Department of Mathematics Tennessee Technological University

Released: November 18, 1998
Last Revised: August 8, 2009

CLIFFORD - A Maple 12 Package for Clifford Algebra Computations
Version 12, August 8, 2009
with Bigebra, Cliplus, Define, code_support, GfG, GTP, Octonion, RJgrobner, SchurFkt, SP, and SymGroupAlgebra

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

(*) Department of Mathematics, Box 5054
Tennessee Technological University, Cookeville, TN 38505
phone: USA (931) 372-3569, fax: USA (931) 372-6353
 rablamowicz@tntech.edu

Version 12 of CLIFFORD for Clifford algebra computations in Maple 12 is distributed as library_M12.zip that contains the following packages:

• CLIFFORD - for computations in Clifford algebras Cl(V,B) where dim V <=9, and B is an arbitrary bilinear form in V
• Bigebra - for computations with Hopf gebras and bi-gebras
• Cliplus - extends CLIFFORD to other bases such as the Clifford basis or the dotted wedge basis
• Define - fixes Maple's 'define'
• GfG - experimental package for Groebner bases in Grassmann algebras
  
• GTP - extends CLIFFORD to graded tensor products of Clifford algebras
• Octonion - for computations with octonions and octonionic matrices
• RJgrobner - a small package (with Jane Liu) for commutative Groebner bases. It does not need CLIFFORD/Bigebra.
• SchurFkt - package for the Hopf Algebra of Symmetric Function
• SymGoupAlgebra - experimental package for computations in the group algebra of the symmetric group S_n (under development: Most recent version is dated January 6, 2009)
  
• SINGULARPLURALlink - package to interface Maple 12 with Singular:Plural to compute Groebner bases in polynomial rings, and Grassmann and Clifford algebras
  
• SP - a  small package for symmetric functions. It does not need CLIFFORD/Bigebra.
• code_support - package to modify, copy, and insert help worksheets into Maple database. It does not need CLIFFORD/Bigebra.

===============
DISCLAIMER: THERE IS NO WARRANTY FOR THE CLIFFORD, Bigebra, Cliplus, Define, GfG, GTP, Octonion, RJGrobner, SchurFkt, SymGroupAlgebra, SINGULARPLURALlink, SP, code_support PACKAGES TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANT ABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
===============

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

HIGHLIGHTS OF NEW FEATURES IN VERSION 12:

• This is an updated version of CLIFFORD ver. 11 from Maple 11 and made for Maple 12.
• There are no additional procedures in CLIFFORD but the package SymGroupAlgebra is new and under development.
• To install all needed libraries, download libraries.
• To see all features, open help pages in Maple 12 (see below how) that are already included in the library files, or download documentation in .pdf format for printing and/or viewing in Adobe Reader.
• To see the Maple code of each module, download codes.
• To view installation instructions, go to section on installation under Windows, Linux, or download README.txt.
• To see how to test your installation go to section on testing installation.
• Total of 280 help worksheets converted to 36MB of pdf help files taking up 1,213 pages of documentation. Clifford alone has about 4,500 lines of code.
  

