A generalization of the matrix transpose map and its relationship to the twist of the polynomial ring by an automorphism

Abstract

A generalization of the notion of symmetric matrix was introduced by Cassidy and Vancliff in 2010 and used by them in a construction that produces quadratic regular algebras of finite global dimension that are generalizations of graded Clifford algebras. In this article, we further their ideas by introducing a generalization of the matrix transpose map and use it to generalize the notion of skew-symmetric matrix. With these definitions, an analogue of the result that every $n\times n$ matrix is a sum of a symmetric matrix and a skew-symmetric matrix holds. We also prove an analogue of the result that the transpose map is an antiautomorphism of the algebra of $n\times n$ matrices, and show that the antiautomorphism property of our generalized transpose map is related to the notion of twisting the polynomial ring on $n$ variables by an automorphism.