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. In 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 satisfies the TQ-equation. We find a new algebraic structure generated by these L-operators and, as a consequence, by the Q-operators.