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 | 7.4.1 G-algebrasDefinition (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 (Poincaré-Birkhoff-Witt basis), if a  -basis of  is Mon  ,
where a power-product  (in this particular order) is called
a monomial. For example,  is a monomial, while  is, in general, not
a monomial. Definition (G-algebra)Let be a field, and let a  -algebra  be given in terms of generators subject to the following relations: 
 
 
 
 
Note: Note that non-degeneracy 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 G-algebras)
Let  
 
 Setting up a G-algebra
In order to set up a  
 
 PLURAL does not check automatically whether the non-degeneracy conditions hold but it provides a procedure ndcond from the library nctools_lib to check this. | 
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