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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 (PoincarĂ©-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)

    1. 235#235, with nonzero entries 210#210 of type number (210#210 for 236#236 will be ignored).

    2. 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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