The Master Field for Large-N Matrix Models and Quantum Groups

In recent works by Singer, Douglas, Gopakumar and Gross an application of results of Voiculescu from noncommutative probability theory to constructions of the master field for large-N matrix field theories have been suggested. It turns out that this master field becomes the Boltzmann field in the free Fock space. In this note we consider interrelations between the master field and quantum semigroups. We define the master field algebra and observe that it is isomorphic to the algebra of functions on the quantum semigroup SUq(2) for q=0. The master field becomes a central element of the quantum group bialgebra. The quantum Haar measure on the SUq(2) for any q gives the Wigner semicircle distribution for the master field. Coherent states on SUq(2) become coherent states in the master field theory.