetric (resp. Novikov) algebra, be a vector space containing as a subspace and be a complement of in .The extending structures problem which asks for the classification of all left-symmetric (resp. Novikov) algebra structures on such that is a subalgebra of is studied. In this paper, the definition of the unified product of left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on such that is a subalgebra of if and only if is isomorphic to a unified product of and . Two cohomological type objects and are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension of left-symmetric (resp. Novikov) algebras, another cohomological…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
