On the algebraic variety of Hom-Lie algebras

The variety of $2$-dimensional algebras over an algebraically closed field

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the variety under consideration. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible, and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

Generators and defining relations for ring of differential operators on smooth affine algebraic variety in prime characteristic

For ring of differential operators on smooth affine algebraic variety over perfect field of prime characteristic a set of algebra generators and a set of defining relations are found explicitly.

反结合代数的代数与几何分类

The algebraic and geometric classification of antiassociative algebras

本文对复数域上4维与5维反结合代数给出完整的代数与几何分类，确定了两类簇的维数、不可约分支数及刚性代数个数。

Hopf代数齐性空间的有理解释：Abe-Kanno定理

Hopf algebraic homogeneous spaces interpreted rationally: the Abe-Kanno theorem

将Abe-Kanno定理推广到Hopf代数，建立子Hopf代数与有理函数域不变子域之间的对应，并应用于剩余有限维Hopf代数的性质研究。

Homologies of Algebraic Structures via Braidings and Quantum Shuffles

In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their super- and co-versions. L…