2007/08/11 by Chengming Bai · 7 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Sphingolipid Metabolism and Signaling #math-ph #math.MP #math.QA #msc:17B #msc:53C #msc:81R
paper · pdf · doi:10.1142/s0219199708002752
published as Communications in Contemporary Mathematics 10 (2008) 221-260 · 44 pages
arxiv created 2007/08/11 · openalex publication_date 2008/04/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.