Leavitt path algebras with bases consisting solely of units

Abstract

An algebra is said to be an invertible algebra if it has a basis consisting solely of units. Given a field K and a finite graph E, we give a condition on E that is equivalent to that of the Leavitt path algebra LK(E) being an invertible K-algebra. We derive from this a necessary and sufficient condition for the Cohn path algebra CK(E) to be an invertible K-algebra. In addition, given a unital commutative ring R, sufficient conditions on E for the Leavitt path algebra LR(E) to be an invertible R-algebra are given.