Related papers: Quantum torus methods for Kauffman bracket skein m…
We construct invertible field theories generalizing abelian prequantum spin Chern-Simons theory to manifolds of dimension 4k+3 endowed with a Wu structure of degree 2k+2. After analysing the anomalies of a certain discrete symmetry, we…
The Reshetikhin - Turaeve approach to topological invariants of three - manifolds is generalized to quantum supergroups. A general method for constructing three - manifold invariants is developed, which requires only the study of the…
Techniques from higher categories and higher-dimensional rewriting are becoming increasingly important for understanding the finer, computational properties of higher algebraic theories that arise, among other fields, in quantum…
The Kashaev-Murakami-Murakami Volume Conjecture connects the hyperbolic volume of a knot complement to the asymptotics of certain evaluations of the colored Jones polynomials of the knot. We introduce a closely related conjecture for…
We resolve two questions of Cauchon and Meriaux on the spectra of the quantum Schubert cell algebras U^-[w]. The treatment of the first one unifies two very different approaches to Spec U^-[w], a ring theoretic one via deleting derivations…
Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of…
The goal of this paper is to establish the classification of all homogeneous surfaces of 3-sphere by using the moving frame method. We will show that such surfaces are 2-spheres and flat torus.
On certain M-theory backgrounds which are a circle fibration over a smooth Calabi-Yau, the quantum theory of M2 branes can be studied in terms of the K-theoretic Donaldson-Thomas theory on the threefold. We extend this relation to…
We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its variants. These algorithms have proven effective in devising higher-order methods for solving the time-dependent Schr\"{o}dinger equation in the…
We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is…
An important step in the determination of the automorphism group of the quantum torus of rank $n$ (or twisted group algebra of $\mathbb Z^n$) is the determination of its so-called non-scalar automorphisms. We present a new algorithimic…
In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and investigate the transformation property of…
In the paper, we show some properties of (reduced) stated $\mathrm{SL}(n)$-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees.…
Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form $q\otimes q \cong \mathbf{1}\oplus\bigoplus^k_{i=1}x_{i}$ in a UFC $\mathcal{C}$, and…
Let $K$ denote the contact Lie superalgebra $K(m,n;\underline{t})$ over a field of characteristic $p>3$, which has a finite $\mathbb{Z}$-graded structure. Let $T_K$ be the canonical torus of $K$, which is an abelian subalgebra of $K_{0}$…
This paper constructs (with challenging obstacles) on the three torus with its cubical decomposition: Firstly, a combinatorial graded intersection algebra (graded by the codimension) which is commutative and associative defined by…
In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which…
The quantum modularity conjecture, first introduced by Don Zagier, is a general statement about a relation between $\mathfrak{sl}_2$ quantum invariants of links and 3-manifolds at roots of unity related by a modular transformation. In this…
We determine the structure of the Kauffman bracket skein module of the connected sum of two genus one handlebodies over the ring of Laurent polynomials $\mathbb Z[q^{\pm 1}]$, thereby proving a conjecture posed by the first and third…
Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Talk aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras.…