Structure of Hochschild cohomology of path algebras and differential formulation of Euler’s polyhedron formula

Abstract

This article studies the Lie algebra Der(kΓ) of derivations on the path algebra kΓ of a quiver Γ and the Lie algebra on the first Hochschild cohomology group HH1(kΓ). We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical basis of Der(kΓ) and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connection matrix and boundary matrix are introduced to study the relations among these derivations, concluding that the space of edge derivations is the direct sum of the spaces of the vertex derivations and the face derivations, while the dimensions of the latter spaces are the largest possible. By taking dimensions, this relation among spaces of derivations recovers Euler’s polyhedron formula. This relation also leads a combinatorial construction of a canonical basis of the Lie algebra HH1(kΓ), together with a new semidirect sum decomposition of HH1(kΓ).