2015/11/27 by Yanyong Hong, Hong, Yanyong · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1511.08571
Let A be a left-symmetric (resp. Novikov) algebra, E be a vector space containing A as a subspace and V be a complement of A in E.The extending structures problem which asks for the classification of all left-symmetric (resp. Novikov) algebra structures on E such that A is a subalgebra of E 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 E such that A is a subalgebra of E if and only if E is isomorphic to a unified product of A and V. Two cohomological type objects HA2(V,A) and H2(V,A) are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension A⊂ E of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of A in E. Several special examples are provided in details.