Invariant polynomial on a Lie algebra

Intuition [Beginner]

A polynomial in matrices that doesn't change when you change basis — that's an invariant polynomial. The trace is the simplest example: $\mathrm{tr}(A)=\mathrm{tr}(P^{-1}AP)$ for any invertible $P$. The determinant is another. Powers of the trace, products of eigenvalue-symmetric expressions, the Pfaffian — all invariant.

Invariant polynomials are the coordinate-free measurements of a matrix. Anyone evaluates them, anywhere, in any basis, and gets the same answer. They are exactly the polynomial functions that survive a change of perspective.

When you have a Lie algebra — a vector space with a bracket — the natural notion of "change of perspective" is the adjoint action. Invariant polynomials with respect to the adjoint action are the generalisation of trace and determinant to any Lie algebra, and they are the input that Chern-Weil theory turns into characteristic classes.

Visual [Beginner]

A matrix being conjugated by a series of changes of basis. The invariant polynomial reads the same value at every step, while individual matrix entries shuffle around.

The dial doesn't move. That's invariance.

Worked example [Beginner]

Take the Lie algebra ${\mathfrak{gl}}_2(\mathbb{R})$ of $2\times 2$ real matrices. Consider the polynomial $f(A)=\mathrm{tr}(A^2)$. Let's check it's invariant.

For $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, compute

$$A^2=\left(\begin{array}{cc}{a}^2+bc& b(a+d)\\ c(a+d)& d^2+bc\end{array}\right),$$

so $\mathrm{tr}(A^2)=a^2+2bc+d^2$. This is a degree-2 polynomial in the matrix entries.

Now conjugate: $A\mapsto P^{-1}AP$. The trace identity $\mathrm{tr}(XY)=\mathrm{tr}(YX)$ gives

$$\mathrm{tr}((P^{-1}AP{)}^2)=\mathrm{tr}(P^{-1}{A}^{2}P)=\mathrm{tr}(A^{2}P{P}^{-1})=\mathrm{tr}(A^2).$$

The polynomial is unchanged. This is adjoint invariance: invariance under the action of conjugation by group elements. The same trick works for $\mathrm{tr}(A^k)$ for any $k$, and for $\det (I+tA)$ as a polynomial in $t$ (whose coefficients give the characteristic-polynomial elementary symmetric functions in eigenvalues).

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $K$ be a field of characteristic zero, $\mathfrak{g}$ a Lie algebra over $K$ 03.04.01, and $G$ a Lie group with Lie algebra $\mathfrak{g}$ (or, more generally, any group acting on $\mathfrak{g}$ by Lie-algebra automorphisms). The space $S^k({\mathfrak{g}}^{\ast })$ of degree-$k$ symmetric multilinear functions

$$f:\underset{k}{\underbrace{\mathfrak{g}\times \cdots \times \mathfrak{g}}}\to K$$

(symmetric meaning unchanged under permutation of inputs) is a $K$-vector space, and the direct sum

$$S^{\bullet }({\mathfrak{g}}^{\ast }):=\underset{k\ge 0}{⨁}{S}^k({\mathfrak{g}}^{\ast })$$

is a graded commutative algebra under the symmetric product.

The adjoint action of $G$ on $\mathfrak{g}$ induces a contragredient action on ${\mathfrak{g}}^{\ast }$, and hence on each $S^k({\mathfrak{g}}^{\ast })$:

$$(g\cdot f)(X_1,\dots ,X_k)=f({\mathrm{Ad}}_{g^{-1}}{X}_1,\dots ,{\mathrm{Ad}}_{g^{-1}}{X}_k).$$

A symmetric multilinear function $f\in S^k({\mathfrak{g}}^{\ast })$ is invariant if $g\cdot f=f$ for every $g\in G$, equivalently

$$f({\mathrm{Ad}}_g{X}_1,\dots ,{\mathrm{Ad}}_g{X}_k)=f(X_1,\dots ,X_k)\text{for all }g\in G,X_i\in \mathfrak{g}.$$

The graded subalgebra of $G$-invariants is denoted $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$. Its elements are the invariant polynomials of degree $k$ on $\mathfrak{g}$ ^{[Kobayashi-Nomizu Vol. II §XII.1]}.

