Invariant Polynomials – Generators of Semisimple Lie Algebra

Question

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?

Answer 1

In $\mathfrak{sl}(2,\mathbb{C})$, there is only one one-dimensional homogeneous space generating $S(\mathfrak g)^{\mathfrak g}\cong \mathbb C[X]$. Hence the canonical choice.

In general, you can start with $l$ homogeneous generators $f_1, \dots f_l$ of respective degrees $d_1, \dots, d_l$ and make replacements of the form $f_i':=f_i+f_jf_k$ whenever $d_i=d_j+d_k$. I don't know any way to state that one basis is more canonical than the other (it does not mean that some basis are not nicer than others).

For instance, here are two classical ways to construct homogeneous invariants in $\mathfrak{sl}(n,\mathbb{C})$.

-Compute the characteristic polynomial of a general matrix. Then the coefficients of the polynomial are homogeneous generators of $S(\mathfrak g)^{\mathfrak g}$

-Compute $Tr(X^k)$ ($2\leqslant k\leqslant n$).

(This is pretty much the same as the difference between the elementary symmetric polynomial and the power sum symmetric polynomials)

Note that the last construction can be generalized (at least, theoretically) for any semisimple Lie algebra. Theorem: The vector space $S^n(\mathfrak g)^{\mathfrak g}$ of homogeneous invariants of degree n is spanned by polynomial functions of the form $x\mapsto Tr(\rho(x)^n)$ where $\rho$ is a finite dimensional representation of $\mathfrak g$. (Ref: Tauvel-Yu, Lie algebras and algebraic groups, 31.2.5)