Can anyone suggest me some books on Leavitt Path algebra?

Question

I am a new one in this world. I want to study Leavitt Path algebra. Can anyone suggest me some books? It would be good to get some motivation and who are the persons currently working in this field?

Answer 1

Here are some references which might be useful. I am giving some more links and yes P. Ara and Molina started this field which has become popular now a days.

http://www.springer.com/in/book/9781447173434

https://arxiv.org/abs/1410.1835

https://academics.uccs.edu/gabrams/documents/versionsenttoSpringer4April2017.pdf

http://www.math.unipd.it/~dottmath/corsi2013/Abrams.pdf

http://archive.schools.cimpa.info/archivesecoles/20160119110553/[email protected]

https://www.researchgate.net/publication/321381712_The_Basics_of_Leavitt_Path_Algebras_Motivations_Definitions_and_Examples

http://www.math.unl.edu/~jorr1/gpots07/talks_pdf/Tomforde.pdf

Answer 2

Leavitt path algebras arise as a generalization from Leavitt algebras of type $(1, n)$ and $C^∗$ -algebras Cuntz On. The authors are inspired by the work of Leavitt, who in the late 50`s builds examples of rings $R$ that do not have the IBN property (Invariant Basis Number); which states: If $m$ and $m_0$ are integers with the property that free modules on the left $R^m$ and $R^{m_0}$ are isomorphic, then $m = m_0$". It is easy to prove that for a unitary ring $R, R^1 \cong R^n$ for some $n> 1$ if and only if there exist elements $x_1, . . . , x_n, y_1, . . . , y_n ∈ R$ such that $x_iy_j = δ_{ij}1_R$ for all $i, j$ and $n$; $\sum_{i=1}^n y_ix_i = 1_R$. For a given field $K$, Leavitt considered the $K$ unital algebra $L(1, n)$ generated by the elements ${x_1, . . . , x_n, y_1, . . . , y_n} $ satisfying these relationships and proved that the algebras of the form $L(1, n)$ are simple.

You can have these notes

1. http://archive.schools.cimpa.info/archivesecoles/20160119110553/[email protected]
2. http://www.springer.com/in/book/9781447173434
3. https://www.sciencedirect.com/science/article/pii/S0022404906001666
4. https://www.math.uh.edu/~tomforde/CopenhagenTalks/Talk6.pdf

Nowadays Pere Ara, Mercedes Siles Molina in Spain, Gene Abrams in the USA work in LPA. There are other mathematicians like Muge Kanuni, Lisa Ordloff Clark and Roozbeh Hazrat work in LPA.