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, 74:32 | Cite as

Extending Structures and Classifying Complements for Left-Symmetric Algebras

  • Yanyong HongEmail author
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Abstract

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 up to an isomorphism which stabilizes A such that A is a subalgebra of E is studied. In this paper, the definition of the unified product for 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. A cohomological type object \(\mathcal {H}_A^2(V,A)\) is constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension \(A\subset E\) of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of A in E. Several examples are provided in detail.

Keywords

Left-symmetric algebra Novikov algebra the extending structures problem matched pair complements quasicentroid 

Mathematics Subject Classification

17A30 17D25 17A60 18G60 
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Notes

Acknowledgements

This work was done during the author’s visit to TongJi University. He would like to thank Professor Yucai Su for the hospitality, comments on this work and valuable discussions on Lie algebras. Moreover, this work was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LQ16A010011) and the National Natural Science Foundation of China (No. 11501515, 11871421).

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Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouPeople’s Republic of China

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