Yuqing Shi: Higher enveloping algebras in monochromatic layers

Abstract.

The universal enveloping algebra functor assigns to a Lie algebra a unital associative algebra, characterised by the property that it sends free Lie algebras to free associative algebras. A spectral Lie algebra in an algebra over the so-called spectral Lie operad in the infinity category of spectra, generalising the classical notion of Lie algebras. Ben Knudsen generalises the universal enveloping algebra construction to a functor from the infinity category of spectral Lie algebras to the infinity category of augmented spectral E_n-algebras for any natural number n, known as the higher enveloping algebra functor. Recall that an E_1-algebra is an associative algebra up to coherent homotopies. A monochromatic layer of height h is the localisation of the infinity category of spectra, where the so-called v_h-periodic equivalences are inverted, generalising the rational localisation of spectra. In my talk, I will first introduce the construction of higher enveloping algebras, using a different approach than that of Knudsen. Then I will discuss a relationship between spectral Lie algebras and E_n algebras in the monochromatic layer of height h, which is exhibited by these higher enveloping algebra functors. This is a joint work with Gijs Heuts.