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A.1 G-algebras
Definition
Let be a field, and let a -algebra be given in terms of generators and relations:
,
where
.
is called a -algebra, if the following conditions hold:
-
there is a monomial well-ordering
such that
,
- non-degeneracy conditions:
, where
Theorem
Let be a -algebra. Then
-
has a PBW (Poincaré-Birkhoff-Witt) basis,
-
is left and right noetherian,
-
is an integral domain,
-
has left and right quotient rings.
In order to set up a -algebra
in PLURAL, one has to do the following steps:
-
define a commutative ring
, equipped
with a global monomial ordering,
-
define strictly
upper triangular matrices
-
, with nonzero entries of type number,
-
, with polynomial entries from .
-
call the initialization function
ncalgebra ( ncalgebra)
with the data
and .
At present, we do not check automatically whether non-degeneracy conditions
hold but provide a procedure ndc from the library nctools.lib instead.
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