Files:	Name:	Size:	Description:
Libraries:	library_M12.zip	1.45MB	A .zip file with all library files containing compiled code for Bigebra, CLIFFORD, Cliplus, Define, code_support, GfG, GTP, Octonion, RJgrobner, SchurFkt, SP, SymGroupAlgebra, and  SINGULARPLURALlink together with help pages for most modules. It includes also a sample maple.ini file that needs to be placed in \Lib or \Users subdirectory of Maple 12 directory on Windows machine. For more detailed installation instructions see README.txt.  Posted/Last updated: 8/08/2009 Note: A small correction was made in matKrepr procedure and in matdatam_M12.zip files..
Documentation:	all_help_pages06i09_M12.pdf	37MB	A single  file with all help pages in pdf format for all functions and procedures in all modules.  It contains 1,213 pages in full color. Posted/Last updated: 1/06/2009
	BigebraHelpPages20xii08_M12.pdf	4.8MB	A single file with all help pages  in pdf format for Bigebra. It contains 160 pages in full color.  Posted/Last updated: 12/20/2008
	BigebraHelpPages20xii08_M12.zip	155KB	A single file with all help pages  in mws format for Bigebra. It contains 38 worksheets.  Posted/Last updated: 12/20/2008
	CliffordHelpPages20xii08_M12.pdf	12.6MB	A single file with all help pages in pdf format for all procedures in CLIFFORD. It contains 448 pages in full color.  Posted/Last updated: 12/20/08
	CliffordHelpPages20xii08_M12.zip	418 KB	A single file with all help pages in mws format for all procedures in CLIFFORD. It contains 110 worksheets. Posted/Last updated: 12/20/2008
	CliplusHelpPages20xii08_M12.pdf	1.8MB	A single file with all help pages in pdf format for all procedures in Cliplus. It contains 74 pages in full color.  Posted/Last updated: 12/20/2008
	CliplusHelpPages20xii08_M12.zip	78KB	A single file with all help pages in mws format for all procedures in Cliplus. It contains 14 files. Posted/Last updated: 12/20/2008
	GfGHelpPages20xii08_M12.pdf	164KB	A single file with all help pages in pdf format for all procedures in GfG. It contains 15 pages in full color.  Posted/Last updated: 12/20/2008
	GfGHelpPages20xii08_M12.zip	12KB	A single file with all help pages in mws format for all procedures in GfG. It contains 1 worksheet with examples.  Posted/Last updated: 12/20/2008
	GTPHelpPages20xii08_M12.pdf	1.2MB	A single file with all help pages in pdf format for all procedures in GTP. It contains 33 pages in full color.  Posted/Last updated: 12/20/2008
	GTPHelpPages20xii08_M12.zip	31KB	A single file with all help pages in mws format for all procedures in GTP. It contains 12 worksheets.  Posted/Last updated: 12/20/2008
	OctonionHelpPages20xii08_M12.pdf	1.5MB	A single file with all help pages in pdf format for all procedures in Octonion. It contains 33 pages in full color.  Posted/Last updated: 12/20/2008
	OctonionHelpPages20xii08_M12.zip	34KB	A single file with all help pages in mws format for all procedures in Octonion. It contains 15 worksheets. Posted/Last updated: 12/20/2008
	RJgrobnerHelpPages20xii08_M12.pdf	4.6MB	A single file with all help pages in pdf format for all procedures in RJgrobner. It contains 35 pages in full color.  Posted/Last updated: 12/20/2008
	RJgrobnerHelpPages20xii08_M12.zip	22KB	A single file with all help pages in mws format for all procedures in RJgrobner. It contains 1 worksheet with examples. Posted/Last updated: 12/20/2008
	SchurFktHelpPages20xii08_M12.pdf	7.1MB	A single file with all help pages in pdf format for all procedures in SchurFkt. It contains 183 pages in full color.  Posted/Last updated: 12/20/2008
	SchurFktHelpPages20xii08_M12.zip	174KB	A single file with all help pages in mws format for all procedures in SchurFkt. It contains 66 worksheets. Posted/Last updated: 12/20/2008
	SINGULARPLURALlinkHelpPages20xii08_M12.pdf	441KB	A single file with all help pages in pdf format for all procedures in SINGULARPLURALlink. It contains 59 pages in full color.  Posted/Last updated: 12/20/2008
	SINGULARPLURALlinkHelpPages20xii08_M12.zip	33KB	A single file with all help pages in mws format for all procedures in SINGULARPLURALlink. It contains 1 worksheet with examples. Posted/Last updated: 12/20/2008
	SPHelpPages20xii08_M12.pdf	2.2MB	A single file with all help pages in pdf format for all procedures in SP. It contains 90 pages in full color.  Posted/Last updated: 12/20/2008
	SPHelpPages20xii08_M12.zip	80KB	A single file with all help pages in mws format for all procedures in SP. It contains 18 worksheets. Posted/Last updated: 12/20/2008
	SymGroupAlgebraHelpPages06i09_M12.pdf	152KB	A single file with all help pages in pdf format for all procedures in SymGroupAlgebra. It contains 21 pages in full color.  Posted/Last updated: 1/06/2009
	SymGroupAlgebraHelpPages06i09_M12.zip	42KB	A single file with all help pages in mws format for all procedures in SymGroupAlgebra. It contains 1 worksheet. Posted/Last updated: 1/06/2009
	code_supportHelpPages20xii08_M12.pdf	529KB	A single file with three help pages in pdf format for all procedures in code_support. It contains 62 pages in full color.  Posted/Last updated: 12/20/2008
	code_supportHelpPages20xii08_M12.zip	36KB	A single file with three help pages in mws format for all procedures in code_support. It contains 3 worksheets. Posted/Last updated: 12/20/2008
	README.txt.	10KB	Installation instructions Posted/Last updated: 8/08/2009
Code:	all_code_mws08viii09_M12.zip	243KB	A .zip file with CLIFFORD, Bigebra, Cliplus, Define, GTP, Octonion, code_support, SchurFkt, SINGULARPLURALlink, GfG, RJgrobner, SP, and SymGroupalgebra  Maple 12 worksheets with codes for all modules.  Posted/Last updated: 8/08/2009 Note: The only new item since the last posting is a small correction in matKrepr procedure and in matdatam_M12.zip files
	Bigebra_1.01_M12.mws	44KB	A Maple 12 worksheet with Bigebra code ver. 1.01.  Posted/Last updated: 12/20/2008
	clifford_12_M12.mws	306KB	A Maple 12 worksheet with CLIFFORD code ver. 12.  Posted/Last updated: 08/08/2009
	Cliplus_02_M12.mws	29KB	A Maple 12 worksheet with Cliplus code ver. 02.  Posted/Last updated: 12/20/2008
	Define_1.00_M12.mws	26KB	A Maple 12 worksheet with Define code ver. 1.00.  Posted/Last updated: 12/20/2008
	GfG_20xii08_M12.mws	72KB	A Maple 12 worksheet with GfG code ver. 20 xii 08. Posted/Last updated: 12/20/2008
	GTP_01_M12.mws	18KB	A Maple 12 worksheet with GTP code ver. 01.  Posted/Last updated: 12/20/2008
	Octonion_03_M12.mws	24KB	A Maple 12 worksheet with Octonion code ver. 03.  Posted/Last updated: 5/27/2009
	RJgrobner_07_M12.mws	58KB	A Maple 12 worksheet with RJgrobner code ver. 07.  Posted/Last updated: 12/20/2008
	SchurFkt-code-ver.1.0.2-9vi08_M12.mws	129KB	A Maple 12 worksheet with SchurFkt code ver.1.0.2. 9 vi 2008.  Posted/Last updated: 12/20/2008
	SINGULARPLURALlink_0515_M12.mws	85KB	A Maple 12 worksheet with SINGULARPLURALlink code ver. 0515.  Posted/Last updated: 12/20//2008
	SP_06_M12.mws	45KB	A Maple 12 worksheet with SP code ver. 06.  Posted/Last updated: 12/20/2008
	SPinitialization_01_M12.mws	13KB	A Maple 12 worksheet with SPinitialization code ver. 01 needed for the SP package.  Posted/Last updated: 12/20/2008
	SymGroupAlgebra_03_M12.mws	56KB	A Maple 12 worksheet with SymGroupAlgebra code ver. 03.  Posted/Last updated: 1/06/2009
	code_support_1.03_M12.mws	37KB	A Maple 12 worksheet with code_support code ver. 1.03.  Posted/Last updated: 12/20/2008

