On \(\mathbb Z_2\)-graded identities of the super tensor product of \(UT_2(F)\) by the Grassmann algebra.
Let \(F\) be a field of characteristic zero and \(E\) the unitary Grassmann algebra generated over an infinite-dimensional vector space \(L\). The author considers arbitrary \(\mathbb Z_2\)-gradings of \(E\): \(\mathcal E=\mathcal E^{(0)}\oplus\mathcal E^{(1)}\) such that the space \(L\) is homogeneous. Let \(\mathcal B=\{e_1,e_2,\dots\}\) be a linear homogeneous basis of \(L\). Then arbitrary gradings are given by a map \(\|*\|\colon\mathcal B\to\mathbb Z_2\). There exist three important possibilities: 1) \(\|e_i\|=0\), \(i=1,\dots,k\), \(\|e_i\|=1\) otherwise, for some \(k\geq 1\); 2) \(\|e_i\|=1\), \(i=1,\dots,k\), \(\|e_i\|=0\) otherwise, for some \(k\geq 1\); 3) \(\|e_i\|=0\) for \(i\) even and \(\|e_i\|=1\) for \(i\) odd. Given an associative superalgebra \(A=A^{(0)}\oplus A^{(1)}\), one defines the superalgebra \(A\widehat\otimes \mathcal E=(A^{(0)}\otimes\mathcal E^{(0)})\oplus (A^{(1)}\otimes\mathcal E^{(1)})\). In case of the canonical grading of \(E\) the algebra \(A\widehat\otimes E\) is the ordinary Grassmann envelope of \(A\). The \(\mathbb Z_2\)-codimensions for the Grassmann algebras with such noncanonical gradings were studied by the author earlier: [\textit{O. M. Di Vincenzo} and \textit{V. Ribeiro Tomaz da Silva}, Linear Algebra Appl. 431, No. 1-2, 56-72 (2009; Zbl 1225.16009); \textit{V. Ribeiro Tomaz da Silva}, Commun. Algebra 37, No. 9, 3342-3359 (2009; Zbl 1192.16021)]. Now the author studies the superalgebras \(UT_2(F)\widehat\otimes \mathcal E\), when the algebra \(UT_2(F)\) of \(2\times 2\) upper triangular matrices over \(F\) is endowed with its canonical grading. The author describes the generators of their graded T-ideals as well as linear bases of the corresponding relatively free algebras. Furthermore, the author describes the \(Y\)-proper codimension sequences and finally gives a complete description of the space of multilinear \(Y\)-proper identities in the language of Young diagrams. As a result, the author finds bases of \(\mathbb Z_2\)-graded identities and describes \(\mathbb Z_2\)-graded codimension and cocharacter sequences for these superalgebras.
scientific article; zbMATH DE number 6105458