Theorems · Definition · nonassociative algebras
killingForm
(R : Type u_1) → (L : Type u_3) → [inst : CommRing R] → [inst_1 : LieRing L] → [inst_2 : LieAlgebra R L] → LinearMap.BilinForm R L
A finite, free (as an R-module) Lie algebra L carries a bilinear form on L.
This is a specialisation of LieModule.traceForm to the adjoint representation of L.
- Defined in
- Mathlib.Algebra.Lie.TraceForm
- Cited by
- 34 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingLieRingLieAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- LieRingstatement and proof · cited by 1,548
- LieAlgebrastatement and proof · cited by 1,246
- LinearMap.BilinFormstatement · cited by 501
- LieModule.traceFormproof · cited by 41
Cited by36
Results whose statement or proof uses this declaration.
- LieIdeal.killingComplproof · cited by 8
- LieAlgebra.IsKilling.ker_killingForm_eq_botstatement · cited by 7
- LieAlgebra.IsKilling.finrank_rootSpace_eq_oneproof · cited by 6
- LieAlgebra.IsKilling.exists_isSl2Triple_of_weight_isNonZeroproof · cited by 5
- IsSl2Triple.h_eq_corootproof · cited by 4
- LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_negstatement · cited by 4
- LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpaceproof · cited by 3
- LieAlgebra.mem_ker_killingForm_of_mem_rootSpace_of_forall_rootSpace_negstatement and proof · cited by 3
- LieIdeal.isSolvable_of_killingForm_apply_lie_eq_zerostatement and proof · cited by 2
- LieAlgebra.IsKilling.traceForm_eq_zero_of_mem_ker_of_mem_span_corootproof · cited by 2
- LieIdeal.mem_killingComplstatement · cited by 2
- LieAlgebra.IsKilling.killingForm_nondegeneratestatement · cited by 2