On Jordan algebras and some unification results

The Tits--Kantor--Koecher construction for Jordan dialgebras

We study a noncommutative generalization of Jordan algebras called Jordan dialgebras. These are algebras that satisfy the identities $[x_1 x_2]x_3= 0$, $(x_1^2,x_2,x_3)=2(x_1,x_2,x_1x_3)$, $x_1(x_1^2 x_2)=x_1^2(x_1 x_2)$; they are related with Jordan algebras in the same way as Leibniz algebras are related to Lie algebras. We present an analogue of the Tits---Kantor---Koecher construction for Jordan dialgebras that provides an embedding of such an algebra into Leibniz algebra.

Jordan 三代数与 post-Jordan 代数

Jordan Trialgebras and Post-Jordan Algebras

本文通过计算三结合与三树状代数中 Jordan 型积的多重线性恒等式极小生成元集，引入 Jordan 三代数与 post-Jordan 代数，并借助计算机代数验证低元数恒等式关系，提出下一独立恒等式元数为 8 的猜想。

C*-代数上多重线性映射的Jordan范数与Grothendieck不等式

Jordan norms for multilinear maps on C*-algebras and Grothendieck's inequalities

该研究提出C*-代数上多重线性映射完全有界范数的推广形式Jordan范数，通过映射分解定义，并证明C*-代数对上的有界双线性型均为Jordan有界，且其Jordan范数不超过原范数的两倍。

Jordan-Hölder theorem for finite dimensional Hopf algebras

We show that a Jordan-Hölder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus' butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Hölder theorem holds as well for lower and upper composition series, even though the f…

Novikov-Jordan algebras

Algebras with identity $(a\star b)\star (c\star d) -(a\star d)\star(c\star b)$ $=(a,b,c)\star d-(a,d,c)\star b$ are studied. Novikov algebras under Jordan multiplication and Leibniz dual algebras satisfy this identity. If algebra with such identity has unit, then it is associative and commutative.

关于Kashuba和Mathieu的自由Jordan代数猜想

On the conjecture of Kashuba and Mathieu about free Jordan algebras

本文通过总结自由Jordan代数的新计算数据，解释了为何在大量正面证据下，Kashuba和Mathieu关于Lie代数同调消没的猜想并不成立。

Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan Schwinger map

The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well known Hopf algebraic structure of the Lie algebras, through a realization of Lie algebras using the parabosonic (and parafermionic) extension of the Jordan Schwinger map. The differences between the Hopf algebraic and the graded Hopf superalgebraic structure on the parabosonic algebra are discussed.