Details:
Title  Computation of noncommutative Gr\"obner bases in Grassmann and Clifford algebras.  Author(s)  Rafal Ablamowicz  Type  Article in Journal  Abstract  Tensor, Clifford and Grassmann algebras belong to a wide class of noncommutative algebras that have a PoincaréBirkhoffWitt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “nondegeneracy condition”. This has led V. Levandovskyy to a rediscovery of the so called Galgebras (previously introduced by J. Apel) and GRalgebras (Gröbnerready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Gröbner bases for commutative and noncommutative algebras. It was T. Stokes who eighteen years ago introduced Gröbner left bases (GLB) and Gröbner left ideal bases (GLIB) for left ideals in Grassmann algebras, with the GLIB bases solving an ideal membership problem. Thus, a natural question is to first seek Gröbner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levandovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Gröbner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise this way. Examples of Galgebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GRalgebras, which are either G algebras or are isomorphic to quotient algebras of a Galgebra modulo a proper twosided ideal, include Grassmann and Clifford algebras. After recalling basic concepts behind the theory of commutative Gröbner bases, a review of the Gröbner bases in PBW algebras, G,and GRalgebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. GLB and GLIB bases will also be computed.  Keywords  PBW bases, monomial order, Gröbner GLB and GLIB bases, normal form, Spolynomial, Grassmann algebra, Clifford algebra, normal form  ISSN  01887009; 16614909/e 
URL 
http://link.springer.com/article/10.1007%2Fs0000601002050 
Language  English  Journal  Adv. Appl. Clifford Algebr.  Volume  20  Number  34  Pages  447476  Publisher  Springer (Birkh\"auser), Basel  Year  2010  Edition  0  Translation 
No  Refereed 
No 
