Casimir operator

In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.

The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931.

The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order; their definition is given last.

Suppose that ${\displaystyle {\mathfrak {g}}}$ is an ${\displaystyle n}$-dimensional semisimple Lie algebra. Let B be a nondegenerate bilinear form on ${\displaystyle {\mathfrak {g}}}$ that is invariant under the adjoint action of ${\displaystyle {\mathfrak {g}}}$ on itself, meaning that ${\displaystyle B(ad_{X}Y,Z)+B(Y,ad_{X}Z)=0}$ for all X,Y,Z in G. (The most typical choice of B is the Killing form.) Let

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