On Z2Z2-graded identities of UT2(E)UT2(E) and their growth

Let F be an infinite field of characteristic different from two and E be the infinite dimensional Grassmann algebra over F  . We consider the upper triangular matrix algebra UT2(E)UT2(E) with entries in E   endowed with the Z2Z2-grading inherited by the natural Z2Z2-grading of E   and we study its ideal of Z2Z2-graded polynomial identities (TZ2TZ2-ideal) and its relatively free algebra. In particular we show that the set of Z2Z2-graded polynomial identities of UT2(E)UT2(E) does not depend on the characteristic of the field. Moreover we compute the Z2Z2-graded Hilbert series of UT2(E)UT2(E) and its Z2Z2-graded Gelfand–Kirillov dimension.