On the Lie algebra of differential operators on a path algebra

Details On the Lie algebra of differential operators on a path algebra On the Lie algebra of differential operators on a path algebra

2010-10-11

arxiv.org/abs/1010.1980v1

Mathematics

Rings and Algebras

29 pages

Scientific paper

In this article we study the Lie algebra Diff(k Gamma) of differential operators on the path algebra k Gamma of a quiver Gamma and relate this Lie algebra to the algebraic and combinatorial properties of the path algebra. We first characterize when a linear operator on a path algebra is a differential operator and thus obtain a canonical basis of Diff(k Gamma). We show that, while the Lie ideal of inner differential operators on k Gamma to a large extent determines the Lie algebra structure on k Gamma, the Lie algebra OutDiff(k Gamma) of outer differential operators, defined to be the quotient of Diff(k Gamma) modulo the inner differential operators, is closely related to the topological and graph theoretic properties of k Gamma. In particular, from a careful analysis of the connection matrix and boundary matrix of a quiver, a canonical basis of OutDiff(k Gamma) is obtained.

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