Liangyun Chen

We study noncommutative Novikov–Poisson algebras, obtained by allowing the associative product of a Novikov–Poisson algebra to be noncommutative. For differential constructions, we establish a radical–annihilator principle, prove without using classification tables that every derivation of a three-dimensional complex noncommutative associative algebra yields such a structure. We also determine all two-dimensional complex examples with noncommutative associative product. We develop representations and isolate a coherent subclass, whose commutator construction produces special Gel’fand–Dorfman algebras. We then formulate coherent bialgebras and characterize them by matched pairs and standard Manin triples. Skew-symmetric solutions of the simultaneous associative and Novikov Yang–Baxter equations yield coboundary bialgebras and admit an O-operator interpretation. Finally, we relate factorizable coboundary bialgebras to quadratic Rota–Baxter coherent algebras and obtain a sharp low-dimensional obstruction.

In this paper, we use the higher derived bracket to give the controlling algebra of pre-LieDer pairs. We give the cohomology of pre-LieDer pairs by using the twist L_{\infty} -algebra of this controlling algebra. In particular, we define the cohomology of regular pre-LieDer pairs. We study infinitesimal deformations of pre-LieDer pairs, which are characterized by the second cohomology group of pre-LieDer pairs. We also define the cohomology of regular pre-LieDer pairs with coefficients in an arbitrary representation and using the second cohomology group to classify abelian extensions of regular pre-LieDer pairs.

In this paper, we first describe the cohomology theory of Lie supertriple systems by using the cohomology theory of the associated Leibniz superalgebras. Then we focus on Lie supertriple systems with superderivations, called LSTSDer pairs. We introduce the notion of representations of LSTSDer pairs and investigate their corresponding cohomology theory. We also construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTSDer pairs. Moreover, we consider the relationship between a LSTSDer pair and the associated LeibSDer pair. Furthermore, we develop the 1-parameter formal deformation theory of LSTSDer pairs and prove that it is governed by the cohomology groups. At last, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups.