
7.5.1 bimodules_lib
 Library:
 bimodules.lib
 Purpose:
 Tools for handling bimodules
 Authors:
 Ann Christina Foldenauer, Christina.Foldenauer@rwthaachen.de
Viktor Levandovskyy, levandov@math.rwthaachen.de
 Overview:
The main purpose of this library is the handling of bimodules
which will help e.g. to determine weak normal forms of representation matrices
and total divisors within noncommutative, nonsimple Galgebras.
We will use modules homomorphisms between a Galgebra and its enveloping algebra
in order to work left Groebner basis theory on bimodules.
Assume we have defined a (noncommutative) Galgebra A over the field K, and an (A,A)bimodule M.
Instead of working with M over A, we define the enveloping algebra A^{env} = A otimes_K A^{opp}
(this can be done with command envelope(A)) and embed M into A^{env} via imap().
Thus we obtain the left A^{env}module M otimes 1 in A^{env}.
This has a lot of advantages, because left module theory has much more commands
that are already implemented in SINGULAR:PLURAL. Two important procedures that we can use are std()
which computes the left Groebner basis, and NF() which computes the left normal form.
With the help of this method we are also able to determine the set of bisyzygies of a bimodule.
A builtin command twostd in PLURAL computes the twosided Groebner basis of an ideal
by using the right completion algorithm of [2]. bistd from this library uses very different
approach, which is often superior to the right completion.
 References:
The procedure bistd() is the implementation of an algorithm M. del Socorro Garcia Roman presented in [1](page 6678).
[1] Maria del Socorro Garcia Roman, Effective methods in Algebras with PBW bases:
Galgebras and YangBaxter Algebras, Ph.D. thesis, Universidad de La Laguna, 2005.
[2] Viktor Levandovskyy, Noncommutative Computer Algebra for polynomial Algebras:
Groebner Bases, Applications and Implementations, Ph.D. thesis, Kaiserlautern, 2005.
[3] N. Jacobson, The theory of rings, AMS, 1943.
[4] P. M. Cohn, Free Rings and their Relations, Academic Press Inc. (London) Ltd., 1971.
Procedures:
7.5.1.0. bistd   computes the twosided Groebner bases of an ideal or module 
7.5.1.0. bitrinity   computes the trinity of M: Groebner basis, lift matrix and bisyzygies 
7.5.1.0. liftenvelope   computes the coefficients of an element g concerning the generators of a bimodule M in the enveloping algebra 
7.5.1.0. CompDecomp   returns an ideal which contains the component decomposition of a polynomial p in the enveloping algebra regarding the right side of the tensors 
7.5.1.0. isPureTensor   checks whether an element p in A^{env} is a pure tensor 
7.5.1.0. isTwoSidedGB   checks whether an ideal I is twosided Groebner basis 
See also:
ncalg_lib;
nctools_lib.
