Differential identities, 2 × 2 upper triangular matrices and varieties of almost polynomial growth

Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

“…The last section of our paper is devoted to work out the small cases 2 and 3. In the former we regain the results of [11] as byproduct of our general description. On the other hand, 2 is too small to depict the effective impact of the -action on the differential polynomial identities in general cases.…”

“…Now let be an -algebra. The following notions have been outlined in [4] in the general settings of Hopf algebra actions, and employed in [11] in the more specific situation of a derivation action. Let be the set constituted by the differential polynomials laying in the kernel of all -homomorphisms .…”

“…Recall that a polynomial is multilinear of degree if each name among 1 occurs exactly once in any monomial of , and no other name occurs. As in [11] we denote span Then is the least -ideal of containing the set . In other words, is generated, as -ideal, by the multilinear -polynomials it contains.…”

“…Also the differential codimensions can be computed from the proper ones (see [11], Theorem 19): , passing to the dimensions we get 2 0 1 2 1 2 .…”

“…So let 2 be the -algebra under the -derivation action, the formal differences between 2 and 2 consist solely in the presence or not of the polynomial among the generators. In [11] the -action has been considered, so is missing.…”