Group gradations on Leavitt path algebras.

Given a directed graph E and an associative unital ring R one may define the Leavitt path algebra with coefficients in R , denoted by L R (E). For an arbitrary group G , L R (E) can be viewed as a G -graded ring. In this paper, we show that L R (E) is always nearly epsilon-strongly G -graded. We also show that if E is finite, then L R (E) is epsilon-strongly G -graded. We present a new proof of Hazrat's characterization of strongly ℤ -graded Leavitt path algebras, when E is finite. Moreover, if E is row-finite and has no source, then we show that L R (E) is strongly ℤ -graded if and only if E has no sink. We also use a result concerning Frobenius epsilon-strongly G -graded rings, where G is finite, to obtain criteria which ensure that L R (E) is Frobenius over its identity component. [ABSTRACT FROM AUTHOR]