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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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