Polynomial identities for the Jordan algebra of upper triangular matrices of order 2

The associative algebras UTn(K) of the upper triangular matrices of order n play an important role in PI theory. Recently it was suggested that the Jordan algebra UJ2(K) obtained by UT2(K) has an extremal behaviour with respect to its codimension growth. In this paper we study the polynomial identities of UJ2(K). We describe a basis of the identities of UJ2(K) when the field K is infinite and of characteristic different from 2 and from 3. Moreover we give a description of all possible gradings on UJ2(K) by the cyclic group Z2 of order 2, and in each of the three gradings we find bases of the corresponding graded identities. Note that in the graded case we need only an infinite field K, .