Related papers: Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: Th…
We establish the existence of several quantum trace maps. The simplest one is an algebra map between two quantizations of the algebra of regular functions on the $SL_n$-character variety of a surface $\mathfrak{S}$ equipped with an ideal…
Quantization of the Teichm\"uller space of a punctured Riemann surface $S$ is an approach to $3$-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop $\gamma$ in $S$ gives rise to…
We define a quantum trace map from the skein module of a 3-manifold with torus boundary components to a module (left and right quotient of a quantum torus) constructed from an ideal triangulation. Our map is a 3-dimensional version of the…
The ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$, studied by Sikora, is an algebra generated by isotopy classes of $n$-webs living in the thickened surface $\mathfrak{S} \times (-1,1)$, where an $n$-web is a union of…
Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed ${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent examples of cluster $\mathscr{X}$-varieties and higher Teichm\"uller spaces. In a…
In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic…
We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the…
We construct the 3d quantum trace map, a homomorphism from the Kauffman bracket skein module of an ideally triangulated 3-manifold to its (square root) quantum gluing module, thereby giving a precise relationship between the two…
We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this…
We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and…
The q-deformed traces and orbits for the two parametric quantum groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$ are defined. They are subsequently used in the construction of $q$-orbit invariants for these groups. General $qp$-(super)oscillator…
We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement $S^3\backslash \mathcal{K}$. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives…
We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$…
Quantum deformations of sets of points of the real and the complexified projective line are constructed. These deformations depend on the deformation parameter q and certain further parameters \lambda_{ij}. The deformations for which the…
This work finishes a classification of $U_q(\mathfrak{sl}_2)$-symmetries on the Laurent extension $\mathbb{C}_q\big[x^{\pm 1},y^{\pm 1}\big]$ of the quantum plane. After reproducing the partial results of a previous paper of the author…
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface.…
We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum…
Quantum invariants like the colored Jones polynomial are algebraic in nature but are conjectured to detect important information about the geometry of links. In this thesis we explore these connections using an enhanced version of the RT…
We define a map from the skein module of a cusped hyperbolic 3-manifold to the ring of Laurent series in one variable with integer coefficients that satisfies two properties: its evaluation at peripheral curves coincides with the…
We study $N^2-1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$ for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on $SL_q(2)$…