Quantum Shuffle Algebras Package

Mathematica + Julia | MIT License

Author: Eremey Valetov
Creation date: 09-Sep-2016
Email: eremey@valetov.com

Description

A package for quantum shuffle algebra calculations, including construction of bases in terms of Lyndon words (primes), element comparison, unique prime factorization, quantum shuffle multiplication, Hopf algebra operations (coproduct, counit, antipode), Lyndon word enumeration, and diagonal braiding.

Available in both Mathematica and Julia.

Requirements

• Mathematica: version 10.4 or later
• Julia: version 1.6 or later

Installation (Mathematica)

Copy QuantumShuffleAlgebra.wl to a directory on Mathematica's search path:

$UserBaseDirectory/Applications/

Then load:

Get["QuantumShuffleAlgebra`"]

Installation (Julia)

using Pkg
Pkg.develop(path="julia/")

using QuantumShuffleAlgebras

Quick Start (Mathematica)

Get["QuantumShuffleAlgebra`"]

(* Check whether a word is a Lyndon word *)
QSAIsPrime[{1, 2}]     (* True *)

(* Unique prime factorization *)
QSAUniquePrimeFactorization[{2, 1}]     (* {{2}, {1}} *)

(* Quantum shuffle product *)
QSAShuffleMultiplication[{{1}, {2}}]

(* Enumerate Lyndon words up to length 3 over {1,2} *)
QSALyndonWords[3, {1, 2}]

(* Hopf algebra coproduct *)
QSACoproduct[{1, 2}]

Functions

Core operations

Function	Purpose
QSARelation[x, y]	Compare words (1 = >, 0 = =, -1 = <)
QSAIsPrime[word]	Test whether a word is a Lyndon word
QSAFirstPrime[word]	Longest Lyndon prefix
QSAUniquePrimeFactorization[word]	Factorization into Lyndon words
QSAX[word, qpar]	Quantum shuffle basis element X_a
QSAShuffleMultiplication[{a, b}]	Quantum shuffle product v_a . v_b
QSAPrimaryCoefficient[word]	Coefficient of v_a in X_a
QSAExpressInLyndonWords[word]	Express v_a in terms of X_c's

Lyndon word enumeration

Function	Purpose
QSALyndonWords[n, alphabet]	All Lyndon words up to length n

Hopf algebra

Function	Purpose
QSACoproduct[word, q]	Deconcatenation coproduct
QSACounit[word]	Counit (1 for empty word, 0 otherwise)
QSAAntipode[word, q]	Antipode via Hopf axiom

Braiding

Function	Purpose
QSABraid[word1, word2, qMatrix]	Diagonal braiding

Tests

wolframscript -file tests/RunTests.wl

Copyright and Citation

© 2016 Eremey Valetov. Released under the MIT License.

This package was developed as part of an internship project at UPMC — Université Pierre et Marie Curie — Paris 6 (now part of Sorbonne Université), titled "Bases of Quantum Group Algebras in Terms of Lyndon Words". The project report is available at https://hal.science/hal-02448969 (DOI: 10.48550/arXiv.2001.10435).

If you use this package in your research, please cite:

Eremey Valetov. Bases of Quantum Group Algebras in Terms of Lyndon Words. [Internship report] Université Pierre & Marie Curie — Paris 6; Université Paris Diderot — Paris 7. 2016. ⟨hal-02448969v2⟩