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Factorization through the Clifford quotient

Statement

Let $q_{Q} : T_{k} (V) \to C l (V, Q)$ be the quotient map from Universal Clifford algebra of a quadratic space. If $A$ is a unital associative $k$ -algebra and $ψ : T_{k} (V) \to A$ is a unital $k$ -algebra homomorphism satisfying $ψ (j (v)^{2} - Q (v) 1) = 0$ for every $v \in V$ , then there is a unique unital $k$ -algebra homomorphism $\overset{ˉ}{ψ} : C l (V, Q) \to A$ with $ψ = \overset{ˉ}{ψ} \circ q_{Q}$ .

Facts & Assumptions

Given: The field, vector space, form, tensor algebra, and Clifford quotient in Universal Clifford algebra of a quadratic space, together with a unital $k$ -algebra $A$ and a homomorphism $ψ$ killing each defining relation.

[F1]

The Clifford algebra is $T_{k} (V) / J_{Q}$ , where $J_{Q}$ is the smallest two-sided ideal containing all $j (v)^{2} - Q (v) 1$ (Universal Clifford algebra of a quadratic space).

[F2]

A two-sided ideal is an additive subgroup stable under multiplication on both sides, and a unital $k$ -algebra has a central unital $k$ -structure map (Left, right and two-sided ideals, Algebras over a commutative ring, central structure maps, and algebra homomorphisms).

Proof

technique · direct

1.1givenF1F2algebra

Since $ψ$ is a homomorphism, its kernel is closed under addition and under left and right multiplication by elements of $T_{k} (V)$ . It is therefore a two-sided ideal in the sense of [F2] containing each defining relation, and minimality in [F1] gives $J_{Q} \subseteq \ker ψ$ .

2.1step 1.1F1construct

Define $\overset{ˉ}{ψ} (t + J_{Q}) = ψ (t)$ . If $t + J_{Q} = t^{'} + J_{Q}$ , then $t - t^{'} \in J_{Q} \subseteq \ker ψ$ , so $ψ (t) = ψ (t^{'})$ and the definition is well defined.

3.1step 2.1F1algebra∎

The induced map preserves addition, multiplication, scalar multiplication, and the unit because $ψ$ does and the quotient operations are induced from $T_{k} (V)$ . Since $q_{Q}$ is surjective, any map factoring $ψ$ agrees with $\overset{ˉ}{ψ}$ on every coset; thus the factor is unique.

Depends on

Used by

Universal property of the Clifford algebra Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean Gallier, Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups, §1.2 (standard reference, not scraped)
M. F. Atiyah, R. Bott, and A. Shapiro, Clifford Modules, Part I, §1 (standard reference, not scraped)