Centers and representations of ${\rm SL}_n$ quantum Teichmüller spaces

TL;DR

This work determines the center and the rank of the balanced Fock–Goncharov algebra at roots of unity, and uses this to classify irreducible representations of the algebra. It then connects these representations to irreducible representations of the projected SL$_n$–skein algebra via the Frobenius map and quantum trace, showing that a large subset of the SL$_n$ character variety can arise as the classical shadows of skein representations. The results yield a robust understanding of triangulation-invariance (naturality) of these representations and provide a concrete description of how central data govern representation theory in quantum Teichmüller–type spaces for SL$_n$. Overall, the paper advances the structural theory of SL$_n$ quantum Teichmüller spaces by linking centers, ranks, and irreducible representations to classical character-theoretic data. These insights have potential implications for the study of SL$_n$–skein algebras and their classical shadows across punctured surfaces.

Abstract

In this paper, we compute the center of the balanced Fock-Goncharov algebra and determine its rank over the center when the quantum parameter is a root of unity. These results have potential applications to the study of the center and rank of the ${\rm SL}_n$-skein algebra. Building on this computation, we classify the irreducible representations of the balanced Fock-Goncharov algebra. Due to the Frobenius homomorphism, every irreducible representation of the (projected) ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$ determines a point in the ${\rm SL}_n$ character variety of $\mathfrak{S}$, known as the classical shadow of the representation. By pulling back the irreducible representations of the balanced Fock-Goncharov algebra via the quantum trace map, we show that there exists a ``large'' subset of the ${\rm SL}_n$ character variety such that, for any point in this subset, there exists an irreducible representation of the (projected) ${\rm SL}_n$-skein algebra whose classical shadow is this point. Finally, we prove that, under mild conditions, the representations of the ${\rm SL}_n$-skein algebra obtained in this way are independent of the choice of ideal triangulation.

Figures (11)

• Figure 1: The central puncture is $p$. The blue line is at distance $1$ from $p$, the red line at distance $2$, and the green line at distance $3$.
• Figure 2: The left is $C(p)_{ij}$ and the right is $\reflectbox{\vec{\reflectbox{C}}}(p)_{ij}$.
• Figure 3: Barycentric coordinates $ijk$ and a $4$-triangulation with its quiver
• Figure 4: (a) The twice punctured sphere $\mathsf A$. (b) A knot in $\mathsf A$. (c) The $n$-web diagram $\alpha_k'$ in $\mathsf{A}$.
• Figure 5: The arc $c_p$ and the enclosed monogon $D_p$.

Theorems & Definitions (108)

• Theorem 1.1: Theorems \ref{['thm-center-balanced-root-of-unity']} and \ref{['thm-rank-Z']}
• Corollary 1.2: Corollary \ref{['lam-decom']}
• Theorem 1.3: Theorem \ref{['thm-representation-Fock']}
• Theorem 1.4: Theorem \ref{['thm-main-3']}
• Theorem 1.5: Theorem \ref{['thm-naturality']}
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5