A new class of supermatrix algebras defined by transitive matrices

TL;DR

This paper introduces a new class of supermatrix algebras defined via transitive matrices and endomorphisms, exploring their algebraic properties, embeddings, and identities, with applications to Grassmann algebras.

Contribution

It defines supermatrix algebras based on transitive matrices and endomorphisms, establishing their properties, embeddings, and Cayley-Hamilton identities, extending the theory of supermatrix algebras.

Findings

M_{n}(R,d,T) is closed under taking the (pre)adjoint.

A Cayley-Hamilton identity with right coefficients in Fix(d) is established.

New supermatrix algebras over Grassmann algebra are constructed.

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.