| ### 7.4.1 G-algebras ### 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: there is a monomial well-ordering on such that , non-degeneracy conditions: , where 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 `C_ij` and `D_ij` as above. ### Theorem (properties of G-algebras) Let be a -algebra. Then has a PBW (Poincaré-Birkhoff-Witt) basis, is left and right noetherian, is an integral domain. ### Setting up a G-algebra In order to set up a -algebra one has to do the following steps: - define a commutative ring , equipped with a monomial ordering (see ring declarations (plural)). This provides us with the information on a field (together with its parameters), variables and an ordering <. From the sequence of variables we will build a G-algebra with the Poincaré-Birkhoff-Witt (PBW) basis . - define strictly upper triangular matrices (of type `matrix`) , with nonzero entries of type number ( for will be ignored). , with polynomial entries from ( for will be ignored). - Call the initialization function `nc_algebra(C,D)` (see nc_algebra) with the data and . 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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