Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters

The split Casimir operator and solutions of the Yang-Baxter equation for the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters

We find the characteristic identities for the split Casimir operator in the defining and adjoint representations of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras. These identities are used to build the projectors onto invariant subspaces of the representation $T^{\otimes 2}$ of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the cases when $T$ is the defining and adjoint representations. For defining representations, the $osp(M|N)$- and $s\ell(M|N)$-invariant solutions of the Yang-Baxter equation are expressed as rational functions of the split Casimir operator. For the adjoint repres…

$\mathsf{ad}$表示的三次方中简单李代数的分裂Casimir算子与Vogel参数

Split Casimir operator for simple Lie algebras in the cube of $\mathsf{ad}$-representation and Vogel parameters

本文研究简单李代数伴随表示三次方中的3-分裂Casimir算子，通过特征恒等式构造不变子空间投影算子，给出高次幂迹的普适表达式并计算子表示维数，结果与Vogel参数描述一致。

Projectors on invariant subspaces of representations $\operatorname{ad}^{\otimes 2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel parametrization

Explicit formulae for the projectors onto invariant subspaces of the $\operatorname{ad}^{\otimes 2}$ representation of the Lie algebras $so(N)$ and $sp(2r)$ have been found by means of the split Casimir operator. These projectors have also been considered from the viewpoint of the universal complex simple Lie algebra description by using the Vogel parametrisation.

在旋量表示中推导李代数so(2r)的分裂Casimir算子特征恒等式、颜色因子和Yang-Baxter方程

Split Casimir Operator of the Lie Algebra so(2r) in Spinor Representations, Colour Factors and Yang-Baxter Equation

本文研究了李代数so(2r)在旋量表示中的分裂Casimir算子特征恒等式，推导了颜色因子表达式，并给出了Yang-Baxter方程的新解形式。

Yang-Baxter 能否蕴含 Lie 代数？

Can Yang-Baxter imply Lie algebra?

本文回顾 Vogel 对 Vassiliev 展开系数的三参数多项式参数化，指出其构造暗示无限维 Lie 代数或可扩展 Chern-Simons 理论，并提供从纽结不变量反推单 Lie 代数分类的逆向公理化路径。

Vogel universality和 beyond

Vogel universality and beyond

本文通过Vogel参数为所有简单李代数（除e_8外）推导了Casimir算子在表示分解中的特征恒等式和投影算子的通用公式。