Theorems · Theorem · nonassociative algebras
LieIdeal.rootSpan_mem_invtRootSubmodule
∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : LieRing L] [inst_2 : LieAlgebra K L]
[inst_3 : FiniteDimensional K L] {H : LieSubalgebra K L} [inst_4 : H.IsCartanSubalgebra] [inst_5 : CharZero K]
[inst_6 : LieAlgebra.IsKilling K L] [inst_7 : LieModule.IsTriangularizable K (↥H) L] (I : LieIdeal K L),
I.rootSpan ∈ (LieAlgebra.IsKilling.rootSystem H).invtRootSubmoduleThe submodule spanned by roots of a Lie ideal is invariant under all root reflections.
- Defined in
- Mathlib.Algebra.Lie.Weights.IsSimple
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites38
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- RingHom.idstatement · cited by 18,349
- Finsetstatement · cited by 13,712
- Fieldstatement and proof · cited by 7,404
- Submodulestatement and proof · cited by 7,192
- Set.imageproof · cited by 5,609
- FiniteDimensionalstatement and proof · cited by 1,854
- LieRingstatement and proof · cited by 1,548
- Submodule.spanproof · cited by 1,504
- LieAlgebrastatement and proof · cited by 1,246
- LinearEquiv.toLinearMapproof · cited by 1,171
- CharZerostatement and proof · cited by 932
Cited by2
Results whose statement or proof uses this declaration.
- LieIdeal.toInvtRootSubmoduleproof · cited by 2
- LieAlgebra.IsKilling.lieIdealOrderIso_left_invstatement and proof · cited by 0