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Polarized relations in a Clifford algebra

Statement

Let $k, V, B, Q$ be as in Universal Clifford algebra of a quadratic space, and let $\overset{ˉ}{v} = i_{Q} (v)$ . Then

$\overset{ˉ}{v} \overset{ˉ}{w} + \overset{ˉ}{w} \overset{ˉ}{v} = 2 B (v, w) 1$

for all $v, w \in V$ .

Facts & Assumptions

Given: A field $k$ of characteristic not $2$ , a finite-dimensional $k$ -vector space $V$ , a symmetric bilinear form $B$ , $Q (v) = B (v, v)$ , and $v, w \in V$ .

[F1]

The quotient map sends each $v \in V$ to an element satisfying $i_{Q} (v)^{2} = Q (v) 1$ (Universal Clifford algebra of a quadratic space).

Proof

technique · direct

1.1givenF1algebra

Applying [F1] to $v + w$ , $v$ , and $w$ , then subtracting the last two defining equalities from the first, gives $\overset{ˉ}{v} \overset{ˉ}{w} + \overset{ˉ}{w} \overset{ˉ}{v} = (Q (v + w) - Q (v) - Q (w)) 1$ .

2.1givenstep 1.1algebra∎

Bilinearity and symmetry give $Q (v + w) - Q (v) - Q (w) = B (v, w) + B (w, v) = 2 B (v, w)$ , proving the relation.

Depends on

Universal Clifford algebra of a quadratic space

Polarized relations in a Clifford algebra

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources