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7.4.1 G-algebrasDefinition (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 (PoincarĂ©-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:
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 Theorem (properties of G-algebras)Let 191#191 be a 190#190-algebra. Then
Setting up a G-algebraIn order to set up a 190#190–algebra one has to do the following steps:
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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