nLab Clifford ring

Contents

Context

Algebra

Group theory

Super-Algebra and Super-Geometry

Contents

Idea

A \mathbb{Z}-Clifford algebra.

Definition

Given a quadratic abelian group GG with a quadratic function q:Gq:G \to \mathbb{Z}, the Clifford ring Cl(G,q)\mathrm{Cl}(G, q) is the quotient ring of the tensor ring T(G)T(G) by the ideal generated by the relations ggq(g)g \cdot g - q(g) for all gGg \in G.

Universal property

Given a quadratic abelian group GG with a quadratic function q:Gq:G \to \mathbb{Z}, the Clifford ring is a ring Cl(G,q)\mathrm{Cl}(G, q) with canonical ring homomorphism j:Cl(G,q)j:\mathbb{Z} \to \mathrm{Cl}(G, q) with a abelian group homomorphism g:GCl(G,q)g:G \to \mathrm{Cl}(G, q) such that

  • for every element a:Ga:G, g(a)g(a)=j(q(g(a)))g(a) \cdot g(a) = j(q(g(a)))

  • for every other ring RR with abelian group homomorphism h:GRh:G \to R where for every element a:Ga:G, h(a)h(a)=j R(q(h(a)))h(a) \cdot h(a) = j_R(q(h(a))), there is a unique ring homomorphism i:Cl(G,q)Ri:\mathrm{Cl}(G, q) \to R such that ig=hi \circ g = h.

See also

Last revised on May 10, 2022 at 22:30:06. See the history of this page for a list of all contributions to it.