No abstract

We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples of Lie algebras to the context of Rota-Baxter Lie algebras. The coboundary case leads to the introduction of the admissible classical Yang-Baxter equation (CYBE) in Rota-Baxter Lie algebras, for which the antisymmetric solutions give rise to Rota-Baxter Lie bialgebras. The notions of O-operators on Rota-Baxter Lie algebras and Rota-Baxter pre-Lie algebras are introduced to produce antisymmetric solutions of the admissible CYBE. Furthermore, extending the well-known property that a Rota-Baxter Lie algebra of weight zero induces a pre-Lie algebra, the Rota-Baxter Lie bialgebra of weight zero induces a bialgebra structure of independent interest, namely the special L-dendriform bialgebra, which is equivalent to a Lie group with a left-invariant flat pseudo-metric in geometry. This induction is also characterized as the inductions between the corresponding Manin triples and matched pairs. Finally, antisymmetric solutions of the admissible CYBE in a Rota-Baxter Lie algebra of weight zero give special L-dendriform bialgebras and in particular, both Rota-Baxter algebras of weight zero and Rota-Baxter pre-Lie algebras of weight zero can be used to construct special L-dendriform bialgebras.

Jacobi algebras, as the algebraic counterparts of Jacobi manifolds, are exactly the unital relative Poisson algebras. The direct approach of constructing Frobenius Jacobi algebras in terms of Manin triples is not available due to the existence of the units, and hence alternatively we replace it by studying Manin triples of relative Poisson algebras. Such structures are equivalent to certain bialgebra structures, namely, relative Poisson bialgebras. The study of coboundary cases leads to the introduction of the relative Poisson Yang-Baxter equation (RPYBE). Antisymmetric solutions of the RPYBE give coboundary relative Poisson bialgebras. The notions of O-operators of relative Poisson algebras and relative pre-Poisson algebras are introduced to give antisymmetric solutions of the RPYBE. A direct application is that relative Poisson bialgebras can be used to construct Frobenius Jacobi algebras, and in particular, there is a construction of Frobenius Jacobi algebras from relative pre-Poisson algebras. Contents 1. Introduction 1 1.1. Generalizations of Poisson algebras 1 1.2. Frobenius Jacobi algebras and relative Poisson bialgebras 3 1.3. Layout of the paper 4 2. Representations and matched pairs of relative Poisson algebras 4 2.1. Representations of relative Poisson algebras and Jacobi algebras 5 2.2. Matched pairs of relative Poisson algebras 9 3. Relative Poisson bialgebras 3.1. Frobenius relative Poisson algebras and Manin triples of relative Poisson algebras 3.2. Relative Poisson bialgebras 4. Coboundary relative Poisson bialgebras 4.1. Coboundary relative Poisson bialgebras 4.2. O-operators of relative Poisson algebras 4.3. Relative pre-Poisson algebras 5. Relative Poisson bialgebras and Frobenius Jacobi algebras References

We introduce the notion of anti-dendriform algebras as a new approach of splitting the associativity. They are characterized as the algebras with two operations whose sum is associative and the negative left and right multiplication operators compose the bimodules of the sum associative algebras, justifying the notion due to the comparison with the corresponding characterization of dendriform algebras. The notions of anti-O-operators and anti-Rota-Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of q-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations. Contents 1. Introduction 1 2. Anti-dendriform algebras 3 2.1. Anti-dendriform algebras 4 2.2. Anti-O-operators and anti-Rota-Baxter operators 9 2.3. Commutative Connes cocycles 13 3. Correspondences between some subclasses of dendriform and anti-dendriform algebras 16 3.1. Correspondences between some subclasses of dendriform and anti-dendriform algebras 16 3.2. More correspondences and their relationships 20 4. General framework: analogues of anti-dendriform algebras and a new splitting of operations 23 References 24