Help For:

'BIGEBRA' - A Maple Package for Clifford and Grassmann Hopf Gebras

Version 0.16 -- designed for Maple V Release 5.1

(Copyright (c) Rafal Ablamowicz * and Bertfried Fauser§, 1999 - 2002. All rights reserved.)

(*) Department of Mathematics, Box 5054
Tennessee Technological University
Cookeville, TN 38501 USA
Phone: (931) 372-3569, Fax: (931) 372-6353
E-mail: rablamowicz@tntech.edu
URL: http://math.tntech.edu/rafal/ (home of this package)

(§) Fachbereich Physik, Universit"at Konstanz
Fach M678, 78457 Konstanz, Germany
Phone: +49 7531 883786, FAX: +49 7531 884266
E-mail: Bertfried.Fauser@uni-konstanz.de
URL: http://clifford.physik.uni-konstanz.de/~fauser/

Last revised: December 21, 2001 (BF. & RA)

The BIGEBRA package supplememts the CLIFFORD package CLIFF5 version 5 for Maple V (R) rel. 5.1. If BIGEBRA is loaded using with(Bigebra); it loads automatically the Cliff5 package. BIGEBRA patches the Maple define/skeleton and define/multilinear routines of Maples define facility to allow a correct implementention of the tensor product .

The main purpose of the BIGEBRA package is to allow computations in tensor products of Clifford and Graßmann algebras. For this purpose, a tensor product `&t` is defined which is linear with respect to all non-Clifford elements (constants). This allows to perform calculations in Graßmann/Clifford modules and Graßmann/Clifford bundles. Bi- and Hopf algebraic structures as co-units, co-products, switches etc. are employed. All structures of Graßmann Hopf algebra and Clifford biconvolution are implemented. However, using this device, Graßmann-Cayley algebras and bracket or Peano algebras are also supported. Especially the meet (of point fields and of plane fiels in Plücker coordinatization) is implemented here in a very effective way. The join (of point fields) is implemented by the wedge of the CLIFFORD package.

There are several functions which allow the usage of linear operators given in a matrix representation w.r.t. the Graßmann basis. Such operators can act on a single tensor slot, i.e. they are from End /\V, or on two adjacent tensor slots, i.e. they are from End (/\V &t /\V), where /\V is the space underlying the Graßmann algebra.

The BIGEBRA package provides a facility to solve tangle equations [6] for linear operators applied to internal lines of the tangle if the tangle equation has n ingoing and one outgoing line ( n ->1 mapping). This simplifies e.g. the search for Clifford antipodes.

The Clifford product can be defined in terms of Hopf algebras [8]. BIGEBRA uses the Clifford product of CLIFFORD cmul which internally uses by default the cmulRS subroutine based on teh Rota-Stein cliffordization thechnique and Hopf algebraic methods. T he Clifford co-product is derived from co-cliffordization inthe same way.

The Clifford co-product needs an additional bilinear form, called co-scalarproduct, which has to be defined as the global dim_V x dim_V matrix BI. The dimension has to be specified using the global variable dim_V of CLIFFORD. The Clifford co-product needs an initialization which is done by calling once the function make_BI_Id . Some caution is needed here, since dim_V is set to the maximal value 9 by CLIFFORD and the initialization may take very long in this case, so that dim_V should be set to a smaler value if possible.

The BIGEBRA package makes use of some global variables, which are stored in the table _CLIENV. Currently in use are:
- _CLIENV[_SILENT], default = unassigned. If `true` it suppresses lots of startup output.
- _CLIENV[_fakenow], a flag used to detect if BIGEBRA was already loaded. Needed for patching define.
- _CLIENV[_QDEF_PREFACTOR], default = -1. Puts q-deformation into the Graßmann coproduct, (beware: ONLY there for now, the q-busines is not yet officially suppoerted and not well tested).

BIGEBRA can also serve to provide the user a possibility to define various multilinear functions, i.e. tensor products over arbitrary rings, see define .

The help pages of BIGEBRA are part of the same Maple database file (maple.hdb) which contains help pages for 'CLIFFORD' and should be located in a directory in Maple's `libname[1]' variable. BIGEBRA is supposed to merge with CLIFFORD in a forthcomming version for Maple ver. 6/7.

BIGEBRA was already successfully used in deriving mathematically and physically relevant results [1,2,3]. Some references are added to provide information about Clifford Hopf gebras.

Literature:
[1] B. Fauser, On the Hopf-algebraic origin of Wick normal-ordering, J. Phys.A. Math. Gen. 34:105-115, 2001.
[2] B. Fauser, Z. Oziewicz, Clifford Hopf gebra for two dimensional space, Miscellanea Algebraicae, 2(1):31-42, 2001.
[3] B. Fauser, A treatise on qunatum Clifford algebras, Habilitation, Uni. Konstanz, Jan. 2002.
[4] F.D. Grosshans, G.-C. Rota, J.A. Stein, Invariant theory and superalgebras, In Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, number 69, pages i-xxi,1-80. American Mathematical Society, 1987. M. Barnabei, A. Brini, G.-C. Rota, On the exterior calculus of invarinat theory, J. Alg. 96:120-160, 1985.
[5] V. Lyubashenko, Tangles and Hopf algebras in braided categories, J. Pure Apl. Alg. 98:245-278, 1995.
[6] J.W. Milnore, J.C. Moore, On the structure of Hopf algebras, Ann. Math., 81:211-264, 1965. M.E. Sweedler, Hopf algebras, W.A. Benjamin, INC, New York, 1969. E. Abe, Hopf algebras, Cambridge Univ. Press, Cambridge, 1980. N. Bourbaki, Elements of Mathematics, Algebra I, Chapters 1-3, Springer-Verlag, Berlin 1989.
[7] Z. Oziewicz, Clifford Hopf gebra and biuniversal Hopf gebra, Czech. J. Phys. 47(12): 1267-1274, 1997. Z. Oziewicz, Guest Editor's Note, Int. J. Theor. Phys. 40(1):1-13, 2001. Z. Oziewicz, Operad of graphs, convolution and Hopf gebra, Contemp. Math. submitted.
[8] G.-C. Rota, J.A. Stein, Plethystic Hopf algebras, Proc. Natl. Acad. SCi USA, 91:13057-13061, 1994.
[9] A. Zaddach, Grassmanns Algebra in der Geometrie mit Seitenblicken auf verwandte Strukturen, Bi.-Wissenschaftsverlag, Mannheim, 1994

