instHasTrivialRadical — Mathlib

English

Assume K is algebraically closed and P.IsReduced, P.IsIrreducible. Then the radical of the Lie algebra lieAlgebra(b) is trivial; equivalently, the Lie algebra is semisimple.

Русский

Пусть K алгебраически закрытое поле, P редуцирована и неприводима. Тогда радикал \n\\operatorname{Rad}(\\mathfrak{g}) у \n\\mathfrak{g}=\\text{lieAlgebra}(b) тривиален; эквивалентно, алгебра Ли semisimple.

LaTeX

$$$\\operatorname{Rad}\\bigl(\\mathrm{lieAlgebra}(b)\\bigr)=\\{0\\}$$$

Lean4

/-- Lemma 4.3 from [Geck](Geck2017). -/
instance instHasTrivialRadical [IsAlgClosed K] : LieAlgebra.HasTrivialRadical K (lieAlgebra b) :=
  by
  cases isEmpty_or_nonempty ι
  · infer_instance
  · exact LieAlgebra.hasTrivialRadical_of_isIrreducible_of_isFaithful K _ _ trace_toEnd_eq_zero

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