
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 simply ensure associativity of multiplication. 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:
At present, PLURAL does not check automatically whether the nondegeneracy conditions hold but it provides a procedure ndcond from the library nctools_lib to check this. 