TITLE: Generators of invariant polynomials of semisimple Lie algebra
QUESTION [14 upvotes]: Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ homogeneous polynomials where $l$ is the rank of $\mathfrak{g}$. The degrees of the generators are known, namely they are the primitive exponents. I am wondering if the generators can be canonically chosen. For example, when we are considering $\mathfrak{sl}(2,\mathbb{C})$, we can take the generator to be the Casimir. Does such kind of canonical choice exist in general?

REPLY [6 votes]: Here are some topological considerations that privilege a choice of generators up to scale. First, Chern-Weil theory gives an isomorphism
$$S(\mathfrak{g}^{\ast})^{\mathfrak{g}} \cong H^{\bullet}(BG, \mathbb{C})$$
sending an invariant polynomial $f$ to the characteristic class of principal $G$-bundles given by evaluating $f$ on the curvature of a connection. Here $G$ is the corresponding simply connected complex Lie group, or if you prefer its maximal compact. 
Second, rational homotopy theory gives an isomorphism
$$H^{\bullet}(BG, \mathbb{C}) \cong S( \left( \pi_{\bullet}(BG) \otimes \mathbb{C} \right)^{\ast} )$$
which singles out some particularly nice choices of generators of $H^{\bullet}(BG, \mathbb{C})$, namely dual bases of graded bases of $\pi_{\bullet}(BG) \otimes \mathbb{C}$. If $\pi_{\bullet}(BG) \otimes \mathbb{C}$ has rank at most $1$ in each degree then this graded basis is unique up to scale, and hence the corresponding dual basis is also unique up to scale. To fix scales up to sign you can take a basis of $\pi_{\bullet}(BG) \otimes \mathbb{C}$ which lifts to a basis of the torsion-free part of $\pi_{\bullet}(BG)$. 
In fact $\pi_{\bullet}(BG) \otimes \mathbb{C}$ has Poincare series $\sum t^{2e_i}$ where the $e_i$ are the exponents of $G$, so the above condition is satisfied whenever the exponents are distinct, and this happens for all simple $\mathfrak{g}$ except sometimes in type $D$ (and this ambiguity can be resolved).
I believe the resulting choice of generators for classical types is given, up to scale, by invariant polynomials of the form 
$$X \mapsto \text{tr}(X^i)$$
as in Humphreys' answer. To see how this works for type $A$ let me work first with the groups $U(n)$ rather than $SU(n)$ or $SL_n(\mathbb{C})$. The sequence of inclusions $U(1) \hookrightarrow U(2) \hookrightarrow \dots$, which has colimit $U$, induces a sequence of maps
$$BU(1) \to BU(2) \to \dots$$
with colimit $BU$. $BU$ has a natural infinite loop space structure given by taking direct sums of complex vector bundles, and so its cohomology is naturally a cocommutative Hopf algebra. Moreover the isomorphism
$$H^{\bullet}(BU, \mathbb{C}) \cong S( \left( \pi_{\bullet}(BU) \otimes \mathbb{C} \right)^{\ast} )$$
is an isomorphism of Hopf algebras, where the RHS should be interpreted as the universal enveloping algebra of an abelian Lie algebra. Hence the distinguished choice of generators above corresponds to a choice of primitive elements in this Hopf algebra, of which there is one in each even degree. 
On the other hand, $H^{\bullet}(BU, \mathbb{C})$ is known to be isomorphic, as a Hopf algebra, to the Hopf algebra of symmetric functions, with the isomorphism given explicitly by sending the $i^{th}$ Chern class $c_i$ to the elementary symmetric function $e_i$. The primitive elements of the Hopf algebra of symmetric functions, over a field of characteristic $0$, are given by linear combinations of the power symmetric functions $p_i$, and tracing back through all of the isomorphisms we've written down these correspond to the trace functions $X \mapsto \text{tr}(X^i)$. The corresponding characteristic classes are the coefficients of the Chern character. 
This is all a long-winded way of saying "among the characteristic classes of complex vector bundles we can single out the ones that behave as nicely as possible with respect to direct sum. The Chern character is additive with respect to direct sum and so it's reasonable to single out its coefficients." 
In the other classical types suitable choices of embedding into $U(n)$ should also reproduce this result. The ambiguity in type $D$ comes from the classifying spaces $BSO(4n)$, where the Euler class has the same degree as a Pontryagin class. The Pontryagin class can be distinguished by the fact that it is stable, or in other words by the fact that it is the pullback of a class on $BSO(4n+1)$, and similarly the Euler class can be distinguished by the fact that it vanishes when pulled back to $BSO(4n - 1)$.