Editing Universal enveloping algebra (section)

==Casimir operators==
{{See also|Harish-Chandra isomorphism}}
The [[center of an algebra|center]] <math>Z(U(\mathfrak{g}))</math> of <math>U(\mathfrak{g})</math> can be identified with the centralizer of <math>\mathfrak{g}</math> in <math>U(\mathfrak{g}).</math> Any element of <math>Z(U(\mathfrak{g}))</math> must commute with all of <math>U(\mathfrak{g}),</math> and in particular with the canonical embedding of <math>\mathfrak{g}</math> into <math>U({\mathfrak {g}}).</math> Because of this, the center is directly useful for classifying representations of <math>\mathfrak{g}</math>. For a finite-dimensional [[semisimple Lie algebra]], the [[Casimir operator]]s form a distinguished basis from the center <math>Z(U(\mathfrak{g}))</math>. These may be constructed as follows.

The center <math>Z(U(\mathfrak{g}))</math> corresponds to linear combinations of all elements <math>z=v\otimes w \otimes \cdots \otimes u \in U(\mathfrak{g})</math> that commute with all elements <math>x\in \mathfrak{g};</math> that is, for which <math>[z,x]=\mbox{ad}_x(z)=0.</math> That is, they are in the kernel of <math>\mbox{ad}_\mathfrak{g}.</math> Thus, a technique is needed for computing that kernel. What we have is the action of the [[adjoint representation]] on <math>\mathfrak{g};</math> we need it on <math>U(\mathfrak{g}).</math> The easiest route is to note that <math>\mbox{ad}_\mathfrak{g}</math> is a [[derivation (abstract algebra)|derivation]], and that the space of derivations can be lifted to <math>T(\mathfrak{g})</math> and thus to <math>U(\mathfrak{g}).</math> This implies that both of these are [[differential algebra]]s.

By definition, <math>\delta:\mathfrak{g}\to\mathfrak{g}</math> is a derivation on <math>\mathfrak{g}</math> if it obeys [[product rule|Leibniz's law]]:

:<math>\delta([v,w])=[\delta(v),w]+[v,\delta(w)]</math>

(When <math>\mathfrak{g}</math> is the space of left invariant vector fields on a group <math>G</math>, the Lie bracket is that of vector fields.) The lifting is performed by ''defining''
:<math>\begin{align}\delta(v\otimes w \otimes \cdots \otimes u)
=& \, \delta(v) \otimes w \otimes \cdots \otimes u \\
&+ v\otimes \delta(w) \otimes \cdots\otimes u \\
&+ \cdots + v\otimes w \otimes \cdots \otimes \delta(u).
\end{align}
</math>

Since <math>\mbox{ad}_x</math> is a derivation for any <math>x\in\mathfrak{g},</math> the above defines <math>\mbox{ad}_x</math> acting on <math>T(\mathfrak{g})</math> and <math>U(\mathfrak{g}).</math>

From the PBW theorem, it is clear that all central elements are linear combinations of symmetric [[homogeneous polynomial]]s in the basis elements <math>e_a</math> of the Lie algebra. The [[Casimir invariant]]s are the irreducible homogeneous polynomials of a given, fixed degree. That is, given a basis <math>e_a</math>, a Casimir operator of order <math>m</math> has the form

:<math>C_{(m)} = \kappa^{ab\cdots c}e_a\otimes e_b\otimes \cdots\otimes e_c</math>

where there are <math>m</math> terms in the tensor product, and <math>\kappa^{ab\cdots c}</math> is a completely symmetric tensor of order <math>m</math> belonging to the adjoint representation. That is, <math>\kappa^{ab\cdots c}</math> can be (should be) thought of as an element of <math>\left(\operatorname{ad}_\mathfrak{g}\right)^{\otimes m}.</math> Recall that the adjoint representation is given directly by the [[structure constants]], and so an explicit indexed form of the above equations can be given, in terms of the Lie algebra basis; this is originally a theorem of [[Israel Gel'fand]]. That is, from <math>[x,C_{(m)}]=0</math>, it follows that

:<math>f_{ij}^{\;\; k} \kappa^{jl\cdots m} 
+ f_{ij}^{\;\; l} \kappa^{kj\cdots m} + \cdots 
+ f_{ij}^{\;\; m} \kappa^{kl\cdots j} = 0
</math>
where the structure constants are
:<math>[e_i,e_j]=f_{ij}^{\;\; k}e_k</math>

As an example, the quadratic Casimir operator is
:<math>C_{(2)} = \kappa^{ij} e_i\otimes e_j</math>
where <math>\kappa^{ij}</math> is the inverse matrix of the [[Killing form]] <math>\kappa_{ij}.</math>  That the Casimir operator <math>C_{(2)}</math> belongs to the center <math>Z(U(\mathfrak{g}))</math> follows from the fact that the Killing form is invariant under the adjoint action.

The center of the universal enveloping algebra of a simple Lie algebra is given in detail by the [[Harish-Chandra isomorphism]].

