
7.4.1 GalgebrasDefinition (PBW basis)Let be a field, and let a algebra be generated by variables subject to some relations. We call an algebra with PBW basis (PoincaréBirkhoffWitt basis), if a basis of is Mon , where a powerproduct (in this particular order) is called a monomial. For example, is a monomial, while is, in general, not a monomial.Definition (Galgebra)Let be a field, and let a algebra be given in terms of generators subject to the following relations:, where . is called a algebra, if the following conditions hold:
Note: Note that nondegeneracy conditions ensure associativity of multiplication,
defined by the relations. It is also proved, that they are necessary and sufficient to
guarantee the PBW property of an algebra, defined via Theorem (properties of Galgebras)Let be a algebra. Then
Setting up a GalgebraIn order to set up a algebra one has to do the following steps:
PLURAL does not check automatically whether the nondegeneracy conditions hold but it provides a procedure ndcond from the library nctools_lib to check this. 