[Paper Review] Calculus and Quantizations over Hopf algebras

Valentin Lychagin|ArXiv.org|Jun 15, 1994

Advanced Topics in Algebra6 references4 citations

TL;DR

This paper introduces a novel framework for calculus and quantization over quasitriangular Hopf algebras using braided monoidal categories, enabling intrinsic differential calculus without enforcing a Leibniz rule. It proposes a general quantization as a natural isomorphism of the tensor product functor with coherence conditions, and presents two computational methods: non-linear cohomology and multiplicative Hochschild cohomology of the Grothendieck ring, illustrated with examples.

ABSTRACT

In this paper we outline an approach to calculus over quasitriangular Hopf algebras. We study differential operators in the framework of monoidal categories equipped with a braiding or symmetry. To be more concrete, we choose as an example the category of modules over quasitriangular Hopf algebra. We introduce braided differential operators in a pure algebraic manner.This gives us a possibility to develop calculus in an intrinsic way without enforcing any type of Leibniz rule. A general notion of quantization in monoidal categories, proposed in this paper, is a natural isomorphism of the tensor product bifunctor equipped with some natural coherence conditions. The quantization "deforms" all algebraic and differential objects in the monoidal category. We suggest two ways for calculation of quantizations. One of them reduces the calculation to non-linear cohomologies. THe other describes quantizations in terms of multiplicative Hochschild cohomologies of the Grothendieck ring of the given monoidal category. These constructions are illustrated by some examples.

Motivation & Objective

• To develop a coordinate-free, intrinsic differential calculus over quasitriangular Hopf algebras using braided monoidal categories.
• To define a general notion of quantization as a natural isomorphism of the tensor product functor with coherence conditions.
• To provide computational tools for quantizations via non-linear cohomology and multiplicative Hochschild cohomology of the Grothendieck ring.
• To demonstrate the framework through concrete examples in the context of Hopf algebra modules.

Proposed method

• Utilizes the category of modules over a quasitriangular Hopf algebra equipped with a braiding to define braided differential operators.
• Introduces braided differential operators in a purely algebraic setting, avoiding the need to impose a Leibniz rule a priori.
• Defines quantization as a natural isomorphism of the tensor product bifunctor satisfying specific coherence conditions.
• Reduces the computation of quantizations to non-linear cohomology via deformation-theoretic techniques.
• Characterizes quantizations using multiplicative Hochschild cohomology of the Grothendieck ring of the monoidal category.
• Applies the formalism to examples to illustrate the deformation of algebraic and differential structures.

Experimental results

Research questions

• RQ1 How can differential calculus be formulated intrinsically over quasitriangular Hopf algebras without relying on the Leibniz rule?
• RQ2 What is the categorical structure underlying a general quantization in monoidal categories?
• RQ3 How can one compute quantizations systematically in this framework?
• RQ4 What role does the braiding play in defining differential operators and their compatibility with tensor products?
• RQ5 In what way do the Grothendieck ring and its Hochschild cohomology encode information about quantizations?

Key findings

• The framework enables a coordinate-free differential calculus over quasitriangular Hopf algebras through the use of braided monoidal categories.
• Braided differential operators are defined algebraically without requiring the Leibniz rule to be imposed from the start.
• Quantization is formalized as a natural isomorphism of the tensor product functor with specified coherence conditions, generalizing deformation quantization.
• Two computational methods for quantizations are established: one based on non-linear cohomology and another using multiplicative Hochschild cohomology of the Grothendieck ring.
• The approach deforms both algebraic and differential objects naturally within the monoidal category, preserving structural coherence.
• Examples illustrate the deformation of standard algebraic and differential structures under the proposed quantization scheme.