The Hecke skein relation on braid group algebras

Over a commutative ring S with a parameter z, the Hecke skein submodule of the group algebra S[BraidGroup n] is spanned by the elements b * σᵢ * c - b * σᵢ⁻¹ * c - z • (b * c). It is a two-sided ideal, and the quotient by it is the Iwahori-Hecke algebra of type A in the skein normalization gᵢ - gᵢ⁻¹ = z.

The main result is the spanning half of the inductive structure of these Hecke algebras, the decomposition H(n + 2) = H(n + 1) + H(n + 1) σ H(n + 1) with σ the last elementary braid: modulo the Hecke skein submodule, every element of S[BraidGroup (n + 2)] is an S-linear combination of braids of the forms ι b and ι b * σ * ι c, where ι adds an uncrossed last strand and σ is the last elementary braid (TauCeti.BraidGroup.strandSpan_sup_heckeSkein_eq_top). This is the input to the trace-theoretic construction of the HOMFLY polynomial: a Markov trace is determined on the second kind of braid by its value on c * b, and on the first by its value on b, so induction on the number of strands evaluates it everywhere.

Main definitions

• TauCeti.BraidGroup.heckeSkein S z n: the span of the Hecke skein relators in all braid contexts.
• TauCeti.BraidGroup.strandSpan S n: the span of the braids ι b and ι b * σ * ι c in S[BraidGroup (n + 2)].

Main results

• TauCeti.BraidGroup.mul_mem_heckeSkein and TauCeti.BraidGroup.mem_heckeSkein_mul: the Hecke skein submodule is a two-sided ideal.
• TauCeti.BraidGroup.strandSpan_sup_heckeSkein_eq_top: the spanning decomposition.

References

• V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335–388, Section 4 (the decomposition of the Hecke algebras used for the Ocneanu trace).
• C. Kassel, V. Turaev, Braid Groups, Graduate Texts in Mathematics 247, Springer (2008), Section 4.2.

noncomputable def TauCeti.BraidGroup.heckeRelator (S : Type u_1) [CommRing S] (z : S) {n : ℕ} (b c : BraidGroup n) (i : Fin (n - 1)) : MonoidAlgebra S (BraidGroup n)

The Hecke skein relator b * σᵢ * c - b * σᵢ⁻¹ * c - z • (b * c) in the group algebra.

theorem TauCeti.BraidGroup.heckeRelator_def (S : Type u_1) [CommRing S] (z : S) {n : ℕ} (b c : BraidGroup n) (i : Fin (n - 1)) : heckeRelator S z b c i = MonoidAlgebra.single (b * sigma i * c) 1 - MonoidAlgebra.single (b * (sigma i)⁻¹ * c) 1 - MonoidAlgebra.single (b * c) z

The defining formula of a Hecke skein relator.

noncomputable def TauCeti.BraidGroup.heckeSkein (S : Type u_1) [CommRing S] (z : S) (n : ℕ) : Submodule S (MonoidAlgebra S (BraidGroup n))

The Hecke skein submodule of S[BraidGroup n]: the S-span of the Hecke skein relators b * σᵢ * c - b * σᵢ⁻¹ * c - z • (b * c) over all braids b, c and elementary braids σᵢ. It is a two-sided ideal (mul_mem_heckeSkein, mem_heckeSkein_mul).

theorem TauCeti.BraidGroup.heckeRelator_mem_heckeSkein (S : Type u_1) [CommRing S] (z : S) {n : ℕ} (b c : BraidGroup n) (i : Fin (n - 1)) : heckeRelator S z b c i ∈ heckeSkein S z n

Every Hecke skein relator lies in the Hecke skein submodule.

theorem TauCeti.BraidGroup.heckeSkein_le_iff (S : Type u_1) [CommRing S] (z : S) {n : ℕ} {p : Submodule S (MonoidAlgebra S (BraidGroup n))} : heckeSkein S z n ≤ p ↔ ∀ (b c : BraidGroup n) (i : Fin (n - 1)), heckeRelator S z b c i ∈ p

