E. Hitzer, J. Helmstetter, R. Ablamowicz
Square roots of -1 in real Clifford algebras
In K. Guerlebeck (ed.), Electronic Proceedings of The 9th International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA9), 15-10 July 2011, Weimar, Germany, (2011).

Abstract: Let Cl(p,q) be the universal Clifford algebra (associative with unit) generated over R by p + q elements e_k (with k = 1, 2, ... , p + q) with the relations e^2_k = 1 if k <= p, e^2_k = -1 if k > p and e_he_k + e_ke_h = 0 whenever h not equal k, see [6]. We set n = p + q and s = p - q. This algebra has dimension 2^n, and its even subalgebra Cl_0(p,q) has dimension 2^{n-1} (if n > 0). We are concerned with square roots of -1 contained in Cl(p,q) or Cl_0(p,q). If the dimension of Cl(p,q) or C_0(p,q) is less or equal 2, it is isomorphic to R or R^2 or C, and it is clear that there is no square root of -1 in R and R2 = R x R, and that there are two squares roots i and -i in C. Therefore we only consider algebras of dimension greater or equal 4. Square roots of -1 have been computed in [4] for algebras of dimension less or equal 16, and for Cl(3,0) in [7].

Keywords: Algebra automorphism, inner automorphism, center, centralizer, Clifford algebra, conjugacy class, determinant, primitive idempotent, trace.

[4] E. Hitzer, R. Ablamowicz, Geometric Roots of -1 in Clifford Algebras Cl(p,q) with p + q less or equal 4, Adv. In Appl. Cliff. Algebras, Vol. 21(1) pp. 121-144, (2011), DOI 10.1007/s00006-010-0240-x.

[6] P. Lounesto, Clifford Algebras and Spinors, London Math. Society Lecture Note 239, Cambridge University Press, 1997.

[7] S. J. Sangwine, Biquaternion (Complexified Quaternion) Roots of -1, Adv. Appl. Cliff. Alg. 16(1), pp. 63-68, 2006.


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