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We prove a family of $n$-th order identities for quantum $R$-matrices ofBaxter-Belavin type in fundamental representation. The set of identitiesincludes the unitarity condition as the simplest one ($n=2$). Our study isinspired by the fact that the third order identity provides commutativity ofthe Knizhnik-Zamolodchikov-Bernard connections. On the other hand the sameidentity gives rise to $R$-matrix valued Lax pairs for the classical integrablesystems of Calogero type. The latter construction uses interpretation ofquantum $R$-matrix as matrix generalization of the Kronecker function. Wepresent a proof of the higher order scalar identities for the Kroneckerfunctions which is then naturally generalized to the $R$-matrix identities.

Higher order analogues of unitarity condition for quantum R-matrices