Theorems · Theorem · nonassociative algebras

LieAlgebra.ad_mem_adjoin_of_isSemisimple

∀ {K : Type u_1} {V : Type u_2} [inst : Field K] [PerfectField K] [inst_2 : AddCommGroup V] [inst_3 : Module K V]
  [FiniteDimensional K V] {n s : Module.End K V},
  Commute n s →
    IsNilpotent n →
      s.IsSemisimple → (LieAlgebra.ad K (Module.End K V)) s ∈ K[(LieAlgebra.ad K (Module.End K V)) (n + s)]

The adjoint of the semisimple part of a JC decomposition lies in the subalgebra generated by the adjoint of the sum.

Defined in: Mathlib.Algebra.Lie.AdjointAction.JordanChevalley
Cited by: 1 results in Mathlib
Foundations: Depth 134 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field PerfectField AddCommGroup Module FiniteDimensional

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DFunLike.coestatement and proof · cited by 62,936
Setstatement · cited by 53,352
Modulestatement and proof · cited by 20,661
RingHom.idstatement · cited by 18,349
AddCommGroupstatement and proof · cited by 12,871
Fieldstatement and proof · cited by 7,404
FiniteDimensionalstatement and proof · cited by 1,854
Subalgebrastatement · cited by 1,353
map_addproof · cited by 964
Module.Endstatement and proof · cited by 774
Commutestatement and proof · cited by 639
Algebra.adjoinstatement and proof · cited by 535

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LieAlgebra.ad_mem_adjoin_of_isNilpotentproof · cited by 0