Definition (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:
, where .
is called a -algebra, if the following conditions hold:
Note: Note that non-degeneracy conditions simply ensure associativity of multiplication.
Theorem (properties of G-algebras)
Let be a -algebra. Then
Setting up a G-algebra
In order to set up a -algebra one has to do the following steps: