Cohomological invariants of odd degree Jordan algebras

Categories of Jordan Structures and Graded Lie Algebras

In the paper we describe the subcategory of the category of Z-graded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the TKK construction. For instance, we prove that a Z-graded Lie algebra can be constructed from a Jordan pair if and only if it is generated by odd graded components and the second graded homology group is trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras.

Unified products for Jordan algebras. Applications

Given a Jordan algebra $A$ and a vector space $V$, we describe and classify all Jordan algebras containing $A$ as a subalgebra of codimension ${\rm dim}_k (V)$ in terms of a non-abelian cohomological type object ${\mathcal J}_{A} \, (V, \, A)$. Any such algebra is isomorphic to a newly introduced object called \emph{unified product} $A \, \natural \, V$. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology ${\rm H}^2_{\rm nab} \…

Jordan 代数与权模

Jordan algebras and weight modules

本文研究 Jordan 代数相关的 ${\mathfrak{sl}}_2(J)$ 模，引入 Weyl 模并证明自由 Jordan 代数情形下存在，猜想其构成最高权范畴，以解决上同调消失问题。

On a type of commutative algebras

We introduce some basic concepts for Jacobi-Jordan algebras such as: representations, crossed products or Frobenius/metabelian/co-flag objects. A new family of solutions for the quantum Yang-Baxter equation is constructed arising from any $3$-step nilpotent Jacobi-Jordan algebra. Crossed products are used to construct the classifying object for the extension problem in its global form. For a given Jacobi-Jordan algebra $A$ and a given vector space $V$ of dimension $\mathfrak{c}$, a global non-abelian cohomological object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is constructed: it classifies,…

On the double of the (restricted) super Jordan plane

We consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in works of N. Andruskiewitsch, I. Angiono, I. Heckenberger, and its restricted version in odd characteristic introduced by the same authors. We show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in their work. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of $\mathfrak{osp}(1|2)$.

Cohomological finite generation for restricted Lie superalgebras and finite supergroup schemes

We prove that the cohomology ring of a finite-dimensional restricted Lie superalgebra over a field of characteristic $p > 2$ is a finitely-generated algebra. Our proof makes essential use of the explicit projective resolution of the trivial module constructed by J. Peter May for any graded restricted Lie algebra. We then prove that the cohomological finite generation problem for finite supergroup schemes over fields of odd characteristic reduces to the existence of certain conjectured universal extension classes for the general linear supergroup $GL(m|n)$ that are similar to the universal exte…

Jacobi-Jordan 共形代数：基础、构造及相关结构

Jacobi-Jordan conformal algebras: Basics, Constructions and related structures

本文引入并研究了 Jacobi-Jordan conformal algebra 的概念，探讨了其共形模、表示、O-算子、共形导子、二次结构及统一积，并通过构造上同调对象为 C[delta]-split extending structures problem 提供了理论解答。

对称 2x2 矩阵的 Jordan 代数的坐标化定理

A Coordinatization Theorem for the Jordan algebra of symmetric 2x2 matrices

本文将 Jacobson 坐标化定理扩展至 n=2 的情形，证明了包含 $H_2(F)$ 且具相同单位元的酉 Jordan 代数 J 可表示为 $H_2(F)\otimes A_0+k\otimes A_1$ 的形式，并给出了具体的乘法规则。

同构奇罗塔-巴克斯特算子在修改后的威特型李超代数上的分类

Classification of Homogeneous Odd Rota--Baxter Operators on a Modified Witt-Type Lie Superalgebra

本文研究了修改后的威特型李超代数上零权重同构奇罗塔-巴克斯特算子的分类，揭示了其结构约束及诱导的超前李代数结构。

The Allison-Faulkner construction of $E_8$

We show that the Tits index $E_8^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. We construct two cohomological invariants, in degrees 6 and 8, of the Tits construction and the more symmetric Allison-Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

初等阿贝尔 $p^2$ 次扩张的上同调核

Cohomological Kernels of Elementary Abelian Degree $p^2$ Extensions

本文研究初等阿贝尔 $p^2$ 次扩张的上同调核，在 $p=3$ 时给出六项正合序列，并描述商群与不可分解除环的联系。

环上的色代数

Colour algebras over rings

本文提出在具有1/2∈R的环上定义色代数，并通过非退化三元Hermitian形式构造色代数，探讨其结构及与八元数代数的关系。