Theorems · Definition · nonassociative algebras

LieModule.posFittingCompOf

(R : Type u_2) →
  {L : Type u_3} →
    (M : Type u_4) →
      [inst : CommRing R] →
        [inst_1 : LieRing L] →
          [inst_2 : LieAlgebra R L] →
            [inst_3 : AddCommGroup M] →
              [inst_4 : Module R M] →
                [inst_5 : LieRingModule L M] → [LieModule R L M] → [LieRing.IsNilpotent L] → L → LieSubmodule R L M

If M is a representation of a nilpotent Lie algebra L, and x : L, then posFittingCompOf R M x is the infimum of the decreasing system range φₓ ⊇ range φₓ² ⊇ range φₓ³ ⊇ ⋯ where φₓ : End R M := toEnd R L M x. We call this the "positive Fitting component" because with appropriate assumptions (e.g., R is a field and M is finite-dimensional) φₓ induces the so-called Fitting decomposition: M = M₀ ⊕ M₁ where M₀ = genWeightSpaceOf M 0 x and M₁ = posFittingCompOf R M x. It is a Lie submodule because L is nilpotent.

Defined in

Mathlib.Algebra.Lie.Weights.Basic

Cited by

10 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieRing.IsNilpotent