Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2007
- DOI:
- arXiv:
- arXiv:0708.1551
- Bibcode:
- 2007arXiv0708.1551B
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematical Physics;
- 17B;
- 53C;
- 81R
- E-Print:
- 44 pages