Title 



The PBW property for associative algebras as an integrability condition
 
Author 


  
Abstract 



We develop an elementary method for proving the PoincaréBirkhoffWitt (PBW) property for associative quadraticlinear 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 noncommutative potential in three variables. Finally, as a third example, we found a criterion, for a general quadratic algebra which is the quotientalgebra of T(V)[ℏ] by the twosided ideal, generated by (xi⊗xj−xj⊗xi−ℏϕij)i,j, with ϕij general quadratic noncommutative 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 quadraticlinear algebra fulfils it.   
Language 



English
 
Source (journal) 



MRL Mathematical research letters.  Cambridge  
Publication 



Cambridge : International Press, 2014
 
ISSN 



10732780
 
Volume/pages 



21:6(2014), p. 14071434
 
ISI 



000353048100011
 
Full text (Publisher's DOI) 


  
