## 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

#### 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
Revised: 5 September 2015
Accepted: 7 September 2015
First available in Project Euclid: 22 November 2017

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

#### 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

#### References

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