Title: Leavitt path algebras: Graded direct-finiteness and graded $Σ$-injective simple modules,
Abstract: In this paper, we give a complete characterization of Leavitt path algebras
which are graded $\Sigma $-$V$ rings, that is, rings over which a direct sum of
arbitrary copies of any graded simple module is graded injective. Specifically,
we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded
$\Sigma $-$V$ ring if and only if it is a subdirect product of matrix rings of
arbitrary size but with finitely many non-zero entries over $K$ or
$K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical
characterization of such a graded $\Sigma $-$V$ ring $L$% . When the graph $E$
is finite, we show that $L$ is a graded $\Sigma $-$V$ ring $\Longleftrightarrow
L$ is graded directly-finite $\Longleftrightarrow L $ has bounded index of
nilpotence $\Longleftrightarrow $ $L$ is graded semi-simple. Examples show that
the equivalence of these properties in the preceding statement no longer holds
when the graph $E$ is infinite. Following this, we also characterize Leavitt
path algebras $L$ which are non-graded $\Sigma $-$V$ rings. Graded rings which
are graded directly-finite are explored and it is shown that if a Leavitt path
algebra $L$ is a graded $\Sigma$-$V$ ring, then $L$ is always graded
directly-finite. Examples show the subtle differences between graded and
non-graded directly-finite rings. Leavitt path algebras which are graded
directly-finite are shown to be directed unions of graded semisimple rings.
Using this, we give an alternative proof of a theorem of Vaš \cite{V} on
directly-finite Leavitt path algebras.