Abelian gradings on upper block triangular matrices. (English) Zbl 1244.16036
Algebras of block triangular matrices turn out to be quite important in the study of PI algebras and their extremal properties. Let \(G\) be a finite Abelian group and let \(F\) be an algebraically closed field of characteristic 0. The main result of the paper under review is a complete description of all \(G\)-gradings on the algebras of block triangular matrices over \(F\).
Let \(A=UT(d_1,\dots,d_m)\) be the algebra of upper block triangular matrices over \(F\); here \(d_1,\dots,d_m\) are the sizes of the blocks on the main diagonal. Suppose \(A\) is \(G\)-graded. The authors prove that there is a decomposition \(d_i=tp_i\), \(i=1,\dots,m\), a subgroup \(H\) of \(G\) and an \(n\)-tuple \((g_1,\dots,g_n)\) of elements of \(G\) where \(n=p_1+\cdots+p_m\) such that \(A\) is isomorphic to \(M_t(F)\otimes UT(p_1,\dots,p_m)\) as \(G\)-graded algebras. Here \(M_t(F)\) is \(H\)-graded with a fine \(H\)-grading (that is, each homogeneous component is of dimension \(\leq 1\)). Also the grading on \(UT(p_1,\dots,p_m)\) is the elementary one defined by the \(n\)-tuple \((g_1,\dots,g_n)\).
Let \(A=UT(d_1,\dots,d_m)\) be the algebra of upper block triangular matrices over \(F\); here \(d_1,\dots,d_m\) are the sizes of the blocks on the main diagonal. Suppose \(A\) is \(G\)-graded. The authors prove that there is a decomposition \(d_i=tp_i\), \(i=1,\dots,m\), a subgroup \(H\) of \(G\) and an \(n\)-tuple \((g_1,\dots,g_n)\) of elements of \(G\) where \(n=p_1+\cdots+p_m\) such that \(A\) is isomorphic to \(M_t(F)\otimes UT(p_1,\dots,p_m)\) as \(G\)-graded algebras. Here \(M_t(F)\) is \(H\)-graded with a fine \(H\)-grading (that is, each homogeneous component is of dimension \(\leq 1\)). Also the grading on \(UT(p_1,\dots,p_m)\) is the elementary one defined by the \(n\)-tuple \((g_1,\dots,g_n)\).
Reviewer: Plamen Koshlukov (Campinas)
MSC:
16W50 | Graded rings and modules (associative rings and algebras) |
16S50 | Endomorphism rings; matrix rings |