可裂实量子群的正Casimir特征标和中心特征标

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…

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…

Positive Representations with Zero Casimirs

In this paper, we construct a new family of generalization of the positive representations of split-real quantum groups based on the degeneration of the Casimir operators acting as zero on some Hilbert spaces. It is motivated by a new observation arising from modifying the representation in the simplest case of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ compatible with Faddeev's modular double, while having a surprising tensor product decomposition. For higher rank, the representations are obtained by the polarization of Chevalley generators of $\mathcal{U}_q(\mathfrak{g})$ in a new realizat…

$\Uq$的量子Casimir算符及其本征值

The quantum Casimir operators of $\Uq$ and their eigenvalues

我们证明了Bracken，Gould和Zhang的早期工作中构造的量子线性群的量子Casimir算符加上一个额外的中心元生成$\Uq$的整个中心.作为证明的副产品，我们得到了这些量子Casimir算子的本征值的有趣的新公式，这些公式用经典一般线性代数的一类有限维不可约表示的特征标表示。

曲线Casimir算子与BGG机器

Curved Casimir Operators and the BGG Machinery

我们证明了作用在广义旗流形上齐次向量丛截面上的Casimir算子自然地推广到任意抛物几何上的不变微分算子。我们研究了所得不变算子的一些性质，并计算了它们在各种特殊类型的自然丛上的作用。作为第一个应用，我们给出了一个非常一般的构造抛物几何的分裂算子。然后讨论了值在牵引丛中的微分形式上的曲Casimir算子，这与BGG序列的机制有很好的关系。这也给出了由BGG序列提供的主列表示（中的光滑向量空间）的有限维表示的分辨率的很好的解释。

高阶Casimir元与钩分区在量子群B、C和D类型中的应用

Higher-order Casimir elements and hook partitions for quantum groups of types B, C and D

本文研究了B、C和D类型量子群的高阶Casimir元，通过钩分区与不可约特征标的关系，确定了中心的生成集，并揭示了Casimir元与钩分区之间的稳定性现象。