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.

More generally, Casimir elements can be used to refer to any element of the center of the universal enveloping algebra. The algebra of these elements is known to be isomorphic to a polynomial algebra through the Harish-Chandra isomorphism.

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.

Suppose that ${\displaystyle {\mathfrak {g}}}$ is an ${\displaystyle n}$-dimensional 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 ${\displaystyle B(\operatorname {ad} _{X}Y,Z)+B(Y,\operatorname {ad} _{X}Z)=0}$ for all X, Y, Z in ${\displaystyle {\mathfrak {g}}}$. (The most typical choice of B is the Killing form if ${\displaystyle {\mathfrak {g}}}$ is semisimple.) Let

be any basis of ${\displaystyle {\mathfrak {g}}}$, and

be the dual basis of ${\displaystyle {\mathfrak {g}}}$ with respect to B. The Casimir element ${\displaystyle \Omega }$ for B is the element of the universal enveloping algebra ${\displaystyle U({\mathfrak {g}})}$ given by the formula

Although the definition relies on a choice of basis for the Lie algebra, it is easy to show that Ω is independent of this choice. On the other hand, Ω does depend on the bilinear form B. The invariance of B implies that the Casimir element commutes with all elements of the Lie algebra ${\displaystyle {\mathfrak {g}}}$, and hence lies in the center of the universal enveloping algebra ${\displaystyle U({\mathfrak {g}})}$.