          ### 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)

1. , with nonzero entries of type number ( for will be ignored).

2. , 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.          User manual for Singular version 4.3.2, 2023, generated by texi2html.