Theorems · Theorem · nonassociative algebras

LieModule.traceForm_eq_sum_finrank_nsmul

∀ (K : Type u_2) (L : Type u_3) (M : Type u_4) [inst : LieRing L] [inst_1 : AddCommGroup M] [inst_2 : LieRingModule L M]
  [inst_3 : Field K] [inst_4 : LieAlgebra K L] [inst_5 : Module K M] [inst_6 : LieModule K L M]
  [inst_7 : FiniteDimensional K M] [inst_8 : LieRing.IsNilpotent L] [inst_9 : LieModule.LinearWeights K L M]
  [LieModule.IsTriangularizable K L M],
  LieModule.traceForm K L M =
    ∑ χ,
      Module.finrank K ↥(LieModule.genWeightSpace M ⇑χ) •
        (LieModule.Weight.toLinear K L M χ).smulRight (LieModule.Weight.toLinear K L M χ)

See also LieModule.traceForm_eq_sum_finrank_nsmul' for an expression omitting the zero weights.

Defined in

Mathlib.Algebra.Lie.TraceForm

Cited by

2 results in Mathlib

Foundations

Depth 129 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingAddCommGroupLieRingModuleFieldLieAlgebraModuleLieModuleFiniteDimensionalLieRing.IsNilpotentLieModule.LinearWeightsLieModule.IsTriangularizable