Clifford and Graßmann Hopf algebras via the BIGEBRA package for Maple ( R ) ∗

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 † 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. 1. Aim of the paper The first aim of this paper is to present the features of BIGEBRA, a Maple V rel. 5 package. BIGEBRA was built to deal with tensored Clifford and Graßmann algebras, and it relies on CLIFFORD (Ab lamowicz and Fauser, 1999-2002, 19962002). While the emphasis here is more on the package, we nevertheless provide novel results in the last section on quantum Yang-Baxter equations derived from a Clifford bi-convolution. CLIFFORD and BIGEBRA are meanwhile available for Maple 6, 7 and Maple 8. Package homepage is located at http://math.tntech.edu/rafal/ while CLIFFORD and BIGEBRA are available for Maple V, Maple 6, 7, and 8. Maple (R) is available from http://www.maplesoft.com. The CLIFFORD package is described in a companion paper by Ab lamowicz and Fauser (2003)