Casimir operator

Let $\\mathfrak{g}$ be a semisimple Lie algebra, and let $(\\cdot,\\cdot)$ denote the Killing form. If $\\{g_{i}\\}$ is a basis of $\\mathfrak{g}$ , then there is a dual basis $\\{g^{i}\\}$ with respect to the Killing form, i.e., $(g_{i},g^{j})=\\delta_{ij}$ .Consider the element $\\Omega=\\sum g_{i}g^{i}$ of the universal enveloping algebra of $\\mathfrak{g}$ . This element, called the Casimir operator is central in the enveloping algebra, and thus commutes with the $\\mathfrak{g}$ action on any representation.

单词	CasimirOperator
释义	Casimir operator Let $\\mathfrak{g}$ be a semisimple Lie algebra, and let $(\\cdot,\\cdot)$ denote the Killing form. If $\\{g_{i}\\}$ is a basis of $\\mathfrak{g}$ , then there is a dual basis $\\{g^{i}\\}$ with respect to the Killing form, i.e., $(g_{i},g^{j})=\\delta_{ij}$ .Consider the element $\\Omega=\\sum g_{i}g^{i}$ of the universal enveloping algebra of $\\mathfrak{g}$ . This element, called the Casimir operator is central in the enveloping algebra, and thus commutes with the $\\mathfrak{g}$ action on any representation.
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