The graded identities of upper triangular matrices of size two
Denote by \(UT_2\) the algebra of \(2\times 2\) upper triangular matrices over a field \(F\) of characteristic 0. This algebra is quite important in the PI theory. The paper under review describes all gradings on this algebra by an arbitrary group \(G\). It turns out that there are two possible gradings only. The first is the trivial one, the second is the usual grading by the group of two elements. Then the author describes the graded identities of \(UT_2\). It is shown that the two identities \(y_1y_2-y_2y_1=0\) and \(z_1z_2=0\) generate the graded T-ideal of \(UT_2\). Here \(y_i\) are 0-variables while \(z_i\) are 1-variables. Furthermore the graded cocharacter is computed. As corollaries, various numerical invariants of the graded identities for \(UT_2\) are described such as the graded codimenison and the graded length. Finally, it is shown that the graded codimension is polynomially bounded for any superalgebra whose graded T-ideal contains strictly all polynomial identities of \(UT_2\).
scientific article; zbMATH DE number 1787479