Spin invariants of multivectors
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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 allu ∈Clk(p, q) ands ∈Spin(p, q).
In the last section we identify the ‘fundamental’ spin invariants on the bivector spacesCl2(p, p) forp=2 andp=3.
KeywordsQuadratic Form Clifford Algebra Matrix Algebra Determinant Function Apply Clifford Algebra
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