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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
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Abstract |
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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 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 matrices, and show that the antiautomorphism property of our generalized transpose map is related to the notion of twisting the polynomial ring on variables by an automorphism. |
Keywords
transpose, automorphism, symmetric, skew-symmetric,
polynomial ring, twist
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Mathematical Subject Classification 2010
Primary: 15A15, 15B57, 16S50, 16S36
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Milestones
Received: 27 May 2015
Revised: 5 September 2015
Accepted: 7 September 2015
Published: 11 October 2016
Communicated by Vadim Ponomarenko
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Authors |
| Andrew McGinnis | |
| Department of Mathematics University of California at Riverside Riverside, CA 92521 United States |
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| Michaela Vancliff | |
| Department of Mathematics University of Texas at Arlington P.O. Box 19408 Arlington, TX 76019 United States |
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