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 UJ(2)(K) obtained by UT2(K) has an extremal behaviour with respect to its codimension growth. In this paper we study the polynomial identities of UJ(2)(K). We describe a basis of the identities of UJ(2)(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 UJ(2)(K) by the cyclic group Z(2) 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, char K not equal 2. (C) 2012 Elsevier B.V. All rights reserved.