  Large-N multi-matrix loop equations are formulated as quadratic difference
equations in concatenation of gluon correlations. Though non-linear, they
involve highest rank correlations linearly. They are underdetermined in many
cases. Additional linear equations for gluon correlations, associated to
symmetries of action and measure are found. Loop equations aren't differential
equations as they involve left annihilation, which doesn't satisfy the Leibnitz
rule with concatenation. But left annihilation is a derivation of the
commutative shuffle product. Moreover shuffle and concatenation combine to
define a bialgebra. Motivated by deformation quantization, we expand
concatenation around shuffle in powers of q, whose physical value is 1. At
zeroth order the loop equations become quadratic PDEs in the shuffle algebra.
If the variation of the action is linear in iterated commutators of left
annihilations, these quadratic PDEs linearize by passage to shuffle reciprocal
of correlations. Remarkably, this is true for regularized versions of the
Yang-Mills, Chern-Simons and Gaussian actions. But the linear equations are
underdetermined just as the loop equations were. For any particular solution,
the shuffle reciprocal is explicitly inverted to get the zeroth order gluon
correlations. To go beyond zeroth order, we find a Poisson bracket on the
shuffle algebra and associative q-products interpolating between shuffle and
concatenation. This method, and a complementary one of deforming annihilation
rather than product are shown to give over and underestimates for correlations
of a gaussian matrix model.
