Umbral Calculus and Hopf Algebras

The present volume represents a unique blending of two fields only recently recognized as related. On one hand lies the field of Combinatorics with roots (at least immediately traceable via generating functions to Umbral Calculus, the specialty at hand) in the 19th century writings of Boole on operator calculus. Both the foundations and much of the history of the Umbral Calculus are explored in great clarity in [R-K-O and R-R} which have extensive bibliographies. On the other hand is the field of Hopf Algebras, which is usually traced to the paper of Milnor and Moore [M-M] but whose first general exposition is little more than a decade old [S]. For some years, Gian-Carla Rota wrote that this theory should be directly app icabe to Combinatorcs, especiafly the Umbra Ca)culus, but the first distinct attention given by specialists is probably the lecture Moss Sweedler save at the 13-th Dennison Algebra Conference, Dennison College, 1978. This did not really appear in print, nor did Sweedler’s Hopf Algebra colleagues sieze the subject and carry it fur ther forward. Both Rota and Sweedler, therefore, were pleased when the University of Oklahoma was able to support their joint appearance at a conference funded by the J. C. Karcher Foundation in May, 1978. The conference centered on lectures they gave, with S. A. Joni assisting Rota. Sweedler lectured first on elementary coalgebra theory aimed at combinatorists, Rota on elementary combinatorics aimed at the algebraists. Both lectures converged toward those who were or would work at the intersection. Sweedler and Warren Nichols prepared notes of Sweedler’s talks and Joni and Rota of Rota’s and a mimeographed version was circulated by the Oklahoma Mathematics Department. The present volume represents an attempt to make these more accessible. The Sweedler notes here are essentially unchanged from those distributed by Oklahoma. They aim, in a direct and elementary way, to give the reader sufficient knowledge of coalgebra
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