Studies on uantum integrable systems : algebraic structure and correlation functions

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We studied the following two subjects.(1)Algebraic representation of correlation functions(i) We have obtained an algebraic representation for correlation functions for the XXZ and XYZ spin chains, in a form applicable to the physically interesting homogeneous chains.(ii) For the XXZ chain, we found that the operator Omega appearing in the above representation can be expressed as a bilinear form of fermion annihilation operators. Further we constructed creation operators satisfying a certain locality property. We showed that, for the (quasi) local operators obetained from the trivial one by applying creation operators, the vacuum expectation values can be expressed explicitly in terms of determinants.The existence of a fermionic structure for generic values of coupling parameter is an unexpected new feature which deserve further investigation(2) Monomial basis and characters in conformal field thepry(i) We obtained a conjectural monomial basis for the Virasoro minimal modules $M(p,p')$ with $p'/p>2$ in terms of the Fourier coefficients of the $(2,1)$-primary field. We proved the conjecture for the unitary case, and verified the consistency of the conjecture with the character.(ii) We obtained a Weyl-type character formula for the principal subspace of sl(3) (submodule $U(\hat(n))v$ of integrable \hat{sl}(3)-modules, where v is the highest vector and \hat{n} is the current algebra for the nilpotent subalgebra n of sl(3)).As it turns out, this character is an eigenfunction of the quantum Toda Hamiltonian.