Advertisement

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
Article

Abstract

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.

Keywords

Clifford algebra Clifford group Pauli matrix quaternion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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

Copyright information

© Springer Basel 2013

Authors and Affiliations

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

Personalised recommendations