Related papers: A quantum shuffle approach to quantum determinants

Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism sigma, and let A be an N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for all n.…

We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally-defined flat connection whose holonomy lies in the group of…

For a commutative cocommutative Hopf algebra we study the relationship between a certain linear map defined via a bicharacter, an exponential of a quadratic differential operator and a $\bullet$ product obtained via twisting by a…

We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincar\'e type determinant for operators on the torus $\Tn$ and deduce formulas for determinants of periodic…

In the project AQV the issue of quantum constraints, KMS-states and algebras of quantum configuration and momentum variables in Loop Quantum Gravity has been argued. There a physical algebra has been required to contain complete observables…

As a particular one parameter deformation of the quantum determinant, we introduce a quantum $\alpha$-determinant and study the $\mathcal{U}_q(\mathfrak{gl}_n)$-cyclic module generated by it: We show that the multiplicity of each…

Twisting process for quantum linear spaces is defined. It consists in a particular kind of globally defined deformations on finitely generated algebras. Given a quantum space (A_1,A), a multiplicative cosimplicial quasicomplex C[A_1] in the…

We define a q-deformation of the Dirac operator, inspired by the one dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace…

We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that…

We propose a quantum lattice version of Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of $\widehat{s\ell}_2$. We construct an action of the…

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants $\int_S\mathcal A$ of a surface $S$, determined by the choice of a braided tensor category $\mathcal A$,…

We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give…

We introduce geometric quantization in the setting of shifted symplectic structures. We define Lagrangian fibrations and prequantizations of shifted symplectic stacks and their geometric quantization. In addition, we study many examples…

Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the…

We prove a conjecture of Kuznetsov stating that the equivariant K-theory of affine Laumon spaces is the universal Verma module for the quantum affine algebra U_q(gl_n^). We do so by reinterpreting the action of the quantum toroidal algebra…

The subject of this thesis is the rigorous construction of QFT models with nontrivial interaction. Two different approaches in the framework of AQFT are discussed. On the one hand, an inverse scattering problem is considered. A given…

Let $V$ be a finite dimensional vector space. Given a decomposition $V\otimes V=\oplus_i^n I_i$, define $n$ quadratic algebras $(V, J_m)$ where $J_m=\oplus_{i\neq m} I_i$. This decomposition defines also the quantum semigroup…

One of the fundamental objects in the $K$-theoretic enumerative geometry of Nakajima quiver varieties is known as the the capping operator. It is uniquely determined as the fundamental solution to a system of $q$-difference equations. Such…

The shuffle algebra on positive integers encodes the usual multiple zeta values (MZVs) (with positive arguments) thanks to the representations of MZVs by iterated Chen integrals of Kontsevich. Together with the quasi-shuffle (stuffle)…

We define a differential graded algebra associated to Legendrian knots in thickened convex surfaces $\Sigma\times \mathbb{R}$. The algebra is defined in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in $\mathbb{R}^3$, but…