$\mathbb{Z}_2$-graded polynomial identities for the Jordan algebra of $2\times 2$ upper triangular matrices

Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3

“…It is worth noting that the Jordan case is even harder. Its polynomial identities, even the ordinary ones, are known only for triangular matrices of size 2 [9,8]. Moreover the Specht property for the corresponding ideals of graded identities is obtained for the same algebra of size 2, for all gradings [2].…”

Gradings on the algebra of triangular matrices as a Lie algebra: revisited

“…Meanwhile, several authors were interested in the study of (graded) PI-related properties of the algebra of 2 × 2 upper triangular matrices viewed as a Jordan algebra [8,9,11]. Also, recent works were dedicated to investigating the same algebra over finite fields [21], and over fields of characteristic 2 [29]. All this leads us to think that it would be interesting to obtain a complete classification of group gradings on the algebra of n × n upper triangular matrices considered as a Lie algebra over arbitrary fields (including characteristic 2).…”