60 research outputs found
Tsygan formality and Duflo formula
We prove the 0-(co)homology part of the conjecture on the cup-products on
tangent cohomology in the Tsygan formality [Sh2]. We discuss its applications
to the Duflo formula.Comment: 12 pages, 6 eps figure
The PBW property for associative algebras as an integrability condition
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 's) to a suitable
PBW-like property for the {\it descending} filtration (with the filtered degree
equal to the power of a polynomial parameter , 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 . At second, we develop some very
constructive method for proving this PBW-like property for the descending
filtration by powers of , 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 by the two-sided ideal, generated by
, with
general quadratic non-commutative polynomials, to be a PBW for generic
specialization . This result seems to be new.Comment: 19 page
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