Related papers: Quantum polynomial functors from $e$-Hecke pairs
We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…
We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter…
Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets…
We define generalized bialgebras and Hopf algebras and on this basis we introduce quantum categories and quantum groupoids. The quantization of the category of linear (super)spaces is constructed. We establish a criterion for the classical…
We define a $q$-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity…
We prove a classification of additive polynomial superfunctors, which allows us to compute some extensions of a superfunctor of the form $F \circ A$ where $F$ is a classical polynomial functor and $A$ is additive. We get a formula which…
We discuss the classical and quantum mechanical evolution of systems described by a Hamiltonian that is a function of a solvable one, both classically and quantum mechanically. The case in which the solvable Hamiltonian corresponds to the…
A class of quantum analogues of compact symmetric spaces of classical type is introduced by means of constant solutions to the reflection equations. Their zonal spherical functions are discussed in connection with $q$-orthogonal…
We introduce the notion of numerical functors to generalise Eilenberg & MacLane's polynomial functors to modules over a binomial base ring. After shewing how these functors are encoded by modules over a certain ring, we record a precise…
It is known that there is an one-to-one correspondence among the space of cusp forms, the space of homogeneous period polynomials and the space of Dedekind symbols with polynomial reciprocity laws. We add one more space, the space of…
We demonstrate that, in certain cases, quantization and the classical limit provide functors that are "almost inverse" to each other. These functors map between categories of algebraic structures for classical and quantum physics,…
The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2-categorical analogue of the theory of polynomials and polynomial functors is given, and…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
Quantum K-theory is a K-theoretic version of quantum cohomology, which was recently defined by Y.-P. Lee. Based on a presentation for the quantum K-theory of the classical flag variety Fl_n, we define and study quantum Grothendieck…
Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being "classical" or "quantum" ceases to be a basic conceptual difference. The dynamics differs,…
A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum…
We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton-Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the…
We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of…
In the framework of Category Theory, we study the association between finite--dimensional representations of a compact quantum group and quantum vector bundles with linear connections for a given quantum principal bundle with a principal…