Steinberg algebra and Leavitt path algebras - Rigby, Simon W (Department of Mathematics and Applied Mathematics, 2019)
Leavitt path algebras are a new and exciting subject in noncommutative ring theory. To each directed graph E, and unital commutative ring R, is associated an R-algebra called the Leavitt path algebra of E with coefficients in R. It was discovered, when the theory of Leavitt path algebras was already quite advanced, that some of the more difficult questions were susceptible to a new approach using topological groupoids. Taking a special kind of groupoid G, one can construct an R-algebra called the Steinberg algebra of G. Many interesting classes of algebras, including Leavitt path algebras, can be obtained from this process. This dissertation is an exposition of the recent advances achieved by the groupoid approach to Leavitt path algebras. New proofs are presented to show that the boundary path groupoid (which underlies the Steinberg algebra model for Leavitt path algebras) has the necessary topological properties. A new theorem is presented, characterising strongly graded Leavitt path algebras in graphical terms. We show that the main results on the structure theory of Leavitt path algebras, including the simplicity and primitivity theorems, can be recovered using the groupoid approach. We demonstrate how these methods lead to an explicit description of the centre of a Leavitt path algebra.