| ### 7.4.1 G-algebras ### Definition (PBW basis) Let 50#50 be a field, and let a 50#50-algebra 190#190 be generated by variables 219#219 subject to some relations. We call 190#190 an algebra with PBW basis (Poincaré-Birkhoff-Witt basis), if a 50#50-basis of 190#190 is Mon 220#220, where a power-product 221#221 (in this particular order) is called a monomial. For example, 222#222 is a monomial, while 223#223 is, in general, not a monomial. ### Definition (G-algebra) Let 50#50 be a field, and let a 50#50-algebra 190#190 be given in terms of generators subject to the following relations: 224#224, where 225#225. 190#190 is called a 189#189–algebra, if the following conditions hold: there is a monomial well-ordering 226#226 on 227#227 such that 228#228, non-degeneracy conditions: 229#229, where 230#230 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 190#190 be a 189#189-algebra. Then 190#190 has a PBW (Poincaré-Birkhoff-Witt) basis, 190#190 is left and right noetherian, 190#190 is an integral domain. ### Setting up a G-algebra In order to set up a 189#189–algebra one has to do the following steps: - define a commutative ring 231#231, equipped with a monomial ordering 226#226 (see ring declarations (plural)). This provides us with the information on a field 50#50 (together with its parameters), variables 232#232and an ordering <. From the sequence of variables we will build a G-algebra with the Poincaré-Birkhoff-Witt (PBW) basis 233#233. - define strictly 234#234 upper triangular matrices (of type `matrix`) 235#235, with nonzero entries 210#210 of type number (210#210 for 236#236 will be ignored). 237#237, with polynomial entries 211#211 from 53#53 (211#211 for 236#236 will be ignored). - Call the initialization function `nc_algebra(C,D)` (see nc_algebra) with the data 78#78 and 238#238. 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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