Split Casimir operator and universal formulation of the simple Lie algebras

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

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in defining (minimal fundamental) and adjoint representations. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T^{\otimes 2} in two cases, when T is the defining and the adjoint representation. In the case when T is the defining representation, these projectors and the split Casimir operator are used to explicitly write down invariant solutions of the Yang-Baxter equatio…

普适的带自旋 Casimir 方程及其解

Universal Spinning Casimir Equations and Their Solutions

本文利用 Harish-Chandra 径向分量映射，为任意维度的带自旋共形块构造了普适 Casimir 微分方程及“移位”算子求解方法，并成功将其推广至缺陷共形场论的两点和三点函数。

$\mathfrak{g}^{\otimes 4}$的一致结构

The uniform structure of $\mathfrak{g}^{\otimes 4}$

本文研究所有单李代数伴随表示四次幂的Casimir本征空间一致分解，给出各项维数与分裂Casimir本征值的Vogel型普适公式，并猜想其可推广至任意次幂。

SU(N) 的 $ad^{\otimes k}$ 上 Casimir 本征值关于 N 是线性的

The Casimir eigenvalues on $ad^{\otimes k}$ of SU(N) are linear on N

本文证明 SU(N) 伴随表示张量幂分解中出现的表示，其 Casimir 本征值关于 N 为线性的充要条件，并由此支持 Vogel 普适分解假设。

二次角动量运算的普适性：精确动力学李代数与Casimir障碍的作用

Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction

本文研究二次角动量运算的普适性，证明在固定粒子数子空间上实现普适控制，但抽象动力学李代数受Casimir障碍限制，并精确刻画了相应代数结构。

Casimir elements from the Brauer-Schur-Weyl duality

We consider Casimir elements for the orthogonal and symplectic Lie algebras constructed with the use of the Brauer algebra. We calculate the images of these elements under the Harish-Chandra isomorphism and thus show that they (together with the Pfaffian-type element in the even orthogonal case) are algebraically independent generators of the centers of the corresponding universal enveloping algebras.

Universal R-matrix of quantum affine gl(1,1)

The universal $R$-matrix of the quantum affine superalgebra associated to the Lie superalgebra $\mathfrak{gl}(1,1)$ is realized as the Casimir element of certain Hopf pairing, based on the explicit coproduct formula of all the Drinfeld loop generators.

Connections and loops intertwinning

On a finite graph, we prove that trace of holonomies determine an intertwining relation between merge-and-split generators on collections of geodesic loops ensembles and Casimir operators on unitary connections. By adding a deformation part to the generator on loops, this result is extended to the Casimir operator modified in order to be self adjoint with respect to Yang- Mills measure.

Casimir elements and Sugawara operators for Takiff algebras

For every simple Lie algebra $\mathfrak{g}$ we consider the associated Takiff algebra $\mathfrak{g}^{}_{\ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $\mathfrak{g}$. We use a matrix presentation of $\mathfrak{g}^{}_{\ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${\rm U}(\mathfrak{g}^{}_{\ell})$. A similar matrix presentation for the affine Kac--Moody algebra $\widehat{\mathfrak{g}}^{}_{\ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the cent…