NDLI: Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras

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Advances in Applied Clifford Algebras : Volume 20, Issue 3-4, October 2010

Preface

Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras

Clifford Algebra Applied to Grover’s Algorithm

Clifford Algebra Bundles to Multidimensional Image Segmentation

Quantum/Classical Interface: Classical Geometric Origin of Fermion Spin

Hyperbolic Quaternion Formulation of Electromagnetism

Degenerate Spin Groups as Semi-Direct Products

Hyperbolic Extensions of Integral Formulas

Clifford Analysis Solution of the Electromagnetic Boundary Value Problem in a Gravitational Background Vacuum

About the Structure of Meson Algebras

Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups

The Energy-Momentum Tensor in Premetric Electrodynamics

On the Spectrum of the Twisted Dolbeault Laplacian on Line Bundles over Kähler Manifolds

Noncommutativity and Parametrization of Fields: The Scalar Electrodynamics Case

Semiclassical Pseudodifferential Operators and the Atiyah-Singer Index Theorem

An Approach to Models of Order-Disorder and Ising Lattices

Variational Formulation for Quaternionic Quantum Mechanics

On the Dirac and Spin-Dirac Operators

On the Unification of Interactions by Clifford Algebra

On Directional Hilbert Operators for Regular Quaternionic Functions on $${\mathbb R}^3$$

Quaternionic Analysis, Elliptic Problems and a Physical Application of the Dbar Formalism

Oscillatory Movements and Dual Quaternions

ELKO, Flagpole and Flag-Dipole Spinor Fields, and the Instanton Hopf Fibration

Killing Vector Fields, Maxwell Equations and Lorentzian Spacetimes

Clifford Algebra, its Generalizations, and Applications to Quantum Information and Computing

Dynamic Geometric Graph Processes: Adjacency Operator Approach

Exponential and Cayley Maps for Dual Quaternions

Clifford Algebras and the Minimal Representations of the 1D N-Extended Supersymmetry Algebra

An Algebraic Remark on Clifford Algebras and Differential Operators

Robot Perception and Handling Actions Using the Conformal Geometric Algebra Framework

Advances in Applied Clifford Algebras : Volume 20, Issue 2, May 2010

Advances in Applied Clifford Algebras : Volume 20, Issue 1, March 2010

Advances in Applied Clifford Algebras : Volume 19

Advances in Applied Clifford Algebras : Volume 18

Advances in Applied Clifford Algebras : Volume 17

Advances in Applied Clifford Algebras : Volume 16

Advances in Applied Clifford Algebras : Volume 15

Advances in Applied Clifford Algebras : Volume 14

Advances in Applied Clifford Algebras : Volume 13

Advances in Applied Clifford Algebras : Volume 12

Advances in Applied Clifford Algebras : Volume 11

Advances in Applied Clifford Algebras : Volume 10

Advances in Applied Clifford Algebras : Volume 9

Advances in Applied Clifford Algebras : Volume 8

Advances in Applied Clifford Algebras : Volume 7

Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras

Content Provider	Springer Nature Link
Author	Abłamowicz, Rafał
Copyright Year	2010
Abstract	Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincaré-Birkhoff-Witt (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 “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Gröbner-ready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Gröbner bases for commutative and non-commutative 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 G-algebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided 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 GR-algebras 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.
Ending Page	476
Page Count	30
Starting Page	447
File Format	PDF
ISSN	01887009
e-ISSN	16614909
Journal	Advances in Applied Clifford Algebras
Issue Number	3
Volume Number	20
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2010-03-09
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	S-polynomial Exterior algebra, Grassmann algebras Theoretical, Mathematical and Computational Physics Grassmann algebra monomial order normal form Physics Mathematical Methods in Physics Gröbner GLB and GLIB bases Noetherian rings and modules Clifford algebras, spinors Clifford algebra PBW bases Applications of Mathematics GrÃ¶bner bases; other bases for ideals and modules (e.g., Janet and border bases)
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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Robust Graph Ideals

Characteristic Modules of Dual Extensions and Gröbner Bases

Some algebraic invariants related to mixed product ideals

Betti numbers of mixed product ideals

Functional Identities on Matrix Algebras

Toric ideals and diagonal 2-minors

Ideals of Graph Homomorphisms

On the real rank of monomials

On the radical of a monomial ideal

Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras

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Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras