  The concept of global conformal invariance (GCI) opens the way of applying
algebraic techniques, developed in the context of 2-dimensional chiral
conformal field theory, to a higher (even) dimensional space-time. In
particular, a system of GCI scalar fields of conformal dimension two gives rise
to a Lie algebra of harmonic bilocal fields, V_m(x,y), where the m span a
finite dimensional real matrix algebra M closed under transposition. The
associative algebra M is irreducible iff its commutant M' coincides with one of
the three real division rings. The Lie algebra of (the modes of) the bilocal
fields is in each case an infinite dimensional Lie algebra: a central extension
of sp(infty,R) corresponding to the field R of reals, of u(infty,infty)
associated to the field C of complex numbers, and of so*(4 infty) related to
the algebra H of quaternions. They give rise to quantum field theory models
with superselection sectors governed by the (global) gauge groups O(N), U(N),
and U(N,H)=Sp(2N), respectively.
