The PBW property for associative algebras as an integrability conditionThe PBW property for associative algebras as an integrability condition
Faculty of Sciences. Mathematics and Computer Science
2014Cambridge :International Press, 2014
MRL Mathematical research letters. - Cambridge
21(2014):6, p. 1407-1434
University of Antwerp
We develop an elementary method for proving the PoincaréBirkhoffWitt (PBW) property for associative quadratic-linear algebras, complementary to Drinfelds results. The method is very transparent and emphasizes the integrability nature of PBW property. We show how the method works in three examples. As a first example, we give a proof of the classical PBW theorem for Lie algebras. As a second, less trivial example, we present a new proof of a result of Etingof and Ginzburg on PBW property of algebras with a cyclic non-commutative potential in three variables. Finally, as a third example, we found a criterion, for a general quadratic algebra which is the quotient-algebra of T(V)[ℏ] by the two-sided ideal, generated by (xi⊗xj−xj⊗xi−ℏϕij)i,j, with ϕij general quadratic non-commutative polynomials, to be PBW for generic specialization ℏ=a. This result seems to be new. Our condition for PBW property is only sufficient and not necessary, whence the Drinfelds result in [D, Theorem 2] gives a necessary and sufficient condition. On the other hand, the Drinfeld condition is a countable sequence of equations, and it may be hard to check all of them in practice. Our criterion is a single equation, and is easily checkable, when a particular quadratic-linear algebra fulfils it.