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I know the formal definition of Leavitt path algebras, but I want know why the relations defining Leavitt Path Algebras are defined in that way? what is special of this relations?

My real hidden question is why Path Algebras is defined also by "that" relations? from what does this come?

Thanks

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    $\begingroup$ Did you read the "What is ... a Leavitt path algebra?" (by Gene Abrams) in the AMS notices? See ams.org/publications/journals/notices/201608/rnoti-p910.pdf. It explains to some extent why the defining relations are natural from the point of view of the Invariant Basis Number property of rings. $\endgroup$
    – Tom De Medts
    Commented Apr 9, 2019 at 8:10
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    $\begingroup$ I think the original motivation come from $C^*$-algebras and specifically from the fact that the Toeplitz algebra can be seen as the universal $C^*$-algebra generated by the unilateral shift on $l^2(\mathbb N)$. This shift can be interpreted as shifting along infinite paths by deleting one edge at each step. $\endgroup$
    – Nicola Ciccoli
    Commented Apr 9, 2019 at 14:18

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I can't say what was the motivation of the original authors, who apart from Leavitt were motivated by graph C*-algebras which in turn were motivated by symbolic dynamics. From a Computer Science point of view it is quite natural. Let's assume the digraph is finite and has no sinks or sources for convenience.

Imagine you store an infinite path in your graph on a stack. You have the operations of pushing your favorite edge onto the front of the path on the stack or popping your favorite edge from the front of the path when these operations are defined. You can build from these operators on a vector space with basis the infinite paths in your graph by interpreting an undefined operation as sending a path to 0. It is easy to check these operators satisfy the Leavitt path algebra relations and under mild conditions generate a copy of the Leavitt path algebra.

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