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Register a new userPBW property for associative universal enveloping algebras over an operad
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Authors
Anton Khoroshkin
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Given a symmetric operad $mathcal{P}$ and a $mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${mathsf{U}_{mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $mathcal{P}$ is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for $mathcal{P}$ is discovered. Moreover, given any symmetric operad $mathcal{P}$, together with a Grobner basis $G$, a condition is given in terms of the structure of the underlying trees associated with leading monomials of $G$, sufficient for the PBW property to hold. Examples are provided.
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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 $(x_iotimes x_j-x_jotimes x_i-hbarphi_{ij})_{i,j}$, with $phi_{ij}$ general quadratic non-commutative polynomials, to be a PBW for generic specialization $hbar=a$. This result seems to be new.
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
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