In the differential-geometric Lie-algebraic version, invariance under ${\mathrm{Ad}}_g$ for $g$ in the connected component of the identity is equivalent to infinitesimal invariance: $f$ is invariant iff for all $X,Y_1,\dots ,Y_k\in \mathfrak{g}$,

$$\sum _{i=1}^{k}f(Y_1,\dots ,[X,Y_i],\dots ,Y_k)=0.$$

This is the Lie-algebraic invariance condition, equivalent to invariance under the adjoint action of the connected Lie group when $G$ is connected.

Key theorem with proof [Intermediate+]

Theorem (Group invariance and Lie-algebra invariance agree, connected case). Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$, and $f\in S^k({\mathfrak{g}}^{\ast })$. The following are equivalent:

(i) $f$ is $G$-invariant: $f({\mathrm{Ad}}_g{X}_1,\dots ,{\mathrm{Ad}}_g{X}_k)=f(X_1,\dots ,X_k)$ for all $g\in G$.

(ii) $f$ is $\mathfrak{g}$-invariant: for all $X\in \mathfrak{g}$ and $Y_i\in \mathfrak{g}$,

$$\sum _{i=1}^{k}f(Y_1,\dots ,[X,Y_i],\dots ,Y_k)=0.$$

Proof. (i) $\Rightarrow$ (ii): take a smooth path $g_t$ in $G$ with $g_0=e$ and ${\dot{g}}_0=X$. Then ${\mathrm{Ad}}_{g_t}:\mathfrak{g}\to \mathfrak{g}$ is a smooth path of linear maps with ${\mathrm{Ad}}_{g_0}=\mathrm{id}$ and $\frac{d}{dt}{|}_{t=0}{\mathrm{Ad}}_{g_t}={\mathrm{ad}}_X$. Differentiating

$$f({\mathrm{Ad}}_{g_t}{Y}_1,\dots ,{\mathrm{Ad}}_{g_t}{Y}_k)=f(Y_1,\dots ,Y_k)$$

at $t=0$ and using multilinearity of $f$ gives

$$\sum _{i=1}^{k}f(Y_1,\dots ,{\mathrm{ad}}_X{Y}_i,\dots ,Y_k)=0,$$

which is (ii) (since ${\mathrm{ad}}_X{Y}_i=[X,Y_i]$).

(ii) $\Rightarrow$ (i): define $\phi (t):=f({\mathrm{Ad}}_{g_t}{Y}_1,\dots ,{\mathrm{Ad}}_{g_t}{Y}_k)$ for a smooth path $g_t$ in $G$. Differentiating with respect to $t$,

$$\dot{\phi }(t)=\sum _{i=1}^{k}f({\mathrm{Ad}}_{g_t}{Y}_1,\dots ,{\mathrm{ad}}_{X(t)}{\mathrm{Ad}}_{g_t}{Y}_i,\dots ,{\mathrm{Ad}}_{g_t}{Y}_k)$$

where $X(t)=({\mathrm{Ad}}_{g_t}{)}^{-1}{\dot{g}}_t\cdot g_t^{-1}$ is the body-frame velocity. By (ii) applied with $X=X(t)$ and $Y_i\mapsto {\mathrm{Ad}}_{g_t}{Y}_i$, this sum is zero. So $\phi$ is constant, and $\phi (t)=\phi (0)$ gives the $G$-invariance for $g$ in the path-connected component of $e$. Since $G$ is connected, that's all of $G$. $\square$

The Lie-algebraic condition (ii) is computationally tractable: for matrix Lie algebras, it reduces to a system of polynomial equations in the matrix entries.

