Advances in Applied Clifford Algebras

, Volume 7, Issue 2, pp 103–111 | Cite as

Spin invariants of multivectors



LetCl(p, q) be a real universal Clifford algebra which is isomorphic to a full matrix algebra ℝ(2m).

In this paper we show that on the linear subspaceClk(p, q) ofk-vectors the determinant can be written as a product of two polynomialsdi of degreem and that on the subset ofdecomposable k-vectors we have det=±Qm for some quadratic formQ.

The polynomialsdi andQ are examples of a spin invariant, the latter being defined as a functionJ:Cl k (p,q) → ℝ for whichJ(sus−1)=J(u) for alluClk(p, q) andsSpin(p, q).

In the last section we identify the ‘fundamental’ spin invariants on the bivector spacesCl2(p, p) forp=2 andp=3.


Quadratic Form Clifford Algebra Matrix Algebra Determinant Function Apply Clifford Algebra 


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    Porteous I. R., Clifford algebras and the classical groups, Cambridge University Press, 1995.Google Scholar

Copyright information

© Birkhäuser-Verlag AG 1997

Authors and Affiliations

  1. 1.Department of MathematicsDelft University of TechnologyGA DelftThe Netherlands

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