MONOID GRADINGS ON ALGEBRAS AND THE CARTAN DETERMINANT CONJECTURE

In this work we tackle the Cartan determinant conjecture for finite-di mensional algebras through monoid gradings. Given an adequate Sigma-gr ading on the left Artinian ring A, where Sigma is a monoid, we constru ct a generalized Cartan matrix with entries in Z Sigma, which is right invertible whenever gl.dimA < infinity. That gives a positive answer to the conjecture when A admits a strongly adequate grading by an aper iodic commutative monoid. We then show that, even though this does not give a definite answer to the conjecture, it strictly widens the clas s of known graded algebras for which it is true.