We find the $ell$-weights and the $ell$-weight vectors for the highest $ell$-weight $q$-oscillator representations of the positive Borel subalgebra of the quantum loop algebra $U_q(mathcal L(mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Having t
his, we establish the explicit relationship between the $q$-oscillator and prefundamental representations. Our consideration allows us to conclude that the prefundamental representations can be obtained by tensoring $q$-oscillator representations.
For the case of quantum loop algebras $mathrm U_q(mathcal L(mathfrak{sl}_{l + 1}))$ with $l = 1, 2$ we find the $ell$-weights and the corresponding $ell$-weight vectors for the representations obtained via Jimbos homomorphism, known also as evaluatio
n representations. Then we find the $ell$-weights and the $ell$-weight vectors for the $q$-oscillator representations of Borel subalgebras of the same quantum loop algebras. This allows, in particular, to relate $q$-oscillator and prefundamental representations.
A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $mathrm U_q(mathcal L(mathfrak{sl}_3))$ is given. The full proof of the functional relations in the form independent of
the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.
We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(mathcal L(mathfrak{sl}_2))$. We give a complete set of the functional relations correcting
inexactitudes of the previous considerations. A special attention is given to the connection of the representations used to construct the universal transfer operators and $Q$-operators.
We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group $U_q(mathcal L(m
athfrak{sl}_2))$ related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.
We continue our exercises with the universal $R$-matrix based on the Khoroshkin and Tolstoy formula. Here we present our results for the case of the twisted affine Kac--Moody Lie algebra of type $A^{(2)}_2$. Our interest in this case is inspired by t
he fact that the Tzitzeica equation is associated with $A^{(2)}_2$ in a similar way as the sine-Gordon equation is related to $A^{(1)}_1$. The fundamental spin-chain Hamiltonian is constructed systematically as the logarithmic derivative of the transfer matrix. $L$-operators of two types are obtained by using q-deformed oscillators.
