Yangian for cotangent Lie algebras and spectral $R$-matrices

Abstract

In this paper, we present a canonical quantization of Lie bialgebra structures on the formal power series $\mathfrak{d}[[t]]$ with coefficients in the cotangent Lie algebra $\mathfrak{d}=T^{\ast }\mathfrak{g}=\mathfrak{g}⋉{\mathfrak{g}}^{\ast }$ to a simple complex Lie algebra $\mathfrak{g}$. We prove that these quantizations produce twists to the natural analog of the Yangian for $\mathfrak{d}$. Moreover, we construct spectral $R$-matrices for these twisted Yangians as compositions of twisting matrices. The motivation for the construction of these twisted Yangians over $\mathfrak{d}$ comes from certain 4d holomorphic-topological gauge theory. More precisely, we show that pertubative line operators in this theory can be realized as representations of these Yangians. Moreover, the comultiplications of these Yangians correspond to the monodial structure of the category of line operators.

Fixed well-definiteness of non-rational twisted R-matrices in Section 4.3.1 and fixed and fleshed out some things in Section 3.2.2