Theorems · Theorem · nonassociative algebras

LieAlgebra.isRegular_iff_coeff_polyCharpoly_rank_ne_zero

∀ (R : Type u_1) {L : Type u_3} {ι : Type u_5} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  [inst_3 : Module.Finite R L] [inst_4 : Module.Free R L] [inst_5 : Fintype ι] (b : Module.Basis ι R L) (x : L)
  [inst_6 : DecidableEq ι],
  LieAlgebra.IsRegular R x ↔
    (MvPolynomial.eval ⇑(b.repr x)) (((↑(LieAlgebra.ad R L)).polyCharpoly b).coeff (LieAlgebra.rank R L)) ≠ 0

Defined in

Mathlib.Algebra.Lie.Rank

Cited by

0 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraModule.FiniteModule.FreeFintypeDecidableEq