An Ordered Tuple Construction of Geometric Algebras

Timothy Myers
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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 $${\mathbb {G}}^{(p,q)}$$ .
几何代数的有序元组构造
在本文中,我们将提出一个具有签名(p, q)的任何实数几何(Clifford)代数$${\mathbb {G}}^{(p,q)}$$的新构造,其中$$p+q=n$$通过在向量空间$${\mathbb {R}}^{(2^n)}$$上定义一个乘积,其方式类似于高斯复数的有序对构造($${\mathbb {C}}$$)和汉密尔顿四元数的有序四元构造($${\mathbb {H}}$$)。我们将通过推广$${\mathbb {C}}$$和$${\mathbb {H}}$$上乘法的有序元组定义来激发$${\mathbb {G}}^{(p,q)}$$上几何乘积的定义。与Gauss从$${\mathbb {C}}$$上的乘法的有序对定义中获得基$$\{1, i\}$$的方式类似,我们同样将通过将生成$${\mathbb {G}}^{(p,q)}$$的有序$$2^n$$元组相乘来获得$${\mathbb {G}}^{(p,q)}$$的单项式基。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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