Department of Mathematics Tennessee Technological University

(Released: November 18, 1998)
(Last Revised: December 26, 2001)

• CLIFFORD (ver. 5) - for computations in Clifford algebras Cl(V,B) where dim V <=9, and B is an arbitrary bilinear form in V
• Bigebra (ver. 0.16) - for computations with Hopf gebras and bi-gebras
• Cli5plus (ver. 5) - extends CLIFFORD to other bases such as the Clifford basis or the dotted wedge basis.
• GTP (ver. 5) - extends CLIFFORD to graded tensor products of Clifford algebras.
• Octonion (ver. 4) - for computations with octonions and octonionic matrices.

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

To see sample help pages try this link to Sample Help Pages.

HIGHLIGHTS OF NEW FEATURES IN VERSION 5:

• Runs up to 75 times faster than version 4.
• There are many new procedures in CLIFFORD while the Bigebra package is completely new.
• The CLIFFORD/Bigebra packages can compute very efficiently with meet and join, even with meet of hyperplanes and joins of hyperplanes (based on Hopf formalism). This can be done even *without* specifying a bilinear form at all, only the dimension has to be specified.
• All essential functions come in dual pairs in Bigebra such as:

(i)   Grassmann product <-> Grassmann co-product
(ii)  Clifford  product <-> Clifford  co-product
(iii) Meet  <-> Join  (of points, fields of points, etc..)
(iv) Meet  <-> Join  (of hyperplanes, fields of hyperplanes, etc..)
• A co-basis and co-vectors are now available (with Bigebra)
• A general tensor product `&t` over a ring (functions, numbers, ...) is available [that was not possible with Maple's original 'define'] which allows the computation in tensor products of Clifford algebras, as is currently discussed in multi-particle Clifford algebras and quantum computing.
• Allows user to select between two built-in routines for Clifford product: cmulNUM, based on the recursive use of Chevalley's definition of the Clifford product, or cmulRS, based on Rota-Stein cliffordization process. User can also define and use his/her own routine.
• Allows for simultaneous computations with many different bilinear forms in the same worksheet. For example, Clifford product 'cmul' can be computed in Cl(V,B) with respect to B or in Cl(V,g) where g is the symmetric part of B. Likewise, reversion can be computed with respect to the entire form B or with respect to its symmetric part g or antisymmetric part F.
• Computations can be performed in different bases rather than the standard Grassmann basis: a Clifford basis that involves un-evaluated Clifford products, for example,  &C_B(e1,e2) where the index 'B' (selectable by the user) denotes the bilinear form used.
• Additional types are defined in this version in order to better merge with auxiliary packages Bigebra, GTP, Cli5plus, Octonion, etc.
• Facility to display and easily change all environmental variables used in all packages.
• All packages and all procedures show up in Maple browser, that is, all help pages can be easily found by looking through the browser.
• Ease of installation: There are essentially only two files that need to be downloaded: One is a zipped library file Cliffordlib.zip and one is a new Maple initialization file maple.ini.
• Revised help pages for all procedures.

INSTALLATION INSTRUCTIONS:

A. For Windows:

• Make sure you run Maple V R. 5.1 dated November 5, 1998 [these packages should run under Maple V R. 5.0 but they have not been tested under this version of Maple. Note: This version won't run under Maple 6 or 7.].
• Download maple.ini file into the \LIB subdirectory in your Maple V directory.
• Create a directory where you will store CLIFFORD's library files and help pages. Default directory name used in maple.ini file is \Cliffordlib in the directory C:\Maplev51. Thus, to use the default names, create first a directory \Maplev51 on the C:\ drive and then a subdirectory \Cliffordlib in the directory C:\Maplev51.
• If you want to install CLIFFORD in a different directory than C:\Maplev51\Cliffordlib, you may do so but you will need to change accordingly the single line in maple.ini. Use the current line shown above as an example. Make sure you type the path correctly and appropriately for your operating system.
• Download file Cliffordlib.zip (just left click on the link and then save to your system) to C:\Maplev51\Cliffordliband unzip it there using, for example, on a Windows machine WinZip 8.1 (32-bit) for Windows Me/NT/2000/XP. If you don't have any unzipper, download file Cliffordlib.exe (again, just left click on the link and then save to your system) to C:\Maplev51\Cliffordlib and then "install" it, that is, run it to let it unzip automatically in the directory C:\Maplev51\Cliffordlib (make sure that you type the path C:\Maplev51\Cliffordlib into the window 'Unzip to folder:').
• If you have any problems with installation under Windows, please contact Dr. Rafal Ablamowicz at rablamowicz@tntech.edu.

