# nLab Clifford ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

group theory

supersymmetry

# Contents

## Idea

A $\mathbb{Z}$-Clifford algebra.

## Definition

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

### Universal property

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

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

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