ON GRADED MATRIX HOM-ALGEBRAS

Consider an (associative) matrix algebra M-I (R) graded by means of an abelian group G, and a graded automorphism phi on M-I (R). By defining a new product by x star y :- phi(x)phi(y) on M-I (R), (M-I (R),star) becomes a hom-associative algebra graded by a twist of G. The structure of (M-I (R),star) is studied, by showing that M-I (R) is of the form M-I (R) = U + Sigma I-j(j) with U an R-submodule of the 0-homogeneous component and any I-j a well described graded ideal of M-I (R), satisfying I-j star I-k = 0 if j not equal k. Under certain conditions, the graded simplicity of an arbitrary graded hom-associative algebra M is characterized and it is shown that M is the direct sum of the family of its simple graded ideals.