---
tag: 5X7NA4TTK8
kind: definition
name: Clifford algebra
msc: [15A66, "11E88"]
defines: A38B1QFK20
uses: [000AYPA9AJ, 0RBQS1JJCR, P4J5JZDW0B, 0CD87SFM2N, 0076HWHGHQ]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford_algebra
---

## Definition

Let $F$ be a field, let $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
Let $T(V)$ be the [[tensor algebra|P4J5JZDW0B]] of $V$.
Let $I_Q$ be the [[two-sided ideal|0CD87SFM2N]] of $T(V)$ generated by the elements $v \otimes v - Q(v) \cdot 1$ with $v \in V$.
The *Clifford algebra* of $(V, Q)$ is the [[quotient ring|0076HWHGHQ]]
$$\mathrm{Cl}(V, Q) = T(V) / I_Q.$$

## Remarks

Universal property: for every associative unital $F$-algebra $A$ and every linear map $f \colon V \to A$ with $f(v)^2 = Q(v) \cdot 1_A$ for all $v \in V$, there is exactly one unital algebra homomorphism $\mathrm{Cl}(V, Q) \to A$ that extends $f$.
If $\dim V = n$, then $\dim \mathrm{Cl}(V, Q) = 2^n$, with basis the products $e_{i_1} \cdots e_{i_k}$, $i_1 < \cdots < i_k$, for a basis $e_1, \ldots, e_n$ of $V$.
Example: $V = \R^2$ with $Q(x, y) = -(x^2 + y^2)$.
The relation $(e_1 + e_2)^2 = Q(e_1 + e_2) = -2$ with $e_1^2 = e_2^2 = -1$ gives $e_1 e_2 = -e_2 e_1$, and then $(e_1 e_2)^2 = -1$; so $\mathrm{Cl}(V, Q)$ is the quaternion algebra with $i = e_1$, $j = e_2$, $k = e_1 e_2$.
For $Q = 0$, the Clifford algebra is the [[exterior algebra|0MDMZ965VE]] of $V$.
Non-example: the tensor algebra $T(V)$ for $V \ne 0$ is not a Clifford algebra of $(V, Q)$, because $v \otimes v$ is not a scalar there.
Some authors use the opposite sign convention $v^2 = -Q(v)$.