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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