INSTALLATION INSTRUCTIONS:

A. For Windows:

• Make sure you run Maple 12.
• Create a directory C:\Maple12\Cliffordlib.
  
• Download library_M12.zip to the directory C:\Maple12\Cliffordlib and unzip it there. It contains four files: maple.hdb, maple.lib, maple.ini, and maple.ind.
• Copy maple.ini to \LIB subdirectory of your Maple main directory (for example, by default Maple installs in C:\Program Files\Maple 12).
• If you want to install CLIFFORD and its supplementary packages in a directory other than C:\Maple12\Cliffordlib, you may do so but you will need to modify accordingly maple.ini. Make sure you type the path correctly and appropriately for your operating system.
• Test that installation has been done correctly (see below)

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

• 1. Download library_M12.zip (just right click on the link and then save to your system).
• 2. Change to root:

> su

• 3. Create a library directory in the maple path: Maple may be installed in /usr/local/maple called $MAPLE_PATH in the following

> mkdir $MAPLE_PATH/Cliffordlib

• 4. Copy the zip-file into this dir and unzip:

> cp library_M12.zip $MAPLE_PATH/Cliffordlib
> cd $MAPLE_PATH/Cliffordlib
> unzip library_M12.zip
Then proceed with 5a) if you have only maple8 running and with 5b) if you  have several maple versions running

• 5a. 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:=`$MAPLE_PATH`,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:

         ==5b==>

• 5b. Stay being root!

> cd ../bin

open the file maple, and look for the place where xmaple12 is called change this line to (no white space after \ or type a single line only)
> exec ${MAPLE}/$MAPLE_SYS_BIN/xmaple12 -km p -b ${MAPLE}/Cliffordlib \ -b ${MAPLE}/lib $INCLUDES $PARAM
and change the line where mapleTTY is loaded to
>  exec ${MAPLE}/$MAPLE_SYS_BIN/mapleTTY -b ${MAPLE}/Cliffordlib \ -b ${MAPLE}/lib $INCLUDES $PARAM

(The option -b makes maple aware of locations of libraries. In this way you might want ot change every start script of several maple versions to find their own local libraries)
> exit
leave root shell.

C. For Linux (if you do not have root access):

• 1. Download library_M12.zip (just right click on the link and then save to your system).
• 2. Create a library directory in your home directory (* if you choose  a name starting with a dot, this dir is not displayed by default *)

> cd           (* go to home dir *)
> mkdir .Cliffordlib

• 3. Copy the zip-file into this dir and unzip:

> cp library_M12.zip ./.Cliffordlib
> cd .Cliffordlib
> unzip library_M12.zip

• 4. Change the .mapleinit file to accept the library as *first* entry in the variable 'libname'

> cd                  (* go to home dir *)
> pico .mapleinit     (* pico / or vi editors *)

• 5. Ad the line:

libname:=`<homedirectory>/.Cliffordlib',libname;
and store the file (create it if necessary) as .mapleinit.

