Second Frobenius–Schur indicators of simple objects

For an object V with a right dual in a k-linear monoidal category, Frobenius reciprocity identifies the invariant space 𝟙_ C ⟶ tensorPow V 2 with Vᘁ ⟶ V (TauCeti.homTensorPowTwoLinearEquiv).

In a k-linear pivotal category with kernels, Schur's lemma controls this hom space whenever V and Vᘁ are simple (which holds automatically for Vᘁ when C is abelian, by TauCeti.simple_rightDual):

• If V is not isomorphic to Vᘁ, then Vᘁ ⟶ V is zero and ν₂(V) = 0 over any commutative ring k (TauCeti.frobeniusSchurIndicator_two_eq_zero_of_isEmpty_iso).
• Over an algebraically closed field k (of any characteristic), if V ≅ Vᘁ and V ⟶ V is finite-dimensional, then 𝟙_ C ⟶ tensorPow V 2 is one-dimensional and E_V^{(2)} is an involution on it, so ν₂(V) = 1 or ν₂(V) = -1 (TauCeti.frobeniusSchurIndicator_two_eq_one_or_eq_neg_one_of_nonempty_iso).

Combining the two cases gives the Ng–Schauenburg trichotomy ν₂(V) ∈ {1, 0, -1} (TauCeti.frobeniusSchurIndicator_two_eq_one_or_eq_zero_or_eq_neg_one) and the self-duality criterion ν₂(V) ≠ 0 ↔ Nonempty (V ≅ Vᘁ) (TauCeti.frobeniusSchurIndicator_two_ne_zero_iff), valid in arbitrary characteristic.

Main declarations

• TauCeti.frobeniusSchurIndicator_two_eq_zero_of_isEmpty_iso and TauCeti.frobeniusSchurIndicator_two_eq_one_or_eq_neg_one_of_nonempty_iso: the non-self-dual and self-dual cases for ν₂(V);
• TauCeti.frobeniusSchurIndicator_two_eq_one_or_eq_zero_or_eq_neg_one: the trichotomy ν₂(V) = 1 ∨ ν₂(V) = 0 ∨ ν₂(V) = -1 for simple objects;
• TauCeti.frobeniusSchurIndicator_two_ne_zero_iff and TauCeti.frobeniusSchurIndicator_two_eq_zero_iff: ν₂(V) ≠ 0 iff V ≅ Vᘁ.

References

• S.-H. Ng, P. Schauenburg, Higher Frobenius–Schur indicators for pivotal categories, in Hopf algebras and generalizations, Contemp. Math. 441 (2007), 63–90, §§4–5
• [P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories][egno15], §4.7

theorem TauCeti.frobeniusSchurIndicator_two_eq_zero_of_isEmpty_iso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Preadditive C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.RigidCategory C] [PivotalCategory C] [CategoryTheory.Limits.HasKernels C] (k : Type w) [CommRing k] [CategoryTheory.Linear k C] [CategoryTheory.MonoidalLinear k C] (V : C) [CategoryTheory.Simple V] [CategoryTheory.Simple Vᘁ] (h : IsEmpty (V ≅ Vᘁ)) : frobeniusSchurIndicator k V 2 = 0

If V and Vᘁ are simple and V is not isomorphic to Vᘁ, then ν₂(V) = 0.

theorem TauCeti.frobeniusSchurIndicator_two_eq_one_or_eq_neg_one_of_nonempty_iso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Preadditive C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.RigidCategory C] [PivotalCategory C] [CategoryTheory.Limits.HasKernels C] (k : Type w) [Field k] [IsAlgClosed k] [CategoryTheory.Linear k C] [CategoryTheory.MonoidalLinear k C] (V : C) [CategoryTheory.Simple V] [FiniteDimensional k (V ⟶ V)] (h : Nonempty (V ≅ Vᘁ)) : frobeniusSchurIndicator k V 2 = 1 ∨ frobeniusSchurIndicator k V 2 = -1

If V is simple with finite-dimensional endomorphisms over an algebraically closed field and V ≅ Vᘁ, then ν₂(V) = 1 or ν₂(V) = -1.

theorem TauCeti.frobeniusSchurIndicator_two_eq_one_or_eq_zero_or_eq_neg_one {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Preadditive C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.RigidCategory C] [PivotalCategory C] [CategoryTheory.Limits.HasKernels C] (k : Type w) [Field k] [IsAlgClosed k] [CategoryTheory.Linear k C] [CategoryTheory.MonoidalLinear k C] (V : C) [CategoryTheory.Simple V] [FiniteDimensional k (V ⟶ V)] [CategoryTheory.Simple Vᘁ] : frobeniusSchurIndicator k V 2 = 1 ∨ frobeniusSchurIndicator k V 2 = 0 ∨ frobeniusSchurIndicator k V 2 = -1

Ng–Schauenburg trichotomy: for a simple object V (with Vᘁ simple and V ⟶ V finite-dimensional over an algebraically closed field k), the second Frobenius–Schur indicator takes values in {1, 0, -1}.

theorem TauCeti.frobeniusSchurIndicator_two_ne_zero_iff {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Preadditive C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.RigidCategory C] [PivotalCategory C] [CategoryTheory.Limits.HasKernels C] (k : Type w) [Field k] [IsAlgClosed k] [CategoryTheory.Linear k C] [CategoryTheory.MonoidalLinear k C] (V : C) [CategoryTheory.Simple V] [FiniteDimensional k (V ⟶ V)] [CategoryTheory.Simple Vᘁ] : frobeniusSchurIndicator k V 2 ≠ 0 ↔ Nonempty (V ≅ Vᘁ)

For a simple object V (with Vᘁ simple and V ⟶ V finite-dimensional over an algebraically closed field k), ν₂(V) ≠ 0 if and only if V is self-dual (Nonempty (V ≅ Vᘁ)).

theorem TauCeti.frobeniusSchurIndicator_two_eq_zero_iff {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Preadditive C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.RigidCategory C] [PivotalCategory C] [CategoryTheory.Limits.HasKernels C] (k : Type w) [Field k] [IsAlgClosed k] [CategoryTheory.Linear k C] [CategoryTheory.MonoidalLinear k C] (V : C) [CategoryTheory.Simple V] [FiniteDimensional k (V ⟶ V)] [CategoryTheory.Simple Vᘁ] : frobeniusSchurIndicator k V 2 = 0 ↔ IsEmpty (V ≅ Vᘁ)

For a simple object V (with Vᘁ simple and V ⟶ V finite-dimensional over an algebraically closed field k), ν₂(V) = 0 if and only if V is not isomorphic to Vᘁ.