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We present a systematic algebraic approach for the weak coupling of Cauchyproblems to multiple Lohe tensor models. For this, we identify an admissibleCauchy problem to the Lohe tensor (LT) model with a characteristic symbolconsisting of four tuples in terms of a size vector, a natural frequencytensor, a coupling strength tensor and admissible initial configuration. Inthis way, the collection of all admissible Cauchy problems to the LT models isequivalent to the space of characteristic symbols. On the other hand, weintroduce a binary operation, namely "{\it fusion operation}" as a binaryoperation between the characteristic symbols. It turns out that the fusionoperation satisfies the associativity and admits the identity element in thespace of characteristic symbols which naturally forms a monoid. By virtue ofthe fusion operation, the weakly coupled system of multiple LT models can beobtained by applying the fusion operation of multiple characteristic symbolscorresponding to the LT models. As a concrete example, we consider a weakcoupling of the swarm sphere model and the Lohe matrix model, and provide asufficient framework leading to emergent dynamics to the proposed weaklycoupled model.

An algebraic approach for the weak coupling of multiple Lohe tensor models