Convention. Bivector normalizations [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000N]
Convention. Bivector normalizations [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000N]
Meinrenken works throughout this chapter with finite-dimensional vector spaces over a characteristic-zero field. The cards below retain his normalization and formulas but follow TauCeti's extensions. Each card states the commutative-ring, module, invertible-\(2\), finite-dimensional, or nondegeneracy assumptions it needs.
Meinrenken starts with a symmetric bilinear form \(B\) and the relation \[uv+vu=2B(u,v)1.\] For the associated quadratic form \(Q(v)=B(v,v)\), the polar form used here is \(B_Q=2B\).
Meinrenken's quantization formula and the half-commutator agree: \[q(u\wedge v)=uv-B(u,v)1 =\frac 12(uv-vu)=:\beta (u,v).\] The induced action on a vector is \[[\beta (u,v),x] =B_Q(v,x)u-B_Q(u,x)v =2\bigl (B(v,x)u-B(u,x)v\bigr ).\] This is Meinrenken's formula \(-2\iota _{B(x,-)}(u\wedge v)\) and Kostant's formula \(-2\iota _x(u\wedge v)\) in the corresponding contraction convention [kostant1997clifford, Section 2.4, (12) and Theorem 8, pp. 283, 286]. The factor of two belongs to the passage from \(B\) to the unhalved polar form \(B_Q\); it is not an additional rescaling of \(\beta \).