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.
In this study we employ Density Functional Theory (DFT) methods to investigate the surface energy barrier for electron emission (surface barrier) and thermodynamic stability of Ba and Ba-O species adsorption under conditions of high temperature (appr
oximately 1200 K) and low pressure (approximately $10^{-10}$ Torr) on the low index surfaces of bixbyite $Sc_2O_3$. The role of Ba in lowering the cathode surface barrier is investigated via adsorption of atomic Ba and Ba-O dimers, where the highest simulated dimer coverage corresponds to a single monolayer film of rocksalt BaO. The change of the surface barrier of a semiconductor due to adsorption of surface species is decomposed into two parts: a surface dipole component and doping component. The lowest surface barrier with atomic Ba on $Sc_2O_3$ was found to be 2.12 eV and 2.04 eV for the (011) and (111) surfaces at 3 and 1 Ba atoms per surface unit cell (0.250 and 0.083 Ba per surface O), respectively. The lowest surface barrier for Ba-O on $Sc_2O_3$ was found to be 1.21 eV on (011) for a 7 Ba-O dimer-per-unit-cell coverage (0.583 dimers per surface O). Generally, we found that Ba in its atomic form on $Sc_2O_3$ surfaces is not stable relative to bulk BaO, while Ba-O dimer coverages between 3 to 7 Ba-O dimers per (011) surface unit cell (0.250 to 0.583 dimers per surface O) produce stable structures relative to bulk BaO. Ba-O dimer adsorption on $Sc_2O_3$ (111) surfaces was found to be unstable versus BaO over the full range of coverages studied. Investigation of combined n-type doping and surface dipole modification showed that their effects interact to yield a reduction less than the two contributions would yield separately.
Recent experimental observations indicate that bulk $Sc_2O_3$ (~200 nm thick), an insulator at room temperature and pressure, must act as a good electronic conductor during thermionic cathode operation, leading to the observed high emitted current de
nsities and overall superior emission properties over conventional thermionic emitters which do not contain $Sc_2O_3$. Here, we employ ab initio methods using both semilocal and hybrid functionals to calculate the intrinsic defect energetics of Sc and O vacancies and interstitials and their effects on the electronic properties of $Sc_2O_3$ in an effort to explain the good conduction of $Sc_2O_3$ observed in experiment. The defect energetics were used in an equilibrium defect model to calculate the concentrations of defects and their compensating electron and hole concentrations at equilibrium. Overall, our results indicate that the conductivity of $Sc_2O_3$ solely due to the presence of intrinsic defects in the cathode operating environment is unlikely to be high enough to maintain the magnitude of emitted current densities obtained from experiment, and that presence of impurities are necessary to raise the conductivity of $Sc_2O_3$ to a high enough value to explain the current densities observed in experiment. The necessary minimum impurity concentration to impart sufficient electronic conduction is very small (approximately $7.5x10^{-3}$ ppm) and is probably present in all experiments.
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.
The Grassmann structure of the critical XXZ spin chain is studied in the limit to conformal field theory. A new description of Virasoro Verma modules is proposed in terms of Zamolodchikovs integrals of motion and two families of fermionic creation op
erators. The exact relation to the usual Virasoro description is found up to level 6.
In this article we unveil a new structure in the space of operators of the XXZ chain. We consider the space of all quasi-local operators, which are products of the disorder field with arbitrary local operators. In analogy with CFT the disorder operat
or itself is considered as primary field. In our previous paper, we have introduced the annhilation operators which mutually anti-commute and kill the primary field. Here we construct the creation counterpart and prove the canonical anti-commutation relations with the annihilation operators. We show that the ground state averages of quasi-local operators created by the creation operators from the primary field are given by determinants.
