| ### 7.4.1 G-algebras ### Definition (PBW basis) Let 50#50 be a field, and let a 50#50-algebra 191#191 be generated by variables 221#221 subject to some relations. We call 191#191 an algebra with PBW basis (Poincaré-Birkhoff-Witt basis), if a 50#50-basis of 191#191 is Mon 222#222, where a power-product 223#223 (in this particular order) is called a monomial. For example, 224#224 is a monomial, while 225#225 is, in general, not a monomial. ### Definition (G-algebra) Let 50#50 be a field, and let a 50#50-algebra 191#191 be given in terms of generators subject to the following relations: 226#226, where 227#227. 191#191 is called a 190#190–algebra, if the following conditions hold: there is a monomial well-ordering 228#228 on 229#229 such that 230#230, non-degeneracy conditions: 231#231, where 232#232 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 191#191 be a 190#190-algebra. Then 191#191 has a PBW (Poincaré-Birkhoff-Witt) basis, 191#191 is left and right noetherian, 191#191 is an integral domain. ### Setting up a G-algebra In order to set up a 190#190–algebra one has to do the following steps: - define a commutative ring 233#233, equipped with a monomial ordering 228#228 (see ring declarations (plural)). This provides us with the information on a field 50#50 (together with its parameters), variables 234#234and an ordering <. From the sequence of variables we will build a G-algebra with the Poincaré-Birkhoff-Witt (PBW) basis 235#235. - define strictly 236#236 upper triangular matrices (of type `matrix`) 237#237, with nonzero entries 211#211 of type number (211#211 for 238#238 will be ignored). 239#239, with polynomial entries 212#212 from 53#53 (212#212 for 238#238 will be ignored). - Call the initialization function `nc_algebra(C,D)` (see nc_algebra) with the data 78#78 and 240#240. 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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