• Download  Cliffordlib.zip or Cliffordlib.tgz (just left click on the link and then save to your system)
• Change to root:

> su
• Create a library directory in the maple path:

> mkdir /opt/mapleVr51/Cliffordlib
• Copy the zip-file into this dir and unzip:

> cp file.zip /opt/mapleVr51/Cliffordlib
> cd /opt/mapleVr51/Cliffordlib
> unzip file.zip
• Change the .mapleinit file to accept the library as *first* entry in 'libname':

> exit        (* close root shell *)
> cd          (* go to home dir *)
> pico .mapleinit   (* pico / or vi editors *)
Add the line:
 libname:=`/opt/mapleVr51/Cliffordlib',libname;
and store the file (create it if necessary) as .mapleinit.
• The initialization is finished. If you have any problems with installation under Linux, please contact Dr. Bertfried Fauser at Bertfried.Fauser@uni-konstanz.de.

• Download  Cliffordlib.zip or Cliffordlib.tgz (just left click on the link and then save to your system).
• Create a library directory in your home directory (* if you choose a dotted name this dir is not displayed by default *)

> cd   (* go to home dir *)
> mkdir .Cliffordlib
• Copy the zip-file into this dir and unzip:

> cp file.zip ./.Cliffordlib
> cd .Cliffordlib
> unzip file.zip
• Change the .mapleinit file to accept the library as *first* entry in 'libname'

> cd          (* go to home dir *)
> pico .mapleinit   (* pico / or vi editors *)
Ad the line:
 libname:=`/opt/mapleVr51/Cliffordlib',libname;
and store the file (create it if necessary) as .mapleinit.
• The initialization is finished. If you have any problems with installation under Linux, please contact Dr. Bertfried Fauser at  Bertfried.Fauser@uni-konstanz.de.

• Start Maple V R. 5.1 session.
• To load the CLIFFORD package, at the Maple prompt, type:

>with(Cliff5);
• When the package is loaded, you should see a list with names of available procedures. If the package has not loaded and you get an error message

Error, (in with) undefined package, Cliff5
make sure that files maple.lib, maple.ind and maple.hdb that are obtained from unzipping Cliffordlib.zip are all located in the directory C:\Maplev51\Cliffordlib and that the path to that directory is included in maple.ini file.
• To load other packages, issue with(packagename); command as well, where 'packagename' is one of Bigebra, Cli5plus, GTP, or Octonion.
• To check if help pages are available in the browser, at the Maple prompt type:

>?Clifford
You should then see the introductory page for CLIFFORD and at the top of the screen you should see Maple browser opening four vertical windows with words Cliff5 and intro highlighted in the two rightmost panels. You should also see, above and below Cliff5 names of other packages for which help should now be available.
• To navigate through the help pages, begin with the introductory help page for CLIFFORD as described above and follow links within. You can also issue this command at the Maple prompt:

>?Cliff5,intro

More information about the use of this package as well as the use of other software packages written in Mathematica, Reduce, Matlab, C++, Fortran, Clical, and Maple for the Clifford algebra research, see the following book and associated with it electronic files located at the Birkhauser Web site: www.birkhauser.com/book/ISBN/0-8176-3907-1/

Title: ``Clifford Algebras with Numeric and Symbolic Computations''
Editors: Rafa\l Ab\lamowicz, Pertti Lounesto, Josep M. Parra
ISBN: 0-8176-3907-1
Birkhauser, Boston
Year: 1996

For more on Clifford algebras and their applications see also:

Title: ``Clifford Algebras and their Applications in Mathematical Physics - Vol. 1 Algebra and Physics, Vol. 2 Clifford Analysis''
Editors: Rafal Ablamowicz and Bertfried Fauser (Vol. 1),  John Ryan and Wolfgang Sproessig (Vol. 2)
ISBN: 0-8176-4182-3 (Vol. 1), 0-8176-4183-1 (Vol. 2) [for more information go to http://math.tntech.edu/rafal/mexico/mexico.html]
Birkhauser, Boston
Year: 2000

Title: ``Clifford Algebras and Spinor Structures -- A Special Volume Dedicated to the Memory of
Albert Crumeyrolle (1919 -- 1992)''.
Editors: Rafal Ablamowicz, Pertti Lounesto
ISBN: 0-7923-3366-7
Kluwer, Dordrecht
Year: 1995

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: November 5, 2002/ra