Vol. 10, No. 1, 2017

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

Andrew McGinnis and Michaela Vancliff

Vol. 10 (2017), No. 1, 43–50

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 × 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 × 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.

transpose, automorphism, symmetric, skew-symmetric, polynomial ring, twist
Mathematical Subject Classification 2010
Primary: 15A15, 15B57, 16S50, 16S36
Received: 27 May 2015
Revised: 5 September 2015
Accepted: 7 September 2015
Published: 11 October 2016

Communicated by Vadim Ponomarenko
Andrew McGinnis
Department of Mathematics
University of California at Riverside
Riverside, CA 92521
United States
Michaela Vancliff
Department of Mathematics
University of Texas at Arlington
P.O. Box 19408
Arlington, TX 76019
United States