Mathematicshard

Adjoint Action of SO(4) on Its Lie Algebra - Equivariant Cohomology and Closed Invariant Submanifolds

Question

Consider the adjoint action of

SO(4)

on its Lie algebra

\mathfrak{so}(4)

. Determine the nonempty closed invariant submanifolds and compute the total rank of the equivariant cohomology.

Lie algebra decomposition, adjoint orbits, and equivariant cohomology.

Key Result

so(3)⊕so(3)

The exceptional isomorphism so(4) ≅ so(3) ⊕ so(3) governs the orbit structure

1Structure of SO(4) and Its Lie Algebra

The Lie algebra

\mathfrak{so}(4)

SO(4)

is 6-dimensional. It decomposes as

\mathfrak{so}(4) \cong \mathfrak{so}(3) \oplus \mathfrak{so}(3)

(an exceptional isomorphism). Every element can be written as a pair of

3 \times 3

skew-symmetric matrices.

Lie Algebra Decomposition

so(3)⁺ self-dual

←

so(4) 6-dim

→

so(3)⁻ anti-self-dual

↓
so(3)⁻ anti-self-dual

so(4) ≅ so(3)⁺ ⊕ so(3)⁻ via self-dual / anti-self-dual split

Exceptional Isomorphism

The decomposition

\mathfrak{so}(4) \cong \mathfrak{so}(3) \oplus \mathfrak{so}(3)

comes from splitting 4D rotations into self-dual and anti-self-dual parts using the Hodge star operator on

\Lambda^2(\mathbb{R}^4)

2Adjoint Orbits

The adjoint action of

SO(4)

\mathfrak{so}(4)

preserves the two

\mathfrak{so}(3)

factors. Each adjoint orbit is characterized by two parameters — the norms

r_+

and

r_-

in each factor.

Orbit Type	Parameters	Topology	Dimension
Origin	r₊ = r₋ = 0	{0}	0
Self-dual sphere	r₊ > 0, r₋ = 0	S²	2
Anti-self-dual sphere	r₊ = 0, r₋ > 0	S²	2
Generic orbit	r₊ > 0, r₋ > 0	S² × S²	4

3Closed Invariant Submanifolds

The nonempty closed invariant submanifolds are **level sets of the two Casimir invariants**

C_+ = r_+^2

and

C_- = r_-^2

  Orbit Space (r₊, r₋) ≥ 0:

  r₋
  │
  │  S²×S²   S²×S²   S²×S²
  │
  │  S² (anti-self-dual spheres along r₋ axis)
  │
  {0}────S² (self-dual spheres along r₊ axis)─── r₊

4Equivariant Cohomology

The

SO(4)

-equivariant cohomology is computed via the Borel model:

H^*_{SO(4)}(M) = H^*(M \times_{SO(4)} ESO(4))

H*_{SO(4)}({0})H*(BSO(4)) = ℤ[p₁,e]

H*_{SO(4)}(S²)Rank depends on embedding

H*_{SO(4)}(S²×S²)Polynomial generators from both factors

TOTAL RANKDepends on chosen submanifold

5Key Concepts

Casimir Invariants

The two Casimir operators

C_\pm = ||\xi_\pm||^2

are the fundamental invariants of the adjoint action. They define the orbit space as the first quadrant

\{(r_+, r_-) : r_+, r_- \geq 0\}

Borel Construction

Equivariant cohomology

H^*_G(M)

is computed as the ordinary cohomology of the homotopy quotient

M \times_G EG

. For free actions, this reduces to

H^*(M/G)

. For fixed points, it equals

H^*(BG)

Quiz

Test your understanding with these questions.

What is the Lie algebra decomposition of

\mathfrak{so}(4)

What is the generic adjoint orbit of

SO(4)

\mathfrak{so}(4)