Department of Mathematics
Tennessee Technological University |

Sample Help Pages for

**'CLIFFORD - A Maple V (R. 5) Package for Clifford Algebra Computations'
(version 5)**
**'Bigebra' - Package for computations with Hopf algebras (version
0.16)**
**'GTP' - Package for computations with graded tensor products of
Clifford algebras (version 5)**
**'Octonion' - Package for computations with octonions (version 4)**
**'Cli5plus' - Package that extends certain features of 'CLIFFORD'
(version 5)**

**by**

**Rafal Ablamowicz (*) and Bertfried Fauser (**)**

Tennessee Technological University, Cookeville, TN 38505

phone: USA (931) 372-3569, fax: USA (931) 372-6353

rablamowicz@tntech.edu

Permission is given to download these files as long as they are for individual use and not for distribution of any kind including posting these files on any other server but this one. All help pages are copyrighted by Rafal Ablamowicz and Bertfried Fauser.

You can find out how I use CLIFFORD by going to my recent publications page.

- Main introductory help page Cliff5[intro].
- Procedure that displays environmental variables used by all packages is called CLIFFORD_ENV.
- Clifford multiplication procedure cmul.
- Internal Clifford multiplication procedure cmulNUM using recursive definition of Chevalley.
- Internal Clifford multiplication procedure cmulRS using Rota-Stein cliffordization procedure.
- Reversion in Clifford algebra Cl(B) using B dependent reversion.
- One of two procedures that gives a scalar product in a spinor space considered as a minimal left ideal is beta_plus.
- Matrix representations of Clifford algebras are accomplished with the procedure matKrepr.
- Multiplication of Clifford matrices is done with rmulm.
- Computation of a spinor basis in a minimal left or right ideal is done with minimalideal.
- Determination if a Clifford matrix is a Vahlen matrix is done with isVahlenmatrix.
- Rotation in three dimensions accomplished with quaternions and a procedure rot3d.
- Exterior wedge multiplication is done with wedge.
- Clifford exponentiation is done with sexp modulo a minimal polynomial.

**SAMPLE HELP PAGES FOR 'Cli5plus' (VERSION
5):**

- Dotted wedge procedure dwedge.
- Procedure clirev extends B dependent reversion in Cl(B) to the Clifford basis.
- Multiplication procedure climul extends Clifford product to the Clifford basis.
- Procedure wedge_to_dwedge converts Clifford polynomials from an undotted wedge basis to a dotted wedge basis.

**SAMPLE HELP PAGES FOR 'Bigebra'** **(VERSION
0.16):**

- Main help page.
- Grassmann co-product `&gco`.
- Clifford co-product `&cco`.
- Dotted Grassmann co-product `&gco_d`.
- Meet product `&v`.
- Graded switch gswitch.
- The antipode map for Grassmann Hopf gebra gantipode.

**SAMPLE HELP PAGES FOR 'GTP'** **(VERSION
5):**

- Clifford product in Clifford algebras Cl(B1), Cl(B2), ..., Cl(Bn) is done with cmulB.
- Basis in the graded tensor product is defined with gbasis.
- Multiplication in the graded tensor product of Clifford algebras is accomplished with gradedprod.

- User may define an octonion multiplication table with def_omultable.
- Octonion multiplication is done with omul.
- Octonionic norm is computed with onorm.
- Octonionic multiplication is defined in terms of Fano_triples which are related to the Fano plane.

Please direct all questions, comments, and reports on bugs to:

Rafal Ablamowicz

rablamowicz@tntech.edu

Department of Mathematics, Box 5054

Tennessee Technological University

Cookeville, TN 38505

phone: USA (931) 372-3569

fax: USA (931) 372-6353

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Last updated: January 5, 2002/ra