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We consider large-N multi-matrix models whose action closely mimics that ofYang-Mills theory, including gauge-fixing and ghost terms. We show that thefactorized Schwinger-Dyson loop equations, expressed in terms of the generatingseries of gluon and ghost correlations G(xi), are quadratic equations S^i G = Gxi^i G in concatenation of correlations. The Schwinger-Dyson operator S^i isbuilt from the left annihilation operator, which does not satisfy the Leibnitzrule with respect to concatenation. So the loop equations are not differentialequations. We show that left annihilation is a derivation of the graded shuffleproduct of gluon and ghost correlations. The shuffle product is the point-wiseproduct of Wilson loops, expressed in terms of correlations. So in the limitwhere concatenation is approximated by shuffle products, the loop equationsbecome differential equations. Remarkably, the Schwinger-Dyson operator as awhole is also a derivation of the graded shuffle product. This allows us toturn the loop equations into linear equations for the shuffle reciprocal, whichmight serve as a starting point for an approximation scheme.

Schwinger-Dyson operator of Yang-Mills matrix models with ghosts and derivations of the graded shuffle algebra