. To the IUM main page . 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.