从立方范数对到 $G_2$- 和 $F_4$- 分级的群和李代数

三次范数对与埃尔米特三次范数结构

Cubic norm pairs and hermitian cubic norm structures

该研究将三次范数结构推广为三次范数对、将埃尔米特三次范数结构拓展至任意含单位元交换环，还为对应斜维1可结构代数构造了李代数及其自同构群。

三次代数的理想与Weyl代数的不变环

Ideals of cubic algebras and an invariant ring of the Weyl algebra

本文对一般三次三维Artin Schelter正则代数的自反分次右理想进行了分类，直至同构和移位。我们还确定了这些理想的可能的希尔伯特函数。这些结果是通过使用类似的方法获得的二次阿廷-Schelter代数。特别是我们的结果适用于包络代数的Heisenberg-李代数，从中我们推导出一个分类的右理想的不变环的第一Weyl代数。

The classification of 3-dimensional noetherian cubic Calabi-Yau algebras

It is known that every 3-dimensional noetherian Calabi-Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S. P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi-Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi-Yau algebras. As an application, we show that if $S$ is a 3-dimensional noetherian cubic Calabi-Yau algebra and $σ$ is a graded algebra automorphism of $S$, then the…

三维立方AS-正则代数的分类：其点方案不为积分的情况

Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral

本文研究三维立方AS-正则代数的分类，针对三种配置，列出代数关系并分类其同态等价，同时给出显式超势能并验证其AS-正则性。

Graded Skew Clifford Algebras that are Twists of Graded Clifford Algebras

We prove that if $A$ is a regular graded skew Clifford algebra and is a twist of a regular graded Clifford algebra $B$ by an automorphism, then the subalgebra of $A$ generated by a certain normalizing sequence of homogeneous degree-two elements is a twist of a polynomial ring by an automorphism, and is a skew polynomial ring. We also present an example that demonstrates that this can fail when $A$ is not a twist of $B$.

Structurable algebras of skew-dimension one and hermitian cubic norm structures

We study structurable algebras of skew-dimension one. We present two different equivalent constructions for such algebras: one in terms of non-linear isotopies of cubic norm structures, and one in terms of hermitian cubic norm structures. After this work was essentially finished, we became aware of the fact that both descriptions already occur in (somewhat hidden places in) the literature. Nevertheless, we prove some facts that had not been noticed before: (1) We show that every form of a matrix structurable algebra can be described by our constructions; (2) We give explicit formulas for the n…

Differential graded algebra over quotients of skew polynomial rings by normal elements

Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements. As a consequence, we give a presentation of the Ext algebra when the elem…

平面三次曲线的几何代数的AS-正则性

AS-regularity of geometric algebras of plane cubic curves

本文分类了三维二次AS-正则代数的超势，给出Type EC的AS-正则判据，并证明任意此类代数与某Calabi-Yau AS-正则代数分次Morita等价，从而简化非交换射影概形的研究。

具有平凡臭氧群的一族代数

A family of algebras with trivial ozone group

研究包含与结点三次曲线相关的二次Artin-Schelter正则代数的一族Calabi-Yau代数，证明其臭氧群平凡，且该族中分次代数在非平凡分次自同构群作用下不变环非Artin-Schelter正则。

Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras

We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$ over arbitrary field, which gives rise to an associative hull $\mathbf A$, a Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$, $\mathbf K$. One has a natural filtration for $\mathbf A$ which associated graded algebra has a structure of a Poisson superalgebra and $\mathrm{gr} \mathbf A\cong\mathbf P$, also $\mathbf P$ admits an algebraic quantization. The Lie superalgebra $\mathbf Q$ is finely $\mathbb Z^3$-graded by multidegree in the generators, $\mathbf A$, $\mathbf P$ are $\mathbb Z^3$-gr…

Induced quotient group gradings of epsilon-strongly graded rings

Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well-known that if $S$ is strongly $G$-graded, then the induced $G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings was recently introduced by Nystedt, Öinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and suffic…

调和三次齐次多项式：其Hessian范数平方是欧几里得二次型的倍数

Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form

本文研究调和三次齐次多项式，其Hessian范数平方为欧几里得二次型的非零倍数，在所有维度构造解并在至多4维分类，提供线性共形不等价的判定技术。