Theorems · Definition · nonassociative algebras

LieModule.rank

(R : Type u_1) →
  (L : Type u_3) →
    (M : Type u_4) →
      [inst : CommRing R] →
        [inst_1 : LieRing L] →
          [inst_2 : LieAlgebra R L] →
            [Module.Finite R L] →
              [Module.Free R L] →
                [inst_5 : AddCommGroup M] →
                  [inst_6 : Module R M] →
                    [inst_7 : LieRingModule L M] → [LieModule R L M] → [Module.Finite R M] → [Module.Free R M] → ℕ

Let M be a representation of a Lie algebra L over a nontrivial commutative ring R, and assume that L and M are finite free as R-module. Then the coefficients of the characteristic polynomial of ⁅x, ·⁆ are polynomial in x. The rank of M is the smallest n for which the n-th coefficient is not the zero polynomial.

Defined in

Mathlib.Algebra.Lie.Rank

Cited by

8 results in Mathlib

Foundations

Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraModule.FiniteModule.FreeAddCommGroupModuleLieRingModuleLieModuleModule.FiniteModule.Free