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# Mathematical Research Letters

## Volume 21 (2014)

### Number 6

### The PBW property for associative algebras as an integrability condition

Pages: 1407 – 1434

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n6.a11

#### Author

#### Abstract

We develop an elementary method for proving the Poincaré–Birkhoff–Witt (PBW) property for associative quadratic-linear algebras, complementary to Drinfeld’s 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 $T(V )[\hbar]$ 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 PBW for generic specialization $\hbar = a$. This result seems to be new.

Our condition for PBW property is only sufficient and not necessary, whence the Drinfeld’s 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.