Higher enveloping algebras

Short introductory talks.

The universal enveloping algebra functor assigns to a Lie algebra over a field k a unital associative algebra over k, characterised by the property that it sends free Lie algebras to free associative algebras. Using the theory of operads, one can generalise this classical construction to a functor assigning to a free Lie algebra a free Eₙ-algebra, for every natural number n. This construction is known as “higher enveloping algebras”, originally due to Knudsen. In this talk I will introduce the higher enveloping algebra functors using the self Koszul duality of the Eₙ-operad and discuss a relationship between Lie algebras and Eₙ-algebras in some stable infinity categories, exhibited by these higher enveloping algebra functors.