The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of O -operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a preJordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of O -operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

The q-discrete two-dimensional Toda lattice equation with self-consistent sources is presented through the source generalization procedure. In addition, the Grammtype determinant solutions of the system are obtained. Besides, a bilinear Bäcklund transformation (BT) for the system is given.

In this paper, we introduce the notions of Hom-Balinskii–Novikov and Hom-Novikov superalgebras, in which the defining identities are twisted by homomorphisms. Then we construct some infinite-dimensional Hom-Lie superalgebras by the affinizations of the above two Hom-superalgebras. Moreover, we apply the bilinear forms with some invariance conditions on the Hom-Balinskii–Novikov and Hom-Novikov superalgebras to construct central extensions of the infinite-dimensional Hom-Lie superalgebras.

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In this paper, we introduce a notion of J-dendriform algebra with two operations as a Jordan algebraic analogue of a dendriform algebra such that the anticommutator of the sum of the two operations is a Jordan algebra. A dendriform algebra is a J-dendriform algebra. Moreover, J-dendriform algebras fit into a commutative diagram which extends the relationships among associative, Lie, and Jordan algebras. Their relations with some structures such as Rota-Baxter operators, classical Yang-Baxter equation, and bilinear forms are given.

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In this paper, we study the operator forms of the classical Yang-Baxter equation (CYBE) in Lie superalgebras. We introduce the notion of super O-operators and give their relationship with the (standard) tensor form. There are close relationships between the CYBE in Lie superalgebras and left-symmetric superalgebras which can be interpreted through the super O-operators. In particular, there are natural and explicit solutions of the CYBE in Lie superalgebras constructed from left-symmetric superalgebras.

We study a twisted generalization of linear Poisson brackets of hydrodynamic type, in which the Lie bracket is replaced by a Hom-Lie bracket. With some natural additional conditions, such structures correspond to the Hom-Novikov algebras introduced by Yau, which is the twisted version of Balinsky-Novikov's approach of constructing a Lie algebra from a Novikov algebra. Certain central extensions of this twisted generalization of linear Poisson brackets of hydrodynamic type are obtained from the bilinear forms on their corresponding Hom-Novikov algebras satisfying some invariance conditions. Finally, we give some examples of the infinite-dimensional Hom-Lie algebras constructed from the Hom-Novikov algebras. In particular, there is an interesting twisted generalization of the Virasoro algebra.