# nLab dynamical extension of a monoidal category

## Idea

Given a monoid $L$ in a monoidal category, there is an induced monad $T^L$ obtained by tensoring with $L$ from the right. Consider the Kleisli category of the monad $T^L$. It is not necessarily a monoidal category; there is a need for an additional structure (distributive law of a sort) in order to make such a lift. If the monoidal category $C$ in question is the monoidal category of left modules over a Hopf algebra $H$, then any commutative algebra $L$ in the center $Z(C)$ provides a canonical lift to the Kleisli category

#### Physical motivation

Dynamical extension or a dynamization of a monoidal category is an abstract framework to aid the construction of so called dynamical quantum groups which are in turn related to dynamical quantum Yang-Baxter equation.

## Properties and remarks

In Donin-Mudrov (2006) a relation to bialgebroids has been exhibited.

## Literature

• J. Donin, A. Mudrov, Dynamical Yang-Baxter equation and quantum vector bundles, Comm. Math. Phys. 254 (2005), no. 3, 719–760, math.QA/0306028, MR2006i:81099, doi
• J. Donin, A. Mudrov, Quantum groupoids and dynamical categories,
J. Algebra 296 (2006), no. 2, 348–384, math.QA/0311316, MR2007b:17022, doi; MPIM-2004-21, dvi with hyperlinks, ps

Last revised on November 24, 2012 at 05:36:00. See the history of this page for a list of all contributions to it.