New Relations in the Algebra of the Baxter $Q$-Operators

We consider irreducible cyclic representations of the algebra of monodromy matrices corresponding to the $R$-matrix of the six-vertex model. At roots of unity, the Baxter $Q$-operator can be represented as a trace of a tensor product of $L$-operators corresponding to one of these cyclic representations, and this operator satisfies the $TQ$ equation. We find a new algebraic structure generated by these $L$-operators and consequently by the $Q$-operators.