A new class of supermatrix algebras defined by transitive matrices

Abstract

We give a natural definition for the transitivity of a matrix. Using an endomorphism d of a base ring R and a transitive nxn matrix over the center Z(R), we construct the subalgebra M_{n}(R,d,T) of the full nxn matrix algebra M_{n}(R) consisting of the so called nxn supermatrices. If the n-th power of d is the identity and T satisfies some extra conditions, then we exhibit an embedding of R into M_{n}(R,d,T). An other result is that M_{n}(R,d,T) is closed with respect to taking the (pre)adjoint. If R is Lie nilpotent and A is in M_{n}(R,d,T), then the use of the preadjoint and the corresponding determinants and characteristic polynomials yields a Cayley-Hamilton identity for A with right coefficients in the fixed ring Fix(d). The presence of a primitive n-th root of unity and the condition that the n-th power of d is the identity guarantee the right integrality of a Lie nilpotent R over Fix(d). We present essentially new supermatrix algebras over the Grassmann algebra.