The Chebyshev–Frobenius homomorphism for stated skein modules of 3-manifolds

Abstract

We study the stated skein modules of marked $3-$manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of $3-$manifolds. We show that there exists a Chebyshev–Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev–Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev–Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of ${\mathcal {O}}_{q^2}(SL(2))$ through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig’s Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev–Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.

A Divided powers and the Hopf pairing

In this appendix we provide the computations required to show that $\varPhi _\omega$ is the Hopf dual of the Frobenius homomorphism.

Note that the Hopf pairing

$$\begin{aligned} \langle , \rangle _{q^2}:U_{q^2}({\mathfrak {sl}}_2) \otimes {\mathcal {O}}_{q^2}(SL(2))\rightarrow {\mathbb {Q}}(q^{1/2}), \end{aligned}$$

is defined on generators as

$$\begin{aligned}&\left\langle \left( \begin{array}{cc} a &{} b\\ c &{} d \end{array}\right) ,K\right\rangle =\left( \begin{array}{cc} q^2 &{} 0\\ 0 &{} q^{-2} \end{array}\right) \\&\left\langle \left( \begin{array}{cc} a &{} b\\ c &{} d \end{array}\right) ,E\right\rangle =\left( \begin{array}{cc} 0 &{} 1\\ 0 &{} 0 \end{array}\right) \\&\left\langle \left( \begin{array}{cc} a &{} b\\ c &{} d \end{array}\right) ,F\right\rangle =\left( \begin{array}{cc} 0 &{} 0\\ 1 &{} 0 \end{array}\right) \end{aligned}$$

This non-degenerate Hopf pairing $\langle , \rangle$ restricted to $U^L_{q^2}({\mathfrak {sl}}_2)$ is integral, meaning

$$\begin{aligned} U^L_{q^2}({\mathfrak {sl}}_2)\otimes {\mathcal {O}}_{q^2} (SL(2))\rightarrow {\mathbb {Z}}[q^{\pm 1/2}]. \end{aligned}$$

Proposition 5

With the notation of Sect. 6, we have for any m and any p.

$$\begin{aligned} \left\langle \left( \begin{array}{cc} a^m &{} b^m \\ c^m &{} d^m \end{array}\right) , K\right\rangle _{q^2} =\left( \begin{array}{cc} q^{2m} &{} 0 \\ 0 &{} q^{-2m} \end{array}\right) \end{aligned}$$

(49)

and

$$\begin{aligned} \left\langle \left( \begin{array}{cc} a^m &{} b^m \\ c^m &{} d^m \end{array}\right) , E^{(p)}\right\rangle _{q^2} =\left( \begin{array}{cc} \delta _{0,p} &{} \delta _{m,p} \\ 0 &{} \delta _{0,p} \end{array}\right) \end{aligned}$$

(50)

and

$$\begin{aligned} \left\langle \left( \begin{array}{cc} a^m &{} b^m \\ c^m &{} d^m \end{array}\right) , F^{(p)}\right\rangle _{q^2} =\left( \begin{array}{cc} \delta _{0,p} &{} 0 \\ \delta _{m,p} &{} \delta _{0,p} \end{array}\right) \end{aligned}$$

(51)

Proof

This result follows from a direct computation based induction using the coproduct in $U_{q^2}({\mathfrak {sl}}_{{\mathfrak {2}}})$. $\square$

Proposition 6

Let $\omega$ be a root of unity with $N=\mathrm{ord}(\omega ^8)$ and $\eta =\omega ^{N^2}$. We have that the Frobenius map f and the map $\varPhi _\omega$ are dual with respect to the Hopf pairings between ${\mathcal {O}}_{\eta ^4}(SL(2))$ and $U_{\eta ^4}({\mathfrak {sl}}_2)$ and between ${\mathcal {O}}_{\omega ^4}(SL(2))$ and $U_{\omega ^4}({\mathfrak {sl}}_2)$, meaning the following diagram commutes:

Proof

This is a direct computation on the generators of the algebras. We will use matrix notation to indicate the map applied to each entry of the matrix.

$$\begin{aligned}&\left\langle \left( \begin{array}{cc} a &{} b \\ c &{} d \end{array}\right) , (-1)^{N+1} K\right\rangle _{\eta ^4}\\&=\left( \begin{array}{cc} (-1)^{N+1}\eta ^{4} &{} 0 \\ 0 &{} (-1)^{N+1}\eta ^{-4} \end{array}\right) =\left( \begin{array}{cc} \omega ^{-4N^2+4N}\omega ^{4N^2} &{} 0 \\ 0 &{} \omega ^{4N^2-4N}\omega ^{-4N^2} \end{array}\right) \\&=\left( \begin{array}{cc} \omega ^{4N} &{} 0 \\ 0 &{} \omega ^{-4N} \end{array}\right) =\left\langle \left( \begin{array}{cc} a^N &{} b^N \\ c^N &{} d^N \end{array}\right) , K\right\rangle _{\omega ^4}\\ \end{aligned}$$

Where the first equality uses the definition of the pairing, the second equality uses that $\eta =\omega ^{N^2}$, and the last equality comes from Equation 49 with $m=N$.

$$\begin{aligned}&\left\langle \left( \begin{array}{cc} a &{} b \\ c &{} d \end{array}\right) , 0\right\rangle _{\eta ^4} =\left( \begin{array}{cc} 0 &{} 0 \\ 0 &{} 0 \end{array}\right) =\left\langle \left( \begin{array}{cc} a^N &{} b^N \\ c^N &{} d^N \end{array}\right) , E\right\rangle _{\omega ^4} \end{aligned}$$

Where the first equality uses the definition of the pairing and the second equality uses Equation 50 with $m=N$ and $p=1$.

$$\begin{aligned}&\left\langle \left( \begin{array}{cc} a &{} b \\ c &{} d \end{array}\right) , 0\right\rangle _{\eta ^4} =\left( \begin{array}{cc} 0 &{} 0 \\ 0 &{} 0 \end{array}\right) =\left\langle \left( \begin{array}{cc} a^N &{} b^N \\ c^N &{} d^N \end{array}\right) , F\right\rangle _{\omega ^4} \end{aligned}$$

Where the first equality uses the definition of the pairing and the second equality uses Equation 51 with $m=N$ and $p=1$.

$$\begin{aligned}&\left\langle \left( \begin{array}{cc} a &{} b \\ c &{} d \end{array}\right) , E\right\rangle _{\eta ^4} =\left( \begin{array}{cc} 0 &{} 1 \\ 0 &{} 0 \end{array}\right) =\left\langle \left( \begin{array}{cc} a^N &{} b^N \\ c^N &{} d^N \end{array}\right) , E^{(N)}\right\rangle _{\omega ^4} \end{aligned}$$

Where the first equality uses the definition of the pairing and the second equality uses Equation 50 with $m=N$ and $p=N$.

$$\begin{aligned}&\left\langle \left( \begin{array}{cc} a &{} b \\ c &{} d \end{array}\right) , F\right\rangle _{\eta ^4} =\left( \begin{array}{cc} 0 &{} 0 \\ 1 &{} 0 \end{array}\right) =\left\langle \left( \begin{array}{cc} a^N &{} b^N \\ c^N &{} d^N \end{array}\right) , F^{(N)}\right\rangle _{\omega ^4}. \end{aligned}$$

Where the first equality uses the definition of the pairing and the second equality uses Equation 51 with $m=N$ and $p=N$. $\square$