
7.6 Graded commutative algebras (SCA)
A Subsystem for Graded Commutative Algebras
This section describes basic mathematical notions, definition, and a little bit the implementation of
the experimental noncommutative kernel extension SCA of SINGULAR
which improves performance of many algorithms in graded commutative algebras.
In order to improve performance of SINGULAR in specific noncommutative algebras
one can extend the internal implementation for them in a virtualmethodoverloadinglike manner.
At the moment graded commutative algebras (SCA) and in particular exterior algebras are implemented this way.
Note that graded commutative algebras require no special user actions apart from
defining an appropriate noncommutative GRalgebra in SINGULAR.
Upon doing that, the suppercommutative structure will be automatically detected and
special multiplication will be used.
Moreover, in most SCAaware (e.g. std ) algorithms special internal improvements will be used
(otherwise standard generic noncommutative implementations will be used).
All considered algebras are assumed to be associative
algebras for some ground field
.
 Definition
Polynomial graded commutative algebras are factors of tensor products of commutative algebras
with an exterior algebra over a ground field
.
These algebras can be naturally endowed with a
grading, where anticommutative algebra generators
have degree
and commutative algebra generators (and naturally scalars) have degree
.
In this particular case they may be considered as supercommutative algebras.
 GRalgebra representation
A graded commutative algebra with
commutative and
anticommutative algebra generators can be represented
as factors of the following GRalgebra by some twosided ideal:
 Distinctive features
Graded commutative algebras are Noetherian.
Graded commutative algebras have zero divisors if and only if
:
.
Unlike other noncommutative algebras one may use any monomial ordering
where only the noncommutative variables are required to be global.
In particular, commutative variables are allowed to be local.
This means that one can work in tensor products of any commutative ring with an exterior algebra.
 Example of defining graded commutative algebras in SINGULAR:SCA and computing with them
Given a commutative polynomial ring
, supercommutative structure on it can be introduced as follows:
 LIB "nctools.lib";
ring r = 0,(a, b, x,y,z, Q, W),(lp(2), dp(3), Dp(2));
// Let us make variables x = var(3), ..., z = var(5) to be anticommutative
// and add additionally a quotient ideal:
def S = superCommutative(3, 5, ideal(a*W + b*Q*x + z) ); setring S; S;
==> // coefficients: QQ
==> // number of vars : 7
==> // block 1 : ordering lp
==> // : names a b
==> // block 2 : ordering dp
==> // : names x y z
==> // block 3 : ordering Dp
==> // : names Q W
==> // block 4 : ordering C
==> // noncommutative relations:
==> // yx=xy
==> // zx=xz
==> // zy=yz
==> // quotient ring from ideal
==> _[1]=xz
==> _[2]=bxyQyz
==> _[3]=aW+bxQ+z
==> _[4]=z2
==> _[5]=y2
==> _[6]=x2
ideal I = a*x*y + z*Q + b, y*Q + a; I;
==> I[1]=axy+b+zQ
==> I[2]=a+yQ
std(I); // Groebner basis is used here since > is global
==> _[1]=yQWz
==> _[2]=yz
==> _[3]=b+zQ
==> _[4]=a+yQ
kill r, S;
// Let's do the same but this time with some local commutative variables:
ring r = 0,(a, b, x,y,z, Q, W),(dp(1), ds(1), lp(3), ds(2));
def S = superCommutative(3, 5, ideal(a*W + b*Q*x + z) ); setring S; S;
==> // coefficients: QQ
==> // number of vars : 7
==> // block 1 : ordering dp
==> // : names a
==> // block 2 : ordering ds
==> // : names b
==> // block 3 : ordering lp
==> // : names x y z
==> // block 4 : ordering ds
==> // : names Q W
==> // block 5 : ordering C
==> // noncommutative relations:
==> // yx=xy
==> // zx=xz
==> // zy=yz
==> // quotient ring from ideal
==> _[1]=xz
==> _[2]=yzbxyQ
==> _[3]=aW+z+bxQ
==> _[4]=x2
==> _[5]=y2
==> _[6]=z2
ideal I = a*x*y + z*Q + b, y*Q + a; I;
==> I[1]=axy+zQ+b
==> I[2]=a+yQ
std(I);
==> _[1]=yQWzbxQ
==> _[2]=zQ+b
==> _[3]=bx
==> _[4]=by
==> _[5]=bz
==> _[6]=b2
==> _[7]=a+yQ

See example of superCommutative from the library nctools.lib .
Reference: Ph.D thesis by Oleksandr Motsak (2010), https://nbnresolving.org/urn:nbn:de:hbz:386kluedo26479.
