Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
| dc.creator | Bai, Chengming | |
| dc.date | 2007-08-11 | |
| dc.date.accessioned | 2026-07-07T09:34:09Z | |
| dc.date.available | 2026-07-07T09:34:09Z | |
| dc.description | 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 "${\cal O}$-operator", whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an ${\cal O}$-operator. Thus a method to construct symmetric solutions of the $S$-equation (hence parakähler Lie algebras) from ${\cal O}$-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. | |
| dc.description | 44 pages | |
| dc.identifier | https://arxiv.org/abs/0708.1551 | |
| dc.identifier | http://arxiv.org/abs/0708.1551 | |
| dc.identifier | Communications in Contemporary Mathematics 10 (2008) 221-260 | |
| dc.identifier | doi:10.1142/S0219199708002752 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/159385 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Mathematical Physics | |
| dc.subject | 17B, 53C, 81R | |
| dc.title | Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation | |
| dc.type | text |