Riemann Tensor Polynomial Canonicalization by Graph Algebra Extension

Tensor expression simplification is an "ancient" topic in computer algebra, a representative of which is the canonicalization of Riemann tensor polynomials. Practically fast algorithms exist for monoterm canonicalization, but not for multiterm canonicalization. Targeting the multiterm difficulty, in this paper we establish the extension theory of graph algebra, and propose a canonicalization algorithm for Riemann tensor polynomials based on this theory.

[1]  L.R.U Manssur,et al.  The Canon package: a fast kernel for tensor manipulators , 2004 .

[2]  Teake Nutma,et al.  xTras: A field-theory inspired xAct package for mathematica , 2013, Comput. Phys. Commun..

[3]  X. Jaén,et al.  TTC: symbolic tensor calculus with indices , 1998 .

[4]  Keith O. Geddes,et al.  Application of genetic algorithms to the algebraic simplification of tensor polynomials , 1997, ISSAC.

[5]  Leonard Parker,et al.  MathTensor - a system for doing Tensor analysis by computer , 1994 .

[6]  Renato Portugal,et al.  An algorithm to simplify tensor expressions , 1998 .

[7]  Kasper Peeters,et al.  Introducing Cadabra: a symbolic computer algebra system for field theory problems , 2007, hep-th/0701238.

[8]  Jiang Liu,et al.  A complete classification of canonical forms of a class of Riemann tensor indexed expressions and its applications in differential geometry , 2013 .

[9]  Classifications and canonical forms of tensor product expressions in the presence of permutation symmetries , 2016, 1604.06156.

[10]  Viatcheslav A. Ilyin,et al.  Symbolic simplification of tensor expressions using symmetries, dummy indices and identities , 1991, ISSAC '91.

[11]  N. Obeid.  On the Simplification of Tensor Expressions , 2001 .

[12]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[13]  Renato Portugal,et al.  The Invar tensor package , 2007, Computer Physics Communications.

[14]  B. Svaiter,et al.  Group-Theoretic Approach for Symbolic Tensor Manipulation , 2001, math-ph/0107032.

[15]  B. G. Wybourne,et al.  Normal forms for tensor polynomials. I. The Riemann tensor , 1992 .

[16]  Jiang Liu,et al.  Normalization in Riemann Tensor Polynomial Ring , 2018, J. Syst. Sci. Complex..

[17]  José M. Martín-García,et al.  xPerm: fast index canonicalization for tensor computer algebra , 2008, Comput. Phys. Commun..

[18]  V.A.Ilyin,et al.  ATENSOR - REDUCE program for tensor simplification , 2018 .