Casimir invariant

In mathematics, a Casimir invariant or Casimir abettor is a acclaimed aspect of the centre of the accepted enveloping algebra of a Lie algebra. A prototypal archetype is the boxlike angular drive operator, which is a Casimir invariant of the three-dimensional circling group.

The Casimir invariant is called afterwards Hendrik Casimir, who articular them in his description of adamant anatomy dynamics in 1931.

Definition

Suppose that \mathfrak{g} is an n-dimensional semisimple Lie algebra. Let

\{X_i\}_{i=1}^n

be any base of \mathfrak{g}, and

\{X^i\}_{i=1}^n

be the bifold base of \mathfrak{g} with account to a anchored invariant bilinear anatomy (e.g. the Killing form) on \mathfrak{g}. The Casimir aspect Ω is an aspect of the accepted enveloping algebra U(\mathfrak{g}) accustomed by the formula

\Omega = \sum_{i=1}^n X_i X^i.

Although the analogue of the Casimir aspect refers to a accurate best of base in the Lie algebra, it is accessible to appearance that the consistent aspect Ω is absolute of this choice. Moreover, the invariance of the bilinear anatomy acclimated in the analogue implies that the Casimir aspect commutes with all elements of the Lie algebra \mathfrak{g}, and appropriately lies in the centermost of the accepted enveloping algebra U(\mathfrak{g}).

Given any representation ρ of \mathfrak{g} on a agent amplitude V, possibly infinite-dimensional, the agnate Casimir invariant is ρ(Ω), the beeline abettor on V accustomed by the formula

\rho(\Omega) = \sum_{i=1}^n \rho(X_i)\rho(X^i).

A appropriate case of this architecture plays an important role in cogwheel geometry and all-around analysis. Suppose that a affiliated Lie accumulation G with the Lie algebra \mathfrak{g} acts on a differentiable assorted M, again elements of \mathfrak{g} are represented by aboriginal adjustment cogwheel operators on M. The representation ρ is on the amplitude of bland functions on M. In this bearings the Casimir invariant is the G-invariant additional adjustment cogwheel abettor on M authentic by the aloft formula.

More accepted Casimir invariants may additionally be defined, frequently occurring in the abstraction of pseudo-differential operators in Fredholm theory.

Properties

The Casimir abettor is a acclaimed aspect of the centermost of the accepted enveloping algebra of the Lie algebra. In added words, it is a affiliate of the algebra of all cogwheel operators that commutes with all the generators in the Lie algebra.

The cardinal of absolute elements of the centermost of the accepted enveloping algebra is additionally the rank in the case of a semisimple Lie algebra. The Casimir abettor gives the abstraction of the Laplacian on a accepted semisimple Lie group; but this way of counting shows that there may be no different alternation of the Laplacian, for rank > 1.

By analogue any affiliate of the centermost of the accepted enveloping algebra commutes with all added elements in the algebra. By Schur's Lemma, in any irreducible representation of the Lie algebra, the Casimir abettor is appropriately proportional to the identity. This connected of arrangement can be acclimated to allocate the representations of the Lie algebra (and hence, additionally of its Lie group). Physical accumulation and circuit are examples of these constants, as are abounding added breakthrough numbers begin in breakthrough mechanics. Superficially, topological breakthrough numbers anatomy an barring to this pattern; although added theories adumbration that these are two facets of the aforementioned phenomenon.

Example

The Lie algebra \mathfrak{so}(3) is the Lie algebra of SO(3), the circling accumulation for three-dimensional Euclidean space. It is semisimple of rank 1, and so it has a distinct absolute Casimir. The Killing anatomy for the circling accumulation is aloof the Kronecker delta, and so the Casimir invariant is artlessly the sum of the squares of the generators L_x,\, L_y,\, L_z of the algebra. That is, the Casimir invariant is accustomed by

L^2=L_x^2+L_y^2+L_z^2.

In an irreducible representation, the invariance of the Casimir abettor implies that it is a assorted of the character aspect e of the algebra, so that

L^2=L_x^2+L_y^2+L_z^2=\ell(\ell+1)e.

In breakthrough mechanics, the scalar amount \ell is referred to as the absolute angular momentum. For finite-dimensional matrix-valued representations of the circling group, \ell consistently takes on accumulation ethics (for bosonic representations) or half-integer ethics (for fermionic representations).

For a accustomed amount of \ell, the cast representation is (2\ell+1)-dimensional. Thus, for example, the three-dimensional representation for so(3) corresponds to \ell\,=\,1, and is accustomed by the generators

L_x= \begin{pmatrix} 0& 0& 0\\ 0& 0& -1\\ 0& 1& 0 \end{pmatrix}, L_y= \begin{pmatrix} 0& 0& 1\\ 0& 0& 0\\ -1& 0& 0 \end{pmatrix}, L_z= \begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 0 \end{pmatrix}.

The boxlike Casimir invariant is then

L^2=L_x^2+L_y^2+L_z^2= 2 \begin{pmatrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix}

as \ell(\ell+1)\,=\,2 back \ell\,=\,1. Similarly, the two dimensional representation has a base accustomed by the Pauli matrices, which accord to circuit 1/2.

Eigenvalues

Given that Ω is central in the enveloping algebra, it acts on simple modules by a scalar. Let \langle,\rangle be our favorite bilinear symmetric non-degenerate form, by which we define Ω. Let L(λ) be the finite dimensional highest weight module of weight λ. Then the Casimir element Ω acts on L(λ) by the constant \langle \lambda, \lambda + 2 \rho \rangle, where ρ is the weight defined by half the sum of the positive roots.