On the PBW Property for Universal Enveloping Algebras
Organizers
Speaker
Anton Khoroshkin
Time
Friday, May 30, 2025 1:00 PM - 2:30 PM
Venue
A3-4-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
The classical Poincaré–Birkhoff–Witt (PBW) theorem claims that the universal enveloping algebra of any Lie algebra admits a canonical filtration such that the associated graded algebra is isomorphic to the symmetric algebra on the underlying vector space.
Viewing the universal enveloping algebra $U(g)$ as an associative algebra whose representations correspond bijectively to those of $g$, one can naturally extend the notion of universal enveloping algebras to broader algebraic settings — including Poisson algebras, Lie algebras with multiple compatible brackets, and other related structures.
In this talk, I will present a necessary and sufficient condition for the PBW property to hold for such generalized enveloping algebras, formulated in the language of (colored) operads and Gröbner basis techniques. In particular, I will show that the PBW property fails for Poisson structures and holds for Lie algebras equipped with two compatible brackets.
All necessary definitions and background will be introduced during the talk.
Viewing the universal enveloping algebra $U(g)$ as an associative algebra whose representations correspond bijectively to those of $g$, one can naturally extend the notion of universal enveloping algebras to broader algebraic settings — including Poisson algebras, Lie algebras with multiple compatible brackets, and other related structures.
In this talk, I will present a necessary and sufficient condition for the PBW property to hold for such generalized enveloping algebras, formulated in the language of (colored) operads and Gröbner basis techniques. In particular, I will show that the PBW property fails for Poisson structures and holds for Lie algebras equipped with two compatible brackets.
All necessary definitions and background will be introduced during the talk.