Knowlpedia

Definition

Invariant polynomial on a Lie algebra

A polynomial on a Lie algebra that is unchanged by the adjoint action of its Lie group.

Definition

Let GG be a with g\mathfrak g. A homogeneous invariant polynomial of degree kk is an element p∈Sym⁡k(g∗)p\in\operatorname{Sym}^k(\mathfrak g^*) fixed by the . Equivalently, regarding pp as a symmetric kk-linear form,

p(Ad⁡gX1,…,Ad⁡gXk)=p(X1,…,Xk)p(\operatorname{Ad}_gX_1,\ldots,\operatorname{Ad}_gX_k)=p(X_1,\ldots,X_k)

for all g∈Gg\in G and Xi∈gX_i\in\mathfrak g. A nonhomogeneous invariant polynomial is a finite sum of homogeneous ones. The acting group is part of the data when GG is disconnected, because invariance under its identity component can be weaker than invariance under all of GG.

Infinitesimal characterization

If GG is connected, invariance is equivalent to

∑i=1kp(X1,…,[Y,Xi],…,Xk)=0\sum_{i=1}^{k}p(X_1,\ldots,[Y,X_i],\ldots,X_k)=0

for every Y,X1,…,Xk∈gY,X_1,\ldots,X_k\in\mathfrak g. This is obtained by differentiating the group invariance condition. For a disconnected group it tests only invariance under the identity component, so the remaining components must be checked separately.

The invariant polynomials form a graded subalgebra

Sym⁡(g∗)G⊆Sym⁡(g∗).\operatorname{Sym}(\mathfrak g^*)^G\subseteq\operatorname{Sym}(\mathfrak g^*).
Examples

For g=gln\mathfrak g=\mathfrak{gl}_n, the functions X↦tr⁡(Xr)X\mapsto\operatorname{tr}(X^r) and the coefficients of the are invariant under conjugation.

An invariant symmetric BB gives the quadratic polynomial X↦B(X,X)X\mapsto B(X,X). The supplies a standard example when it is nonzero.

On so(2m)\mathfrak{so}(2m), the Pfaffian is invariant under the adjoint action of SO(2m)SO(2m). It is the polynomial used to construct the Euler form.

Role in characteristic classes

Applying a degree-kk invariant polynomial to the curvature of a produces a of degree 2k2k. Adjoint invariance is exactly what permits this curvature expression to descend from the principal bundle to the base; the then gives closedness Bott and Tu, chapter 11.

Conventions and scope

Authors alternate between a polynomial function P(X)P(X) and its polarized symmetric multilinear form p(X1,…,Xk)p(X_1,\ldots,X_k). Over R\mathbb R or C\mathbb C these descriptions are equivalent, but normalization factors such as k!k! vary.

References
  1. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: chapter 11, invariant polynomials and the Chern–Weil homomorphism.
  2. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. II, Wiley, 1969. Publisher record. Relevant: chapter XII, invariant polynomials and characteristic forms.