2021/08/12 by Yanyong Hong, Hong, Yanyong
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2108.05586
openalex publication_date 2021/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (\mathfrakg, [⋅,⋅], δ_\mathfrakg) be a fixed Lie bialgebra, E be a vector space containing \mathfrakg as a subspace and V be a complement of \mathfrakg in E. A natural problem is that how to classify all Lie bialgebraic structures on E such that (\mathfrakg, [⋅,⋅], δ_\mathfrakg) is a Lie sub-bialgebra up to an isomorphism of Lie bialgebras whose restriction on \mathfrakg is the identity map. This problem is called the extending structures problem. In this paper, we introduce a general co-product on E, called the unified co-product of (\mathfrakg,δ_\mathfrakg) by V. With this unified co-product and the unified product of (\mathfrakg, [⋅,⋅]) by V developed in \citeAM1, the unified bi-product of (\mathfrakg, [⋅,⋅], δ_\mathfrakg) by V is introduced. Moreover, we show that any E in the extending structures problem is isomorphic to a unified bi-product of (\mathfrakg, [⋅,⋅], δ_\mathfrakg) by V. Then an object HBI_\mathfrakg2(V,\mathfrakg) is constructed to classify all E in the extending structures problem. Moreover, several special unified bi-products are also introduced. In particular, the unified bi-products when dim V=1 are investigated in detail.