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

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 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 H_ (gl (A) ) has a Lie bialgebra structure which is induced from the Poincar\'e duality of A and deforms H_ (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 BGL to the tangent complex of K-theory.