• 6. The initialization is finished.
• 7. Unfortunately there is only one .mapleinit file for every maple version. This means that other versions of maple will use this directory too,

which might cause PROBLEMS. Currently there is no workaround for this and we recommend to insert the line
libname:=`<homedirectory>/.Cliffordlib',libname;
in the first line of any worksheet which use Clifford/Bigebra for Maple 12
Such a maple file looks like:
> restart: libname:=`<homedirectory>/.Cliffordlib',libname;
> with(Clifford):with(Bigebra): ## optional:  with(Cliplus):with(Octonion):with(GTP):
In this case no library path is needed. If you run several versions of  maple you might contact your local administrator 'root' to change the startup scripts as described above.

• Start Maple 12.
• To load the main module, at the Maple prompt, type:

>with(Clifford);

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 make sure that files maple.hdb, maple.lib, and maple.ind extracted from library_M12.zip are all located in the directory such as C:\Maple12\Cliffordlib and that the path to that directory is included in maple.ini file. To see if maple.ini has been executed, type
>libname;
and Maple should respond with

            "C:\\Maple12/Cliffordlib", "C:\\Maple12/lib"

if maple.ini had this line:

libname:="C:\\Maple12/Cliffordlib",libname:
 

• To load other packages, issue with(packagename); command as well, where 'packagename' is one of Bigebra, Cliplus, GTP, Octonion, or code_support, etc...
• 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 Clifford and intro highlighted in the two rightmost panels. You should also see, above and below Clifford, 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:

>?Clifford,intro

Most recent papers with CLIFFORD, Bigebra, etc.
 

• "Gröbner Bases Algorithms for Grassmann Algebras in a Maple Package", Troy Brachey, Department of Mathematics TTU, Tech. Report No. 2008-1, October 2008, downloadable file, 204KB,.pdf
• "Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras", Rafal Ablamowicz, Department of Mathematics TTU, Tech. Report No. 2008-2, December 2008, downloadable file, 364KB,.pdf
• "Some Applications of Gröbner Bases in Robotics and Engineering", Rafal Ablamowicz, Department of Mathematics TTU, Tech. Report No. 2008-3, December 2008, downloadable file, 754KB,.pdf
• "Mathematics of CLIFFORD - a Maple package for Clifford and Grassmann algebras", Rafal Ablamowicz, Department of Mathematics at TTU, and Bertfried Fauser, Universitat Konstanz, Germany, Tech. Report No. 2002-4, December 2002, http://www.math.tntech.edu/techreports/TR_2002_4.pdf
• "Clifford and Grassmann Hopf algebras via the bigebra package for Maple(R)", Rafal Ablamowicz, Department of Mathematics at TTU, and Bertfried Fauser, Universitat Konstanz, Germany, Tech. Report No. 2002-5, December 2002, http://www.math.tntech.edu/techreports/TR_2002_5.pdf

RECENT BOOKS ON CLIFFORD ALGEBRAS:

• "Clifford Algebras - Applications to Mathematics, Physics, and Engineering", Progress in Mathematical Physics, Volume 34, Rafa\l Ab\lamowicz, ed., Birkhauser, Boston 2004, ISBN 0-8176-3525-4, To order, More information
• "Lectures on Clifford (Geometric) Algebras and Applications", Rafa\l Ab\lamowicz and Garret Sobczyk, editors, Birkhauser, Boston 2004, ISBN 0-8176-3257-3, To order, More information
• ``Clifford Algebras and their Applications in Mathematical Physics - Vol. 1 Algebra and Physics, Vol. 2 Clifford Analysis'', Rafa\l Ab\lamowicz and Bertfried Fauser (Vol. 1),  editors; and John Ryan and Wolfgang Sproessig (Vol. 2), editors, Birkhauser Boston 2000, ISBN 0-8176-4182-3 (Vol. 1), 0-8176-4183-1 (Vol. 2), To order vol. 1, To order vol. 2
  
• ``Clifford Algebras with Numeric and Symbolic Computations'', Rafa\l Ab\lamowicz, Pertti Lounesto, Josep M. Parra, editors, Birkhauser, Boston, 1996, ISBN 0-8176-3907-1, To order
  

Please direct all questions, comments, and bug reports to:

Rafal Ablamowicz rablamowicz@tntech.edu
Department of Mathematics, Box 5054
Tennessee Technological University
Cookeville, TN 38505
phone: USA (931) 372-3622
fax: USA (931) 372-6353

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Last updated:  August 8, 2009/ra