Poisson structure and second quantization of quantum cluster algebras

Fang Li, Jie Pan

Published 2020-03-27Version 1

Motivated by the phenomenon that compatible Poisson structure on a cluster algebra plays a key role on its quantization (that is, quantum cluster algebra), we introduce the second quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its secondly quantized cluster algebras. In this reason, we characterize the inner and compatible Poisson structures of a quantum cluster algebra without coefficients which are proved to be a standard Poisson structure and a piecewise standard Poisson structure respectively. Following this, it is shown that the second quantization of a quantum cluster algebra without coefficients is in fact trivial. Meanwhile, we give a class of quantum cluster algebras with coefficients which possess a non-trivial second quantization and moreover, we establish a way to obtain a cluster extension admitting non-trivial second quantization from any quantum cluster algebra.