Load Bigebra in the following way, Cliff5 (CLIFFORD ver. 5) is then loaded automatically!

There is a silent option to suppress the startup message, see Bigebra[init] .

> restart:with(Bigebra); #no silent option here

Warning, new definition for drop_t

Warning, new definition for gco_d_monom

Warning, new definition for gco_monom

Warning, new definition for init

To initialize the Clifford coproduct type:

> dim_V:=2:
BI:=linalg[matrix](dim_V,dim_V,[a,b,c,d]);
make_BI_Id();

BI is the dim_V x dim_V matrix of the co-scalarproduct on co-one-vectors, from which the Clifford co-product `&cco` is derived by Rota-Stein co-cliffordization, [2,7,8]. The tensor product `&t` is already defined and ready to use:

> &t(e1,&t(e2,e3),e4); ## associativity, i.e. drop 'parentheses'

> &t(a*e1+sin(theta)*e3,b*e3-1/x*e1); ## multilinearity

>

Alphabetic listing of available procedures in 'BIGEBRA':

&cco -- Clifford co-product on

&gco -- The Graßmann co-product w.r.t. the wedge product.

&gco_d -- dotted Graßmann co-product acting on the undotted wedge basis.

&gpl_co -- Gra\ss mann-Plücker co-product acting on hyperplanes in Plücker coordinatization.

&map -- &map maps a product, i.e. a Clifford valued function of two Clifford polynoms (a 2->1 mapping) onto two adjaicent slots of a tensor .

&t -- The tensor product defined in BIGEBRA during loading of the package.

&v -- Defined the vee-product, i.e. the meet .

tensor polynoms.

bracket -- Defines a bracket in the sense of a Peano space [8].

cco_monom -- internal use only.

contract -- Contract maps a cliscalar valued function of two Clifford polynoms onto two adjaicent tensor slots.

define -- Maple 'define' has bugs, so 'define' had to be replaced by a patched code.

drop_t -- Drops the tensor sign &t in expresions like &t(e1), projects on the first argument in &t(p1,p2,...).

eps -- no longer supported.

EV -- EV is the evaluation of a multi-co-vector on a multivector. Multi-co-vectors are described currently (we are sorry to say) by the same Graßmann basis elements. The user is responsible to take care in which tensor slot co-vectors reside.

gantipode -- Applies the Graßmann antipode to a tensor slot.

gco_unit -- The Graßmann Hopf algebra co-unit.

gswitch -- Graded switch of two adjacent slots of a tensor .

help -- This page.

init -- A quite tricky init function, which loads Cliff5 and patches define .

linop -- Linop defines a linear operator acting on the Graßmann algebra, having a 2^dim_V x 2^dim_V co-contra-variant matrix representing it.

linop2 -- Linop2 defines a linear operator acting on a tensor product of rank two of the Graßmann algebra, having a 4^dim_V x 4^dim_V co-contra-variant matrix representating it.

make_BI_Id -- Initialization routine for the Clifford co-product.

&map -- &map maps a product, i.e. a Clifford valued function of two Clifford polynoms (a 2->1 mapping) onto two adjaicent slots of a tensor .

mapop -- Mapop applies a linear operator (element of End V) defined by linop onto one single slot of a tensor .

mapop2 -- Mapop2 applies a linear tensor-operator (element of End V &t V) defined by linop2 onto two slots of a tensor .

meet -- The meet is equivalent to the &v-(vee)-product .

pairing -- A pairing of two Clifford polynoms .

peek -- Peek gets a Clifford polynom from a tensor at a certain position.

poke -- Poke puts a Clifford polynom into a tensor at a certain position.

remove_eq -- Helper function, which allows to remove trivial equations if tangle equations are solved manually.

switch -- Switch two adjaicent slots of a tensor (Just a swap).

tcollect -- Tcollect collects cliscalar coefficients in a tensor expression.

tsolve1 -- Tsolve1 solves tangle equations with n ingoing and one outgoing line (n--> 1 mappings). It has the ability to solve for operators applied to internal lines of the tangle. Such operators can be defined algebraically or using linop and linop2 .

VERSION -- Displays informations about the current version of BIGEBRA.

New Types in 'BIGEBRA':

type/tensobasmonom - A tensor basis monom having no prefactor.

type/tensormonom - A tensor monom which may have a prefactor of type cliscalar .

type/tensorpolynom - A sum of tensor monoms.

See Also: Cliff5[init] , Cliff5[version] , Bigebra[VERSION] , Bigebra[init]

(c) Copyright December-16-1999, by Rafal Ablamowicz & Bertfried Fauser, all rights reserved.
Last modified: December 26, 2001/BF/RA.