Colored brackets and 2-manifolds

Frequently, although a set of matrix differential operators will not be clo sed under, say, a Lie bracket, it might be closed under more general Lie Ga mma -graded brackets. Some of these operator products are defined using com mutation factors for Gamma. The classification of commutation factors, and hence Lie Gamma -graded brackets, is one of the stepping stones in an attem pt to classify algebras of matrix differential operators, generalizing a qu estion outline in [Am. J. Math. 114 (6) (1992) 1163]. Many examples of Lie superalgebras of matrix differential operators exist, but there is still a question of what other possible algebra products may close a space of matri x differential operators. Studying Lie color algebras and Lie color superal gebras represents an attempt in this direction. It is a curious fact that c ommutation factors for groups of the form (Z/2Z)(n) coincide with homeomorp hism-equivalence classes of connected compact 2-manifolds, so this coincide nce is also given in Section 4 of this paper. (C) 2001 Elsevier Science B.V . All rights reserved.