Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator for the homogeneous XXX model as integral operator in standard representation of SL(2).

The connection between Q-operator and local Hamiltonians is discussed.

It is shown that operator of Lipatov's duality symmetry arises naturally as leading term of the asymptotic expansion of Q-operator for large values of spectral parameter.

Consider the universal difference operator $D = \sum_{k=0}^N (-1)^k T_k(u) e^{-k\partial_u}$ whose coefficients $T_k(u): M \to M$ are the XXX transfer matrices associated with $M$.

We show that the difference equation $Df = 0$ for an $M$-valued function $f$ has a basis of solutions consisting of quasi-exponentials.

We prove the same for the universal differential operator $D = \sum_{k=0}^N (-1)^k S_k(u) \partial_u^{N-k}$ whose coefficients $S_k(u) : M \to M$ are the Gaudin transfer matrices associated with the tensor product $M$ of finite-dimensional polynomial evaluation $gl_N[x]$-modules.

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Like any other reference manual, this document is dry in places.