  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.
