Editing Universal enveloping algebra (section)

===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>