Casimir invariant

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In mathematics, a Casimir invariant or Casimir operator, $Ω$ , of an $n$ -dimensional semisimple Lie algebra $\mathfrak{g}$ is defined as follows: Let

$(X_i)_{i=1}^n$

be a basis of $\mathfrak{g}$ , and

$(Y_i)_{i=1}^n$

be a basis of $\mathfrak{g}^*$ , where $\mathfrak{g}^*$ is the dual of $\mathfrak{g}$ with respect to the Killing form, $B$ .

The Casimir operator, $Ω$ , is given by

$\Omega = \sum_{i,j = 1}^n B(X_i,X_j) Y_i Y_j$

[edit] Properties

The Casimir operator is a distinguished element of the center of the universal enveloping algebra of the Lie algebra. In other words, it is a member of the algebra of all differential operators that commutes with all the generators in the Lie algebra.

The number of independent elements of the center of the universal enveloping algebra is also the rank in the case of a semisimple Lie algebra. The Casimir operator gives the concept of the Laplacian on a general semisimple Lie group; but this way of counting shows that there may be no unique analogue of the Laplacian, for rank > 1.

In any irreducible representation of the Lie algebra, by Schur's Lemma, any member of the center of the universal enveloping algebra commutes with everything and thus is proportional to the identity. This constant of proportionality can be used to classify the representations of the Lie algebra (and hence, also of its Lie group). Physical mass and spin are examples of these constants, as are many other quantum numbers found in quantum mechanics. Topological quantum numbers form an exception to this pattern.

[edit] Examples

The Lie algebra so(3) is the Lie algebra of the rotation group. It is semisimple of rank 1, and so it has a single independent Casimir. A basis for so(3) is given by the generators

$L_{12}= \begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 0 \end{pmatrix}, L_{23}= \begin{pmatrix} 0& 0& 0\\ 0& 0& -1\\ 0& 1& 0 \end{pmatrix}, L_{13}= \begin{pmatrix} 0& 0& -1\\ 0& 0& 0\\ 1& 0& 0 \end{pmatrix}.$

In terms of this basis, the quadratic Casimir invariant is

$L^2=L_{12}^2+L_{23}^2+L_{13}^2.$

In physics, these operators provide the components of the angular momentum, and the quadratic Casimir provides the magnitude squared of the angular momentum.