Right ideals of free group algebras

DefinitionTwisting character of the free group of rank two#✓fully proved

Let \(F\) be the free group on \(x, y\). For nonzero complex numbers \(a, b\), this is the homomorphism \(\chi \colon F \to \mathbb{C}^\times\) with \(\chi(x) = a\) and \(\chi(y) = b\).

noncomputable def twistCharacter (a b : ℂˣ) : FreeGroup (Fin 2) →* ℂˣ :=
  FreeGroup.lift ![a, b]

Used by 1

Used byTwisting automorphism of the free group algebra

DefinitionTwisting automorphism of the free group algebra#✓fully proved

Let \(F\) be the free group on \(x, y\), let \(\mathbb{C}F\) be its complex group algebra, and for nonzero complex numbers \(a, b\) let \(\chi\) be the character of \(F\) with \(\chi(x) = a\) and \(\chi(y) = b\). The map \(\vartheta_{a,b}\) sends \(\alpha = \sum_{g \in F} c_g\, g \in \mathbb{C}F\) to \(\sum_{g \in F} \chi(g)\, c_g\, g\).

Since \(\chi\) is a homomorphism, \(\vartheta_{a,b}\) is the algebra automorphism of \(\mathbb{C}F\) with \(\vartheta_{a,b}(x) = a x\) and \(\vartheta_{a,b}(y) = b y\).

noncomputable def theta (a b : ℂˣ) (α : MonoidAlgebra ℂ (FreeGroup (Fin 2))) :
    MonoidAlgebra ℂ (FreeGroup (Fin 2)) :=
  α.coeff.sum fun g c => MonoidAlgebra.single g ((twistCharacter a b g : ℂ) * c)

Uses 1 · Used by 1

Statement usesTwisting character of the free group of rank two

Used byLinnell's twisted right ideal question (Kourovka 12.46)

DefinitionLinnell's twisted right ideal question (Kourovka 12.46)#✓fully proved

Let \(F\) be the free group on \(x, y\), let \(\mathbb{C}F\) be its complex group algebra, and for nonzero complex numbers \(a, b\) let \(\vartheta_{a,b}\) be the twisting automorphism of \(\mathbb{C}F\) with \(\vartheta_{a,b}(x) = a x\) and \(\vartheta_{a,b}(y) = b y\). The assertion is that for every nonzero \(\alpha \in \mathbb{C}F\) there are complex numbers \(a, b \neq 1\) with \(|a| = |b| = 1\) such that the right ideals \(\alpha\,\mathbb{C}F\) and \(\vartheta_{a,b}(\alpha)\,\mathbb{C}F\) intersect nontrivially, that is, \(\alpha\beta = \vartheta_{a,b}(\alpha)\gamma \neq 0\) for some \(\beta, \gamma \in \mathbb{C}F\).

def twistedRightIdealsIntersect : Prop :=
  ∀ α : MonoidAlgebra ℂ (FreeGroup (Fin 2)), α ≠ 0 →
    ∃ a b : ℂˣ, ‖(a : ℂ)‖ = 1 ∧ ‖(b : ℂ)‖ = 1 ∧ (a : ℂ) ≠ 1 ∧ (b : ℂ) ≠ 1 ∧
      ∃ β γ : MonoidAlgebra ℂ (FreeGroup (Fin 2)), α * β = theta a b α * γ ∧ α * β ≠ 0

Uses 1 · Used by 1

Statement usesTwisting automorphism of the free group algebra

Used byLinnell's twisted right ideal question has a negative answer (Kourovka 12.46)

TheoremLinnell's twisted right ideal question has a negative answer (Kourovka 12.46)#●ready to prove

Let \(F\) be the free group on \(x, y\), let \(\mathbb{C}F\) be its complex group algebra, and for nonzero complex numbers \(a, b\) let \(\vartheta_{a,b}\) be the twisting automorphism of \(\mathbb{C}F\) with \(\vartheta_{a,b}(x) = a x\) and \(\vartheta_{a,b}(y) = b y\). There is a nonzero \(\alpha \in \mathbb{C}F\) such that for all complex numbers \(a, b \neq 1\) with \(|a| = |b| = 1\) the right ideals \(\alpha\,\mathbb{C}F\) and \(\vartheta_{a,b}(\alpha)\,\mathbb{C}F\) intersect only in \(0\): whenever \(\beta, \gamma \in \mathbb{C}F\) satisfy \(\alpha\beta = \vartheta_{a,b}(\alpha)\gamma\), then \(\alpha\beta = 0\).

The Kourovka Archive answers 12.46 "No, not always" (A. V. Tushev, 1995).

public theorem_wanted twistedRightIdealsIntersect_refuted : ¬ twistedRightIdealsIntersect

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 12.46 (P. A. Linnell); resolution: A. V. Tushev, Ukrain. Mat. Zh. 47 (1995), no. 4, 571-572, Kourovka Archive, Problem 12.46: No, not always.

Uses 1

Statement usesLinnell's twisted right ideal question (Kourovka 12.46)

3 fully proved1 ready to prove

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