===Rank===
The number of algebraically independent Casimir operators of a finite-dimensional [[semisimple Lie algebra]] is equal to the rank of that algebra, i.e. is equal to the rank of the [[Chevalley basis|Cartan–Weyl basis]]. This may be seen as follows. For a {{math|''d''}}-dimensional vector space {{math|''V''}}, recall that the [[determinant]] is the [[completely antisymmetric tensor]] on <math>V^{\otimes d}</math>. Given a matrix {{math|''M''}}, one may write the [[characteristic polynomial]] of {{math|''M''}} as
:<math>\det(tI-M)=\sum_{n=0}^d p_nt^n</math>

For a {{math|''d''}}-dimensional Lie algebra, that is, an algebra whose [[Adjoint representation of a Lie algebra|adjoint representation]] is {{math|''d''}}-dimensional, the linear operator
:<math>\operatorname{ad}:\mathfrak{g}\to\operatorname{End}(\mathfrak{g})</math>
implies that <math>\operatorname{ad}_x</math> is a {{math|''d''}}-dimensional endomorphism, and so one has the characteristic equation
:<math>\det(tI-\operatorname{ad}_x)=\sum_{n=0}^d p_n(x)t^n</math>
for elements <math>x\in \mathfrak{g}.</math>  The non-zero roots of this characteristic polynomial (that are roots for all {{math|''x''}}) form the [[root system]] of the algebra. In general, there are only {{math|''r''}} such roots; this is the rank of the algebra. This implies that the highest value of {{math|''n''}} for which the <math>p_n(x)</math> is non-vanishing is {{math|''r''.}}

The <math>p_n(x)</math> are [[homogeneous polynomial]]s of degree {{math|''d''&nbsp;−&nbsp;''n''.}}  This can be seen in several ways: Given a constant <math>k\in K</math>, ad is linear, so that <math>\operatorname{ad}_{kx}=k\,\operatorname{ad}_x.</math> By [[plug and chug|plugging and chugging]] in the above, one obtains that

:<math>p_n(kx)=k^{d-n}p_n(x).</math>

By linearity, if one expands in the basis,
:<math>x=\sum_{i=1}^d x_i e_i</math>
then the polynomial has the form
:<math>p_n(x)=x_ax_b\cdots x_c \kappa^{ab\cdots c}</math>
that is, a <math>\kappa</math> is a tensor of rank <math>m=d-n</math>. By linearity and the commutativity of addition, i.e. that <math>\operatorname{ad}_{x+y}=\operatorname{ad}_{y+x},</math>, one concludes that this tensor must be completely symmetric. This tensor is exactly the Casimir invariant of order {{math|''m''.}}

The center <math>Z(\mathfrak{g})</math> corresponded to those elements <math>z\in Z(\mathfrak{g})</math> for which <math>\operatorname{ad}_x(z)=0</math> for all {{math|''x'';}}  by the above, these clearly corresponds to the roots of the characteristic equation. One concludes that the roots form a space of rank {{math|''r''}} and that the Casimir invariants span this space. That is, the Casimir invariants generate the center <math>Z(U(\mathfrak{g})).</math>

===Example: Rotation group SO(3)===
The [[rotation group SO(3)]] is of rank one, and thus has one Casimir operator. It is three-dimensional, and thus the Casimir operator must have order (3&nbsp;−&nbsp;1)&nbsp;=&nbsp;2 i.e. be quadratic. Of course, this is the Lie algebra of <math>A_1.</math> As an elementary exercise, one can compute this directly. Changing notation to <math>e_i=L_i,</math> with <math>L_i</math> belonging to the adjoint rep, a general algebra element is <math>xL_1+yL_2+zL_3</math> and direct computation gives

:<math>\det\left(xL_1+yL_2+zL_3-tI\right)=-t^3-(x^2+y^2+z^2)t</math>

The quadratic term can be read off as <math>\kappa^{ij}=\delta^{ij}</math>, and so the squared [[angular momentum operator]] for the rotation group is that Casimir operator. That is,
:<math>C_{(2)} = L^2 = e_1\otimes e_1 +  e_2\otimes e_2 + e_3\otimes e_3</math>
and explicit computation shows that
:<math>[L^2, e_k]=0</math>
after making use of the [[structure constants]]
:<math>[e_i, e_j]=\varepsilon_{ij}^{\;\;k}e_k</math>

===Example: Pseudo-differential operators===
A key observation during the construction of <math>U(\mathfrak{g})</math> above was that it was a differential algebra, by dint of the fact that any derivation on the Lie algebra can be lifted to <math>U(\mathfrak{g})</math>. Thus, one is led to a ring of [[pseudo-differential operator]]s, from which one can construct Casimir invariants.

If the Lie algebra <math>\mathfrak{g}</math> acts on a space of linear operators, such as in [[Fredholm theory]], then one can construct Casimir invariants on the corresponding space of operators. The quadratic Casimir operator corresponds to an [[elliptic operator]].

If the Lie algebra acts on a differentiable manifold, then each Casimir operator corresponds to a higher-order differential on the cotangent manifold, the second-order differential being the most common and most important.

If the action of the algebra is [[Isometry group|isometric]], as would be the case for [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]]s endowed with a metric and the symmetry groups [[SO(N)]] and [[indefinite orthogonal group|SO (P, Q)]], respectively, one can then contract upper and lower indices (with the metric tensor) to obtain more interesting structures. For the quadratic Casimir invariant, this is the [[Laplacian]]. Quartic Casimir operators allow one to square the [[stress–energy tensor]], giving rise to the [[Yang-Mills action]]. The [[Coleman–Mandula theorem]] restricts the form that these can take, when one considers ordinary Lie algebras. However, the [[Lie superalgebra]]s are able to evade the premises of the Coleman–Mandula theorem, and can be used to mix together space and internal symmetries.