Color Lie Bialgebras: Big Bracket, Cohomology and Deformations

2017 ◽  
pp. 69-115 ◽  
Author(s):  
Benedikt Hurle ◽  
Abdenacer Makhlouf
Keyword(s):  
Lie Bialgebras

scholarly journals Contractions of Poisson-Lie groups, Lie bialgebras and quantum deformations

1997 ◽  
Vol 40 (1) ◽  
pp. 261-271
Author(s):  
Angel Ballesteros ◽  
Mariano del Olmo
Keyword(s):  
Lie Groups ◽  
Lie Bialgebras ◽  
Quantum Deformations ◽  
Poisson Lie Groups

scholarly journals Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

2009 ◽  
Vol 15 (1) ◽  
pp. 1-59 ◽  
Author(s):  
Benjamin Enriquez ◽  
Nathan Geer
Keyword(s):  
Lie Bialgebras ◽  
Quantization Functors

scholarly journals Quantum BV-manifolds and quasi-Lie bialgebras

2010 ◽  
Vol 28 (2) ◽  
pp. 194-204
Author(s):  
Johan Granåker
Keyword(s):  
Lie Bialgebras

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

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.

scholarly journals Hamiltonian type Lie bialgebras

2007 ◽  
Vol 50 (9) ◽  
pp. 1267-1279 ◽  
Author(s):  
Bin Xin ◽  
Guang-ai Song ◽  
Yu-cai Su
Keyword(s):  
Lie Bialgebras
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