Singular Manual: G-algebras

| ### 7.4.1 G-algebras ### Definition (PBW basis) ### Definition (G-algebra) non-degeneracy conditions: 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) ### Setting up a G-algebra - 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 -Birkhoff-Witt (PBW) basis - define strictly upper triangular matrices (of type `matrix`) - Call the initialization function `nc_algebra(C,D)` (see nc_algebra) with the data 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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