Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 1 (2017), 43-50.

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

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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 × 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.

Article information

Source
Involve, Volume 10, Number 1 (2017), 43-50.

Dates
Received: 27 May 2015
Revised: 5 September 2015
Accepted: 7 September 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511371081

Digital Object Identifier
doi:10.2140/involve.2017.10.43

Mathematical Reviews number (MathSciNet)
MR3561728

Zentralblatt MATH identifier
1352.15009

Subjects
Primary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 15B57: Hermitian, skew-Hermitian, and related matrices 16S50: Endomorphism rings; matrix rings [See also 15-XX] 16S36: Ordinary and skew polynomial rings and semigroup rings [See also 20M25]

Keywords
transpose automorphism symmetric skew-symmetric polynomial ring twist

Citation

McGinnis, Andrew; Vancliff, Michaela. A generalization of the matrix transpose map and its relationship to the twist of the polynomial ring by an automorphism. Involve 10 (2017), no. 1, 43--50. doi:10.2140/involve.2017.10.43. https://projecteuclid.org/euclid.involve/1511371081


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References

  • M. Artin, J. Tate, and M. Van den Bergh, “Modules over regular algebras of dimension $3$”, Invent. Math. 106:2 (1991), 335–388.
  • T. Cassidy and M. Vancliff, “Generalizations of graded Clifford algebras and of complete intersections”, J. Lond. Math. Soc. $(2)$ 81:1 (2010), 91–112.
  • M. Nafari and M. Vancliff, “Graded skew Clifford algebras that are twists of graded Clifford algebras”, Comm. Algebra 43:2 (2015), 719–725.
  • M. Vancliff and P. P. Veerapen, “Generalizing the notion of rank to noncommutative quadratic forms”, pp. 241–250 in Noncommutative birational geometry, representations and combinatorics, edited by A. Berenstein and V. Retakh, Contemp. Math. 592, Amer. Math. Soc., Providence, RI, 2013.