Restricts quantum representations of mapping class groups to integral coefficients.
problem Integrality of non-semisimple quantum representations of mapping class groups.
method Exhibits explicit bases of states spaces that span Z[ζ]-lattices invariant under mapping class groups. result Restricts quantum representations to integral coefficients from Q(ζ) to Z[ζ]. We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum…
The paper explores mapping class groups and their quantum field theory representations.
problem Understanding finite dimensional representations of mapping class groups.
method Survey of topological quantum field theory aspects.
result Discussion of finite dimensional representations in quantum field theory.
This work constructs a finite-dimensional projective representation for a quantum Teichmüller model.
problem Quantum Teichmüller theory and its finite-dimensional representation.
method Explicit construction using cyclic quantum dilogarithm and mutations of coefficients.
result Reconstruction of quantum Teichmüller space with explicit intertwiners.
Quantum representations of mapping class groups are locally rigid at prime levels.
problem Locally rigid properties of quantum representations of mapping class groups.
method Proving local rigidity for Fibonacci representations of mapping class groups at prime levels.
result Local rigidity of Fibonacci representations of mapping class groups at prime levels.
Proves rigidity of SU(2) and SO(3) quantum representations at prime levels.
problem Quantum representations of mapping class groups at prime levels.
method Ocneanu rigidity of modular categories and harmonic representatives in Hodge theory.
result Rigidity of SU(2) and SO(3) quantum representations at all prime levels for closed surfaces of genus at least 7.
We generalize the asymptotic faithfulness of the skein quantum SU(2) representations of mapping class groups of orientable closed surfaces to skein SU(3). Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…
GQML uses symmetries from representation theory to improve quantum machine learning.
problem Creating quantum models with symmetries to improve performance.
method Introduction to representation theory for quantum learning, focusing on group actions and symmetries.
result Effective implementation of GQML requires knowledge of group representation theory.
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor produ…
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
New structure for quantum algebra representations.
problem Understanding representations of quantum algebras.
method Constructing a quotient category of annular quantum gln webs. result Equivalent to finite dimensional representations of quantum Levi subalgebras.
The paper establishes isomorphisms and constructs colored versions of Lawrence representations.
problem Understanding isomorphisms and colored versions of Lawrence representations.
method Explicit isomorphisms and construction of colored versions.
result Matrices for colored versions of BKL and Lawrence representations provided.
QCML uses quantum geometry to represent data.
problem Data representation and the curse of dimensionality.
method QCML encodes data as Hermitian matrices in Hilbert space.
result Data geometry reveals intrinsic dimension and topological properties.
Study of webs in quantum type C, proving equivalence to quantum representations.
problem Equivalence of webs and quantum representations in type C.
method Defined a linear pivotal category diagrammatically, conjectured equivalence, and proved functorial results.
result Full, essentially surjective functor between web and quantum representation categories.
Homological model for quantum representations of mapping class groups.
problem Investigate linearity of mapping class groups using quantum representations.
method Homological action on configuration space with twisted coefficients.
result Identify subrepresentation equivalent to quantum sl2 representation. Quantum theory uses modular group representations to assign invariants to 3-manifolds.
problem Assigning invariants to 3-manifolds via modular group representations.
method Projective representations of the modular group derived from a noncommutative torus.
result Computed traces and determinants of matrices associated with modular group elements.
We adapt some of the methods of quantum Teichmüller theory to construct a family of representations of the pure braid group of the sphere.
Researchers define new quantum representations for a Lorentz algebra and study their Clebsch-Gordan decomposition.
problem Quantum representations of a Lorentz algebra and their Clebsch-Gordan decomposition.
method Defined new infinite-dimensional irreducible representations using quantum torus algebra and quantized Chern-Simons theory.
result The Clebsch-Gordan decomposition of tensor product representations reduces to problems in Fenchel-Nielson length operators in quantized Chern-Simons theory.
Extended quantum state result for gl_n weight systems.
problem Quantum states associated with gl_n weight systems.
method Extended Corfield et al. result to all gl_n weight systems.
result All gl_n weight systems are quantum states.
Quantum groups created from disk configuration space homologies.
problem Creating quantum groups from algebraic structures.
method Reconstructing quantum groups from homologies of configuration spaces of disks.
result New combinatorics and actual submanifolds of configuration spaces.
Quantum circuits represent binary classification trees with binary features.
problem Classifying data using binary classification trees with binary features.
method Quantum circuits and probabilistic approach for traversing decision trees.
result First realization of a decision tree classifier on a quantum device.
