Anton Khoroshkin: On generalizations of PBW property

A talk at the Midsummer Day Dream Workshop at the Weizmann Institute of Science.
Abstract:
The Poincare-Birkhoff-Witt theorem states that the associated graded to the universal enveloping of a Lie algebra is isomorphic to the symmetric algebra of the underlying Lie algebra.

First, I will explain the categorical definition of the PBW property and why it is not true in positive characteristics.

Second I describe the generalizations of the PBW theorem to different algebraic structures and suggest a useful criterion based on the theory of Grobner bases for coloured operads.

All necessary definitions related to the operad theory will be recalled.

Finally, if time permits, generalizations to the derived setting will be mentioned.

Talk is based on arXiv:1807.05873 and arXiv:2003.06055