## 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

#### 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.

#### 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