
Department of Mathematics
Tennessee Technological University 
(Released: November 18, 1998)
(Last Revised: December 26, 2001)
'CLIFFORD  A Maple V (Rel. 5.1) Package for Clifford
Algebra
Computations'
(version 5, December 26, 2001)
with 'Bigebra' ver. 0.16
Rafal Ablamowicz (*) and Bertfried Fauser (**)
(*) Department of Mathematics, Box 5054
Tennessee Technological University, Cookeville, TN 38505
phone: USA (931) 3723569, fax: USA (931) 3726353
rablamowicz@tntech.edu
(**) Universität Konstanz, Fachbereich Physik, Fach M678
78457 Konstanz, Germany
Bertfried.Fauser@unikonstanz.de
http://kaluza.physik.unikonstanz.de/~fauser/
[For 'CLIFFORD' ver. 6, 7, and 8 for Maple 6, 7, and 8 go to
http://math.tntech.edu/rafal/index.html]
Files listed below contain Version 5 of the package CLIFFORD
for Clifford algebra computations in Maple V R. 5.1 along with some
sample
Maple help pages. Here is the listing of all available packages:

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
bigebras

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.
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. It is expected that appropriate
acknowledgment
will be given to CLIFFORD, Bigebra, Cli5plus, GTP,
Octonion,
and their authors, Rafal Ablamowicz and Bertfried Fauser, when results
are derived. All packages and their extensive 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.
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 coproduct
(ii) Clifford product <> Clifford coproduct
(iii) Meet <> Join (of points, fields of points,
etc..)
(iv) Meet <> Join (of hyperplanes, fields of
hyperplanes,
etc..)
 A cobasis and covectors 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 multiparticle Clifford algebras and quantum computing.

Allows user to select between two builtin routines for Clifford
product:
cmulNUM, based on the recursive use of Chevalley's definition of the
Clifford
product, or cmulRS, based on RotaStein 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 unevaluated 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.
DESCRIPTION OF AVAILABLE FILES:
Name: 
Type: 
Size: 
Description: 
Cliffordlib.zip 
binary 
1,061KB 
A .zip file with all CLIFFORD library files and help
pages.
Includes also library files and help pages for Bigebra, Cli5plus,
GTP
and Octonion packages. It contains three files: maple.lib,
maple.ind
and maple.hdb. You can download free evaluation copy of WinZip
8.1 (32bit) for Windows Me/NT/2000/XP.
Last updated: 3/24/2002 
maple.ini 
binary 
1KB 
Maple initialization file. It contains only one line:
libname:="C:\\Maplev51/Cliffordlib", libname:
Its role is to augment Maple's global variable libname with
a
path to the directory C:\\Maplev51/Cliffordlib where CLIFFORD's
help pages and library files are located. See instructions below how to
modify this path, if needed. If you already have your own maple.ini
file,
just add this new path to the 'libname'.

Cliffordlib.exe 
binary 
1,124KB 
A selfexecutable file made from Cliffordlib.zip
above.
Last updated: 3/24/2002 
Cliffordlib.tgz 
binary 
1,032KB 
A .tgz file made from Cliffordlib.zip above.
Last updated: 1/7/2002 
clifford9M.mws 
ASCII 
300KB 
Maple V Rel. 5.1 worksheet containing the source code of CLIFFORD
version 5.
Last updated: 3/24/2002 
Bigbrcd0.16.mws 
ASCII 
40KB 
Maple V Rel. 5.1 worksheet containing the source code of Bigebra
ver. 0.16
Last updated: 1/7/2002 
Cli5plus.mws 
ASCII 
24KB 
Maple V Rel. 5.1 worksheet containing the source code of Cli5plus
ver. 5
Last updated: 12/26/2001 
GTP5.mws 
ASCII 
12KB 
Maple V Rel. 5.1 worksheet containing the source code of GTP
ver.
5
Last updated: 12/26/2001 
Octonion4.mws 
ASCII 
19KB 
Maple V Rel. 5.1 worksheet containing the source code of Octonion
ver.
4
Last updated: 12/26/2001 
INSTALLATION INSTRUCTIONS:
A. For Windows:
The following installation instructions should be general enough to
allow
for loading and installation of the package under any (hopefully)
operating
system. If you encounter any problems, please let me know.

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 (32bit) 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.
B. For Linux (if you have access to the root directory):

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 zipfile 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@unikonstanz.de.
C. For Linux (if you do not have access to the root directory):

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 zipfile 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@unikonstanz.de.
HOW TO LOAD THE PACKAGE AND CHECK THAT INSTALLATION WAS DONE
CORRECTLY:

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:
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/0817639071/
Title: ``Clifford Algebras with Numeric
and
Symbolic Computations''
Editors: Rafa\l Ab\lamowicz, Pertti Lounesto,
Josep M. Parra
ISBN: 0817639071
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: 0817641823 (Vol. 1), 0817641831 (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: 0792333667
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) 3723569
fax: USA (931) 3726353
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