The PBW property for associative algebras as an integrability condition

Abstract

We develop an elementary method for proving the PBW theorem for associative algebras with an ascending filtration. The idea is roughly the following. At first, we deduce a proof of the PBW property for the {\it ascending} filtration (with the filtered degree equal to the total degree in $x_i$'s) to a suitable PBW-like property for the {\it descending} filtration (with the filtered degree equal to the power of a polynomial parameter $\hbar$, introduced to the problem). This PBW property for the descending filtration guarantees the genuine PBW property for the ascending filtration, for almost all specializations of the parameter $\hbar$. At second, we develop some very constructive method for proving this PBW-like property for the descending filtration by powers of $\hbar$, emphasizing its integrability nature. We show how the method works in three examples. As a first example, we give a proof of the classical Poincar\'{e}-Birkhoff-Witt theorem for Lie algebras. As a second, much less trivial example, we present a new proof of a result of Etingof and Ginzburg [EG] on PBW property of algebras with a cyclic non-commutative potential in three variables. Finally, as a third example, we found a criterium, for a general quadratic algebra which is the quotient-algebra of $T(V)[\hbar]$ by the two-sided ideal, generated by $(x_i\otimes x_j-x_j\otimes x_i-\hbar\phi_{ij})_{i,j}$, with $\phi_{ij}$ general quadratic non-commutative polynomials, to be a PBW for generic specialization $\hbar=a$. This result seems to be new.