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7.4.1 G-algebrasDefinition (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 (PoincarĂ©-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:
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 190#190 be a 189#189-algebra. Then
Setting up a G-algebraIn order to set up a 189#189–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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