Deformation and quantization of the Loday-Quillen-Tsygan isomorphism for Calabi-Yau categories
Xiaojun Chen, Farkhod Eshmatov, Maozhou Huang
TL;DR
The work develops a deformation and quantization theory for the Loday–Quillen–Tsygan isomorphism in the setting of Koszul Calabi–Yau algebras and related Calabi–Yau categories. By establishing a Lie bialgebra and, after Koszul duality, a co‑Poisson bialgebra structure on the primitive cyclic data, the authors construct a Hopf algebra quantization that lifts the LQT isomorphism to the quantum level. This quantization aligns with Hennion’s tangent map from the BGL‑to‑K theory framework, providing a quantized tangent correspondence between algebraic K‑theory and cyclic homology. The results apply to important examples such as preprojective algebras and Fukaya categories, offering a versatile toolkit for noncommutative geometry and symplectic topology. Overall, the paper ties Calabi–Yau duality, Koszul duality, and quantum deformation into a coherent quantization of tangent structures in K‑theory and loop‑space contexts.
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 Poincaré 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.