Abstract: The origin of Leavitt path algebras can be traced back to the pioneering work of Leavitt in his quest for findinguniversal rings failing to satisfy the IBN property, although they have only come to life as such in recent years. On the otherhand, they are the algebraic counterpart of graph C*-algebras.
As with their analytic relatives, Leavitt path algebras provide a source of examples of rings whose algebraic structure is determined by highly visual properties of the underlying graph. For example, conditions on the graph allow us to decide when the corresponding Leavitt path algebra is simple, purely infinite simple, exchange, etc. Some of the graph conditions parallel the corresponding structural properties that one encounters in C*-algebras, but the routes towards the proofs are in general quite different.
A whole range of examples of algebras arise as Leavitt path algebras. Besides the (now already) classical examples investigated by Leavitt, we also find matrix rings over a field, the Laurent polynomial ring K[x, x^-1] over a field K, or the (algebraic) Toeplitz algebra.
In this talk we will travel through the world of Leavitt path algebras by discovering some of the more recent results on their structure. Concretely, we will state the clasification of Leavitt path algebras which have not cycles in the finite and in the infinite dimensional cases. Leavitt path algebras with essential socle will be treated too.
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