Bridge. The construction here builds toward 03.06.06 (chern-weil homomorphism), where the same data is upgraded, and the symmetry side is taken up in 03.06.04 (pontryagin and chern classes). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none — Mathlib has Lie algebras and symmetric tensor powers separately but no bundled API for $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$.

[object Promise]

A formalisation needs the dual representation of $\mathrm{ad}$ on ${\mathfrak{g}}^{\ast }$, the induced action on each $S^k({\mathfrak{g}}^{\ast })$, and the fixed-point subspace under that action. The downstream Chern-Weil theorem (03.06.06) then evaluates these on a curvature 2-form.

Advanced results [Master]

For a compact connected Lie group $G$ with Lie algebra $\mathfrak{g}$, the algebra of invariants $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$ has a clean structural description (Chevalley's restriction theorem above). For ${\mathfrak{gl}}_n$, the invariants are generated by $\mathrm{tr}(A^k)$ for $k=1,2,\dots ,n$, or equivalently by the coefficients of $\det (I+tA)$, or by the elementary symmetric functions in eigenvalues. These three generating sets are related by Newton's identities.

The matrix-Lie-algebra invariants give the natural inputs to characteristic-class constructions:

• ${\mathfrak{gl}}_n(\mathbb{C})$: $\det (I+\frac{i}{2\pi }A)$ — coefficients are the Chern classes 03.06.04.
• ${\mathfrak{so}}_n(\mathbb{R})$, $n$ even: the Pfaffian of the skew-symmetric matrix gives the Euler class; $\det (I+\frac{1}{2\pi }{A}^2)$ coefficients give the Pontryagin classes.
• $\mathfrak{u}(n)$: as in ${\mathfrak{gl}}_n(\mathbb{C})$, with the Hermitian-conjugate condition forcing real-valued classes.
• $\mathfrak{spin}(n)$: matches ${\mathfrak{so}}_n$ via the double cover, giving the same Pontryagin classes plus the half-Pfaffian / Â-class refinement.

The Chern-Weil homomorphism 03.06.06 sends each invariant polynomial of degree $k$ to a closed $2k$-form on the base manifold. The invariance condition is exactly what makes this $2k$-form descend from the principal bundle to the base; without it, $f(\Omega )$ would only be a form on the total space.

A subtler Master-tier result: the Cartan-Eilenberg / Chevalley-Eilenberg cohomology of a Lie algebra $H_{CE}^{\bullet }(\mathfrak{g};K)$ relates to invariant polynomials via the transgression. For compact Lie groups, transgression maps the algebra-generators of $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$ in degree $2k$ to the algebra-generators of $H_{CE}^{2k-1}(\mathfrak{g};\mathbb{R})\cong H^{2k-1}(G;\mathbb{R})$ — the de Rham cohomology of the group itself. This is the source of the Chern-Simons forms appearing in the Chern-Weil proof 03.06.06.

Synthesis. This construction generalises the pattern fixed in 03.04.01 (lie algebra), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Symmetric-power decomposition. Every multilinear function ${\mathfrak{g}}^k\to K$ decomposes uniquely as a sum of irreducibles under the $S_k$ action (permutation of inputs). The symmetric component is $S^k({\mathfrak{g}}^{\ast })$; the antisymmetric component is ${\Lambda }^k({\mathfrak{g}}^{\ast })$ (which underlies the Chevalley-Eilenberg complex). For invariant theory under $G$, only the symmetric part contributes to the standard Chern-Weil construction, since the curvature 2-form is symmetric in its $\mathfrak{g}$-valued slots once antisymmetrised in the form-degree slots.

