Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex

TL;DR

This paper extends the concept of Manin triples to dg Lie algebras, introducing examples related to complex punctured surfaces and higher current algebras, using a novel rectilinear space framework and adelic-like complexes.

Contribution

It introduces homotopy Manin triples for dg Lie algebras and constructs models of derived sections in rectilinear spaces, connecting to higher current algebras and adelic complexes.

Findings

Examples associated with punctured complex surfaces

Construction of a rectilinear space framework

Development of adelic-like complexes for derived sections

Abstract

The notion of a Manin triple of Lie algebras admits a generalization, to dg Lie algebras, in which various properties are required to hold only up to homotopy. This paper introduces two classes of examples of such homotopy Manin triples. These examples are associated to analogs in complex dimension two of, respectively, the punctured formal 1-disc, and the complex plane with multiple punctures. The dg Lie algebras which appear include certain higher current algebras in the sense of Faonte, Hennion and Kapranov arXiv:1701.01368. We work in a ringed space we call rectilinear space, and one of the tools we introduce is a model of the derived sections of its structure sheaf, whose construction is in the spirit of the adelic complexes for schemes due to Parshin and Beilinson.