The algebra of 2 × 2 upper triangular matrices as a commutative algebra: Gradings, graded polynomial identities and Specht property

Let UJ2 be the Jordan algebra of 2×2 upper triangular matrices. This paper is devoted to continue the description given by recent works about the gradings and graded polynomial identities on UJ2(K) when K is an infinite field of characteristic 2. Due to the definition of Jordan algebras in terms of the commutative and Jordan identities being unsuitable in characteristic 2, we decided to study the gradings of the non-associative commutative algebra of 2×2 upper triangular matrices UT2=(UT2(K),∘) with the product x∘y=xy+yx. More precisely, fixed K a field of characteristic 2, we classify the gradings of (UT2(K),∘) and also, given an arbitrary grading, we calculate the generators of the ideals of graded identities and give a positive answer to the Specht property for the variety of commutative algebras generated by (UT2(K),∘) in each grading when K is infinite.