## Double shuffle relations for multiple Dedekind zeta values

### Volume 180 / 2017

#### Abstract

We define two types of shuffle relations for multiple Dedekind zeta values over an imaginary quadratic field. Multiple Dedekind zeta values were defined by the author (2014). We give examples of integral shuffle relations in terms of iterated integrals over membranes, and of infinite sum shuffle relations. Then we establish both types of shuffles in general, which represent a product of two ordinary multiple Dedekind zeta values as a sum of such in two ways. This leads to a linear relation among ordinary multiple Dedekind zeta values for imaginary quadratic fields. We also present explicit formulas for the two types of shuffle for the product $\zeta _{K}(2)\zeta _{K}(2)$.