We consider irreducible cyclic representations of the algebra of monodromy matrices correseponding 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 appropriate to one of such representations. It satisfies the TQ-equation. We find a new algebraic structure generating by these L-operators, and as consequence, by the Q-operators.
