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 center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $\mathfrak{g}^{}_{\ell}$.

中文翻译：

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Takiff 代数的 Casimir 元素和 Sugawara 算子

对于每个简单的李代数 $\mathfrak{g}$ 我们考虑相关的 Takiff 代数 $\mathfrak{g}^{}_{\ell}$ 定义为截断多项式当前李代数，系数在 $\mathfrak{g} $. 我们使用 $\mathfrak{g}^{}_{\ell}$ 的矩阵表示来给出通用包络代数 ${\rm U}(\mathfrak{g} ^{}_{\ell})$。仿射 Kac 的类似矩阵表示——穆迪代数 $\widehat{\mathfrak{g}}^{}_{\ell}$ 然后用于证明 Feigin--Frenkel 定理的类似物对应的临界水平的仿射顶点代数。证明依赖于李代数 $\mathfrak{g}^{}_{\ell}$ 的完整 Segal-Sugawara 向量集的显式构造。