We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on the corresponding representation spaces. Moreover, we prove that they are in one-to-one correspondence with local lattice double Poisson algebras, a new important class among Van den Bergh’s double Poisson algebras. We derive several classification results, and we exhibit their relation to non-abelian integrable differential-difference equations. A rigorous definition of double multiplicative Poisson vertex algebras in the non-local and rational cases is also provided.

中文翻译：

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双乘法泊松顶点代数

我们发展了双乘泊松顶点代数的理论。这些在关联代数级别定义的结构被证明是这样的，它们在相应的表示空间上引入了乘法泊松顶点代数的经典结构。此外，我们证明了它们与局部格双泊松代数一一对应，这是范登伯格双泊松代数中的一个新的重要类。我们得出了几个分类结果，并展示了它们与非阿贝尔可积微分方程的关系。还提供了非局部和有理情况下双乘泊松顶点代数的严格定义。