Polarisation and degree filtration. Any polynomial function $\mathfrak{g}\to K$ that is homogeneous of degree $k$ corresponds bijectively to a symmetric $k$-multilinear function via polarisation:

$$\tilde{f}(X_1,\dots ,X_k):=\frac{1}{k!}\frac{{\partial }^k}{\partial t_1\cdots \partial t_k}{|}_{t=0}f(t_1{X}_1+\cdots +t_k{X}_k).$$

The recovery map is $f(X)=\tilde{f}(X,\dots ,X)$. This identifies the polynomial functions on $\mathfrak{g}$ with $S^{\bullet }({\mathfrak{g}}^{\ast })$.

Equivalence of group and Lie-algebra invariance (connected case). Proved in §"Key theorem". For non-connected $G$, group invariance is strictly stronger than Lie-algebra invariance: it picks up constraints from the component group ${\pi }_0(G)$.

Generators for $S^{\bullet }({\mathfrak{gl}}_n^{\ast }{)}^{{\mathrm{GL}}_n}$. By Chevalley's restriction theorem, this is isomorphic to symmetric polynomials in eigenvalues, which is freely generated by the elementary symmetric polynomials $e_1,e_2,\dots ,e_n$ — equivalently by the power sums $p_k(\lambda )={\sum }_i{\lambda }_i^k=\mathrm{tr}(A^k)$ for $k=1,\dots ,n$. Newton's identities relate these two generating sets.

Real-valuedness for unitary Lie algebras. Proved in Exercise 6 above.

Connections [Master]

• Lie algebra 03.04.01 — supplies the underlying space and bracket.

• Bilinear / quadratic form 01.01.15 — invariant polynomials of degree 2 are invariant bilinear forms; the Killing form is the canonical example.

• Chern-Weil homomorphism 03.06.06 — applies invariant polynomials to curvature 2-forms to produce closed differential forms on the base manifold; this is the construction's downstream payoff.

• Pontryagin and Chern classes 03.06.04 — specific invariant polynomials (det, Pf, $\mathrm{tr}(A^k)$ with normalisation) produce the named characteristic classes.

• Yang-Mills action 03.07.05 — the action functional uses the invariant inner product on $\mathfrak{g}$, which is a degree-2 element of $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$ when $G$ is compact.

Historical & philosophical context [Master]

The theory of invariants on Lie algebras emerged from Cayley and Sylvester's classical-invariant-theory programme of the 1850s–1870s, generalised to arbitrary linear groups by Hilbert in his finiteness theorem (1890). Chevalley and Eilenberg (1948) gave the cohomological framework that connected invariant polynomials to the de Rham cohomology of the corresponding Lie group, the bridge that Chern-Weil theory then exploited ^{[Chevalley-Eilenberg 1948]}.

The realisation that adjoint-invariant polynomials evaluated on curvature produce characteristic classes is due to Chern and Weil in the 1940s, made systematic by Kobayashi and Nomizu's monograph in the 1960s ^{[Kobayashi-Nomizu Vol. II §XII]}. The structure of $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$ for compact $G$ — Chevalley's restriction theorem and the freely-generated polynomial-algebra structure — connects this construction directly to the cohomology of compact Lie groups (Pontryagin, Borel) and to the cohomology of classifying spaces $BG$ (Cartan, Borel).

Bibliography [Master]

• Kobayashi, S. & Nomizu, K., Foundations of Differential Geometry, Vol. II, Wiley, 1969. Ch. XII.
• Milnor, J. & Stasheff, J., Characteristic Classes, Princeton University Press, 1974. Appendix C.
• Chevalley, C. & Eilenberg, S., "Cohomology Theory of Lie Groups and Lie Algebras", Transactions of the American Mathematical Society 63 (1948), 85–124.
• Borel, A., Topology of Lie Groups and Characteristic Classes, Bull. Amer. Math. Soc. 61 (1955), 397–432.
• Chevalley, C., "Invariants of Finite Groups Generated by Reflections", American Journal of Mathematics 77 (1955), 778–782.

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Wave 2 Phase 2.3 unit #2. Bridges Lie algebras to characteristic-class theory; the input space for Chern-Weil.