Valence bond ground states in quantum antiferromagnets and quadratic algebras

The wavefunctions corresponding to the zero-energy eigenvalue of a one-dime nsional quantum chain Hamiltonian can be written in a simple way using quad ratic algebras. Hamiltonians describing stochastic processes have stationar y states given by such wavefunctions and various quadratic algebras have be en found and applied to several diffusion processes. We show that similar m ethods can also be applied for equilibrium processes. As an example, for a class of q-deformed O(N) symmetric antiferromagnetic quantum chains, we giv e the zero-energy wavefunctions for periodic boundary conditions correspond ing to momenta zero and pi. We also consider free and various non-diagonal boundary conditions and give the corresponding wavefunctions. All correlati on lengths are derived.