Advances in Applied Clifford Algebras

, Volume 23, Issue 4, pp 965–980 | Cite as

Matrix Representations of the Low Order Real Clifford Algebras

  • Youngkwon Song
  • Doohann Lee


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 ≤ rsn ≤ 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 Γ0,3.


Clifford algebra Clifford group Pauli matrix quaternion 


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  1. 1.
    Atiyah M.F., Bott R., Shapiro A.: Clifford modules. Topology 3, 3–38 (1964)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brihaye Y., Maslanka P., Giler S., Kosinski P.: Real representations of Clifford algebras. J. Math. Phys. 33(5), 1579–1581 (1992)MathSciNetADSCrossRefMATHGoogle Scholar
  3. 3.
    Lee D., Song Y.: The matrix representation of Clifford algebra. J. Chungcheong Math. Soc. 23, 363–368 (2010)Google Scholar
  4. 4.
    Lee D., Song Y.: Applications of matrix algebra to Clifford groups. Adv. Appl. Clifford Algebras 22, 391–398 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Lee D., Song Y.: Explicit matrix realization of Clifford algebras. Adv. Appl. Clifford Algebras 23, 441–451 (2013)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Poole C.P. Jr., Farach H.A.: Pauli–Dirac matrix generators of Clifford algebras. Found. of Phys. 12, 719–738 (1982)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    C. P. Poole, Jr., H. A. Farach, Y. Aharonov A vector product formulation of special relativity and electromagnetism. Found. of Phys. 10 (1980) 531–553Google Scholar
  8. 8.
    Tian Y.: Universal similarity factorization equalities over real Clifford algebras. Adv. Appl. Clifford Algebras 8, 365–402 (1998)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsKwangwoon UniversitySeoulRepublic of Korea
  2. 2.College of Global General EducationGachon UniversitySungnamRepublic of Korea

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