Quantum gravity yields mapping class group representations.
problem Quantization of 3-manifold metrics and mapping class group invariance.
method Quantum dilogarithm functions and mapping class group action.
result Families of unitary representations of mapping class groups.
Researchers create projective representations of Hecke groups using TQFT.
problem Constructing projective representations of Hecke groups.
method Using Witten-Reshetikhin-Turaev topological quantum field theory of higher genus surfaces.
result The representation's image group is infinite at low levels in genus 2.
The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
"Thick" or "microformal" morphisms of supermanifolds generalize ordinary maps. They were discovered as a tool for homotopy algebras. Namely, the corresponding pullbacks provide L∞-morphisms for S∞ or Batalin--Vilkovisky algebras. It was clear from the start that constructions used for thick morphism…
New proof and formula linking fusion trees to quantum knot invariants.
problem Quantum knot invariants encoding in non-semisimple TQC.
method Connection between fusion trees and Lawrence representations, using graphical calculus.
result Explicit encoding of quantum knot invariants via fusion trees.
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting R-matrix of Uq(sl2) as cluster transformation, introducing auxiliary parameter ε. result Derives perturbed-Alexander invariants with higher-order terms in ε. Quantum invariants derived from Uq(sl2) link holonomy.
problem Quantum invariants of links and their relations.
method Using quantum groups and Schur-Weyl duality, constructing quantum holonomy invariants.
result Quantum invariants of links can be derived from Uq(sl2) representations. New properties established for SO(3) quantum representations, showing density and surjectivity.
problem Properties of SO(3) quantum representations of mapping class groups.
method Analyzing roots of unity and maximal ideals of Z[ζ_p] to establish properties.
result SO(3) quantum representations have dense image and are surjective modulo unramified maximal ideals.
Researchers link knot Floer homology, Burau representation, and quantum gl(1|1).
problem Understanding the Burau representation and its relation to knot Floer homology.
method Developed a Heegaard Floer homology theory and associated a bordered sutured Heegaard Floer homology group to any tangle.
result Established a connection between the Burau representation and quantum gl(1|1), leading to a geometric proof of the braid representation.
We develop a diagrammatic calculus for representations of unrolled quantum sl2 at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given h…
We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).
We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum repr…
The paper proves properties of quantum representations and their Toledo invariants.
problem Proving properties of quantum representations and their Toledo invariants.
method Computing Toledo invariants for specific quantum representations and extending the concept to a series of cohomological invariants.
result The proof of properties of quantum representations and their Toledo invariants, including the computation of the R-matrix at first order. Simpler equations derived for knot polynomials coefficients, forming a ring.
problem Complexity of knot polynomials colored with symmetric representations.
method Deriving two difference equations for quantum C-polynomials coefficients.
result Quantum C-polynomials form a ring and are much simpler than colored polynomials.
We establish various results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat SU(n)-bundle. Working with the metaplectic correction, we proved that these projective representations lift to asymptotic representations. We show that…
We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid grou…
R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group Uqsl2 at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin-Turae…
New quantum algebra connects 3D gravity to complex plane.
problem Quantize 3D gravity with positive cosmological constant.
method Introduced quantum pseudo-Kähler plane and studied its representations.
result Found new operators for 3D gravity quantization.
We study the asymptotic behaviour of the quantum representations of the modular group in the large level limit. We prove that each element of the modular group acts as a Fourier integral operator. This provides a link between the classical and quantum Chern-Simons theories for the torus. From this result we deduce the …
We study the TQFT mapping class group representations for surfaces with boundary associated with the SU(2) gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental valu…
We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein …
The Kashaev invariants of 3-manifolds are based on 6j-symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of $U_q(sl(2,\C))$. In this paper, we show that Kashaev's 6j-symbols are intertwining operators of local representations of quantum Teichmüller space…
For each oriented surface Σ of genus g we study a limit of quantum representations of the mapping class group arising in TQFT derived from the Kauffman bracket. We determine that these representations converge in the Fell topology to the representation of the mapping class group on $\boH(Σ)$, the space of regular f…
Quantum circuits reveal pathways to dequantization in machine learning models.
problem Navigating the complex landscape of quantum machine learning models and algorithms.
method Introducing a framework connecting quantum circuit structure to function representability.
result Fundamental properties of quantum circuits determine classical simulability of models.
We present state sums for quantum link invariants arising from the representation theory of Uq(glN∣M). We investigate the case of the N-th exterior power of the standard representation of Uq(glN∣1) and explicit the relation with Kashaev invariants.
We show how to define invariants of graphs related to quantum sl(2) when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.