We say a unital ring $R$ has the Invariant Basis Number (IBN) property in case, for each pair of positive integers $i,j$ if the left $R$-modules $R^i$ and $R^j$ are isomorphic, then $i=j$. The first examples of non IBN rings were studied by William Leavitt in the 1950s and he defined (what are now known as) Leavitt algebras which are â€˜universalâ€™ with non IBN property. In 2004 the algebraic structures arising from directed (multi)graphs known as Leavitt path algebras (LPA for short) were initially developed as algebraic analogues of graph $C^*$ algebras. LPAs generalize a particular class of Leavitt algebras.

During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in $C^*$-algebras, and symbolic dynamicists as well. The goal of this talk is to introduce the notion of Leavitt path algebras and to present some results on LPAs arising from weighted Cayley graphs of finite cyclic groups.

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Last updated: 06 Mar 2020