{"title":"An Ordered Tuple Construction of Geometric Algebras","authors":"Timothy Myers","doi":"10.1007/s44007-023-00068-9","DOIUrl":null,"url":null,"abstract":"In this paper we will present a new construction of any real geometric (Clifford) algebra $${\\mathbb {G}}^{(p,q)}$$ with signature (p, q) where $$p+q=n$$ by defining a product on the vector space $${\\mathbb {R}}^{(2^n)}$$ in a manner similar to Gauss’ ordered pair construction of the complex numbers ( $${\\mathbb {C}}$$ ) and Hamilton’s ordered quadruple construction of the quaternions ( $${\\mathbb {H}}$$ ). We will motivate the definition of a geometric product on $${\\mathbb {G}}^{(p,q)}$$ by generalizing the ordered tuple definition of multiplication on each of $${\\mathbb {C}}$$ and $${\\mathbb {H}}$$ . Similar to the way in which Gauss obtains the basis $$\\{1, i\\}$$ from the ordered pair definition of multiplication on $${\\mathbb {C}}$$ , we will likewise derive a basis of monomials for $${\\mathbb {G}}^{(p,q)}$$ by multiplying those ordered $$2^n$$ tuples that generate $${\\mathbb {G}}^{(p,q)}$$ .","PeriodicalId":74051,"journal":{"name":"La matematica","volume":"67 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"La matematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s44007-023-00068-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we will present a new construction of any real geometric (Clifford) algebra $${\mathbb {G}}^{(p,q)}$$ with signature (p, q) where $$p+q=n$$ by defining a product on the vector space $${\mathbb {R}}^{(2^n)}$$ in a manner similar to Gauss’ ordered pair construction of the complex numbers ( $${\mathbb {C}}$$ ) and Hamilton’s ordered quadruple construction of the quaternions ( $${\mathbb {H}}$$ ). We will motivate the definition of a geometric product on $${\mathbb {G}}^{(p,q)}$$ by generalizing the ordered tuple definition of multiplication on each of $${\mathbb {C}}$$ and $${\mathbb {H}}$$ . Similar to the way in which Gauss obtains the basis $$\{1, i\}$$ from the ordered pair definition of multiplication on $${\mathbb {C}}$$ , we will likewise derive a basis of monomials for $${\mathbb {G}}^{(p,q)}$$ by multiplying those ordered $$2^n$$ tuples that generate $${\mathbb {G}}^{(p,q)}$$ .