### 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`)

  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.

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