Deformation and quantization of the Loday-Quillen-Tsygan isomorphism for Calabi-Yau categories

Abstract

For an associative algebra $A$, the famous theorem of Loday, Quillen and Tsygan says that there is an isomorphism between the graded symmetric product of the cyclic homology of $A$ and the Lie algebra homology of the infinite matrices $\mathfrak{gl}(A)$, as commutative and cocommutative Hopf algebras. This paper aims to study a deformation and quantization of this isomorphism. We show that if $A$ is a Koszul Calabi-Yau algebra, then the primitive part of the Lie algebra homology $\mathrm{H}_\bullet (\mathfrak{gl}(A))$ has a Lie bialgebra structure which is induced from the Poincar\'e duality of $A$ and deforms $\mathrm{H}_\bullet (\mathfrak{gl}(A))$ to a co-Poisson bialgebra. Moreover, there is a Hopf algebra which quantizes such a co-Poisson bialgebra, and the Loday-Quillen-Tsygan isomorphism lifts to the quantum level, which can be interpreted as a quantization of the tangent map from the tangent complex of $\mathrm{BGL}$ to the tangent complex of K-theory.