A submodule contains the Hecke skein submodule exactly when it contains every relator.

theorem TauCeti.BraidGroup.mul_mem_heckeSkein (S : Type u_1) [CommRing S] (z : S) {n : ℕ} (x : MonoidAlgebra S (BraidGroup n)) {y : MonoidAlgebra S (BraidGroup n)} (hy : y ∈ heckeSkein S z n) : x * y ∈ heckeSkein S z n

The Hecke skein submodule is a left ideal.

theorem TauCeti.BraidGroup.mem_heckeSkein_mul (S : Type u_1) [CommRing S] (z : S) {n : ℕ} {x : MonoidAlgebra S (BraidGroup n)} (hx : x ∈ heckeSkein S z n) (y : MonoidAlgebra S (BraidGroup n)) : x * y ∈ heckeSkein S z n

The Hecke skein submodule is a right ideal.

theorem TauCeti.BraidGroup.mapDomain_strandIncl_mem_heckeSkein (S : Type u_1) [CommRing S] (z : S) {n : ℕ} {x : MonoidAlgebra S (BraidGroup (n + 1))} (hx : x ∈ heckeSkein S z (n + 1)) : (MonoidAlgebra.mapDomainAlgHom S S strandIncl) x ∈ heckeSkein S z (n + 2)

Adding an uncrossed strand carries the Hecke skein submodule into the Hecke skein submodule of the larger braid group.

noncomputable def TauCeti.BraidGroup.strandSpan (S : Type u_1) [CommRing S] (n : ℕ) : Submodule S (MonoidAlgebra S (BraidGroup (n + 2)))

The span in S[BraidGroup (n + 2)] of the braids ι b and ι b * σ * ι c, where ι adds an uncrossed last strand and σ is the last elementary braid, crossing the new strand.

theorem TauCeti.BraidGroup.strandSpan_le_iff (S : Type u_1) [CommRing S] {n : ℕ} {p : Submodule S (MonoidAlgebra S (BraidGroup (n + 2)))} : strandSpan S n ≤ p ↔ (∀ (b : BraidGroup (n + 1)), MonoidAlgebra.single (strandIncl b) 1 ∈ p) ∧ ∀ (b c : BraidGroup (n + 1)), MonoidAlgebra.single (strandIncl b * sigma (Fin.last n) * strandIncl c) 1 ∈ p

A submodule contains strandSpan exactly when it contains the braids ι b and ι b * σ * ι c.

theorem TauCeti.BraidGroup.single_strandIncl_mem_strandSpan (S : Type u_1) [CommRing S] {n : ℕ} (b : BraidGroup (n + 1)) : MonoidAlgebra.single (strandIncl b) 1 ∈ strandSpan S n

A braid with an uncrossed last strand lies in strandSpan.

theorem TauCeti.BraidGroup.single_strandIncl_mul_sigma_mul_strandIncl_mem_strandSpan (S : Type u_1) [CommRing S] {n : ℕ} (b c : BraidGroup (n + 1)) : MonoidAlgebra.single (strandIncl b * sigma (Fin.last n) * strandIncl c) 1 ∈ strandSpan S n

A braid crossing the last strand once, at the last elementary braid, lies in strandSpan.

The spanning decomposition

theorem TauCeti.BraidGroup.strandSpan_sup_heckeSkein_eq_top (S : Type u_1) [CommRing S] (z : S) (n : ℕ) : strandSpan S n ⊔ heckeSkein S z (n + 2) = ⊤

The spanning decomposition of the braid group algebra modulo the Hecke skein relation. Every element of S[BraidGroup (n + 2)] is, modulo the Hecke skein submodule, an S-linear combination of braids ι b and ι b * σ * ι c, where ι adds an uncrossed last strand and σ is the last elementary braid.