标题： 塞加尔- Sugawara 向量对于类型$G_2$的李代数 显示英文标题

标题： Segal-Sugawara vectors for the Lie algebra of type $G_2$

摘要： 与类型$G_2$的简单李代数$\mathfrak{g}$相关的 Segal-Sugawara 向量的显式公式是通过计算机辅助计算找到的。 这导致了对描述相应仿射顶点代数在临界水平下的中心的 Feigin-Frenkel 定理的直接证明。 作为应用，我们通过提供$U(\mathfrak{g})$的极大交换子代数生成元的公式，给出了 Vinberg 量化问题的显式解。 我们还计算了与$\mathfrak{g}$相关的 Gaudin 模型中 Bethe 向量上的哈密顿量的本征值。 显示英文摘要

摘要： Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra $\mathfrak{g}$ of type $G_2$ are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. As an application, we give an explicit solution of Vinberg's quantization problem by providing formulas for generators of maximal commutative subalgebras of $U(\mathfrak{g})$. We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with $\mathfrak{g}$.