Polynomial identities of algebras in positive characteristic.
Let \(E\) (\(E'\)) be the Grassmann algebra with 1 (without 1) over an infinite field \(K\), \(A_{a,b}\) is the subalgebra of \(M_{a+b}(E)\) consisting of all matrices \((a_{ij})\), such that \(a_{ij}\in E\) if either \(1\leq i,j\leq a\) or \((a+1)\leq i,j\leq a+b\) and \(a_{ij}\in E'\) if either \(1\leq i\leq a\), \(a+1\leq j\leq a+b\), or \(1\leq j\leq a\), \(a+1\leq i\leq a+b\). Let also \(M_{a,b}(E)\) be the subalgebra of \(M_{a+b}(E)\) of the block matrices with blocks \(a\times a\) and \(b\times b\) on the main diagonal with entries from \(E_0\) and off diagonal entries from \(E_1\) (\(E=E_0+E_1\) is the natural grading on \(E\)). The authors compute the Gelfand-Kirillov dimensions of the free algebras in the varieties determined by \(E\otimes E\), \(M_{1,1}(E)\otimes E\), \(A_{2,1}\), \(A_{2,2}\) and prove the following main results (\(\text{char\,}K=p>2\)): (i) if \(a+b=c+d\), \(a\leq b\), \(c\leq d\), \(c>a\) then \(M_{a,b}(E)\otimes E\) is not PI equivalent to \(M_{c,d}(E)\otimes E\); (ii) if \(a+b=c+d\), \(a\geq b\), \(c\geq d\), \(a\neq c\) then \(T(A_{a,b})\neq T(A_{c,d})\); (iii) \(M_{1,1}(E)\otimes M_{1,1}(E)\) and \(M_{2,2}(E)\) are not PI equivalent.
scientific article; zbMATH DE number 5078325