Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple

TL;DR

This paper introduces the BIGEBRA package for Maple, enabling symbolic computation of Hopf algebraic structures like tensor products, Grassmann, and Clifford algebras, facilitating research in various advanced mathematical fields.

Contribution

The paper presents the development of the BIGEBRA package for Maple, a tool that automates symbolic calculations of complex algebraic structures such as Hopf, Grassmann, and Clifford algebras.

Findings

Successfully implemented tensor and algebraic operations in CAS

Demonstrated applications in invariant theory and quantum algebra

Enabled new research approaches through computational experimentation

Abstract

Hopf algebraic structures will replace groups and group representations as the leading paradigm in forthcoming times. K-theory, co-homology, entanglement, statistics, representation categories, quantized or twisted structures as well as more geometric topics of invariant theory, e.g., the Grassmann-Cayley bracket algebra are all covered by the Hopf algebraic framework. The new branch of 'experimental mathematics' allows one to easily enter these fields through direct calculations using symbolic manipulation and computer algebra system (CAS). We discuss problems which were solved when building the BIGEBRA package for Maple and CLIFFORD [see math-ph/0212031] to handle tensor products, Grassmann and Clifford algebras, coalgebras and Hopf algebras. Recent results showing the usefulness of CAS for investigating new and involved mathematics provide us with examples. An outlook on further developments is given.