Quantum traces of U_q(sl₂)-modules

The square of the antipode of U_q(sl₂) is conjugation by K⁻¹, so a finite-dimensional module V is identified with its double dual by v ↦ (ξ ↦ ξ(K⁻¹ v)). The two traces of a module endomorphism φ induced by this identification are the twisted traces tr_V(K⁻¹ φ) and tr_V(K φ). This file proves that they coincide, for every module and every module endomorphism, as soon as q - q⁻¹ ≠ 0; no finiteness, genericity or semisimplicity is needed.

Main results

• TauCeti.UqSl2.trace_lsmul_k_comp_eq_trace_lsmul_kInv_comp: tr_V(K φ) = tr_V(K⁻¹ φ) for every module endomorphism φ.

References

• C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Chapter XIV, for quantum traces and quantum dimensions of modules over Hopf algebras.
• A. Virelizier, Quantum groups, Section 2.3, for the rank-one Hopf convention.

theorem TauCeti.UqSl2.trace_lsmul_k_comp_eq_trace_lsmul_kInv_comp {q : ℂ} {V : Type u_1} [AddCommGroup V] [Module ℂ V] [Module (UqSl2 q) V] [IsScalarTower ℂ (UqSl2 q) V] (hq : q - q⁻¹ ≠ 0) (φ : V →ₗ[UqSl2 q] V) : (LinearMap.trace ℂ V) ((Algebra.lsmul ℂ ℂ V) (k q) ∘ₗ ↑ℂ φ) = (LinearMap.trace ℂ V) ((Algebra.lsmul ℂ ℂ V) (kInv q) ∘ₗ ↑ℂ φ)

The two quantum traces agree. For q - q⁻¹ ≠ 0 and every UqSl2 q-linear endomorphism φ of a module V, the traces of K ∘ φ and of K⁻¹ ∘ φ coincide.