Matrix Representations of the Low Order Real Clifford Algebras

In this paper we construct the matrix subalgebras ${L_{r,s}(mathbb{R})}$ of the real matrix algebra ${M_{2^{r+s}} (mathbb{R})}$ when 2 ≤ r + s ≤ 3 and we show that each ${L_{r,s}(mathbb{R})}$ is isomorphic to the real Clifford algebra ${mathcal{C} ell_{r,s}}$ . In particular, we prove that the algebras ${L_{r,s}(mathbb{R})}$ can be induced from ${L_{0,n}(mathbb{R})}$ when 2 ≤ r + s = n ≤ 3 by deforming vector generators of ${L_{0,n}(mathbb{R})}$ to multiply the specific diagonal matrices. Also, we construct two subalgebras ${T_4(mathbb{C})}$ and ${T_2(mathbb{H})}$ of matrix algebras ${M_4(mathbb{C})}$ and ${M_2(mathbb{H})}$ , respectively, which are both isomorphic to the Clifford algebra ${mathcal{C} ell_{0,3}}$ , and apply them to obtain the properties related to the Clifford group Γ_0,3.