---
title: "Classification of Clifford algebras"
chunk: 6/7
source: "https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:08:16.280956+00:00"
instance: "kb-cron"
---

and hence also 
  
    
      
        Spin
        ⁡
        (
        3
        ,
        1
        )
        ≅
        
          SL
          
            2
          
        
        ⁡
        (
        
          C
        
        )
      
    
    {\displaystyle \operatorname {Spin} (3,1)\cong \operatorname {SL} _{2}(\mathbf {C} )}
  
.

== General fields ==
Let F be a field of characteristic not 2, and let 
  
    
      
        q
      
    
    {\displaystyle q}
  
 be a nondegenerate quadratic form on a finite-dimensional F-vector space 
  
    
      
        V
      
    
    {\displaystyle V}
  
. Over such a field, the classification of Clifford algebras is naturally expressed in terms of the center and a Brauer class rather than by a periodic matrix table.
If 
  
    
      
        dim
        ⁡
        V
        =
        2
        m
      
    
    {\displaystyle \dim V=2m}
  
 is even, then the full Clifford algebra 
  
    
      
        Cl
        ⁡
        (
        V
        ,
        q
        )
      
    
    {\displaystyle \operatorname {Cl} (V,q)}
  
 is a central simple algebra over 
  
    
      
        F
      
    
    {\displaystyle F}
  
. Its Brauer class

  
    
      
        c
        (
        q
        )
        :=
        [
        Cl
        ⁡
        (
        V
        ,
        q
        )
        ]
        ∈
        Br
        ⁡
        (
        F
        )
      
    
    {\displaystyle c(q):=[\operatorname {Cl} (V,q)]\in \operatorname {Br} (F)}
  

is called the Clifford invariant of 
  
    
      
        q
      
    
    {\displaystyle q}
  
. The center of the even Clifford algebra 
  
    
      
        
          Cl
          
            0
          
        
        ⁡
        (
        V
        ,
        q
        )
      
    
    {\displaystyle \operatorname {Cl} ^{0}(V,q)}
  
 is the quadratic étale 
  
    
      
        F
      
    
    {\displaystyle F}
  
-algebra

  
    
      
        Z
        (
        q
        )
        =
        F
        [
        x
        ]
        
          /
        
        (
        
          x
          
            2
          
        
        −
        δ
        (
        q
        )
        )
      
    
    {\displaystyle Z(q)=F[x]/(x^{2}-\delta (q))}
  
,
where 
  
    
      
        δ
        (
        q
        )
        =
        (
        −
        1
        
          )
          
            m
          
        
        det
        (
        q
        )
      
    
    {\displaystyle \delta (q)=(-1)^{m}\det(q)}
  
 is the signed discriminant of 
  
    
      
        q
      
    
    {\displaystyle q}
  
. Thus 
  
    
      
        Z
        (
        q
        )
      
    
    {\displaystyle Z(q)}
  
 is either a separable quadratic extension field of 
  
    
      
        F
      
    
    {\displaystyle F}
  
 or the split algebra 
  
    
      
        F
        ⊕
        F
      
    
    {\displaystyle F\oplus F}
  
.
If 
  
    
      
        dim
        ⁡
        V
        =
        2
        m
        +
        1
      
    
    {\displaystyle \dim V=2m+1}
  
 is odd, then the even Clifford algebra 
  
    
      
        
          Cl
          
            0
          
        
        ⁡
        (
        V
        ,
        q
        )
      
    
    {\displaystyle \operatorname {Cl} ^{0}(V,q)}
  
 is central simple over 
  
    
      
        F
      
    
    {\displaystyle F}
  
. In this case the relevant Clifford invariant is

  
    
      
        c
        (
        q
        )
        :=
        [
        
          Cl
          
            0
          
        
        ⁡
        (
        V
        ,
        q
        )
        ]
        ∈
        Br
        ⁡
        (
        F
        )
        ,
      
    
    {\displaystyle c(q):=[\operatorname {Cl} ^{0}(V,q)]\in \operatorname {Br} (F),}
  

while the full Clifford algebra has center 
  
    
      
        Z
        (
        q
        )
      
    
    {\displaystyle Z(q)}
  
 and satisfies

  
    
      
        Cl
        ⁡
        (
        V
        ,
        q
        )
        ≅
        
          Cl
          
            0
          
        
        ⁡
        (
        V
        ,
        q
        )
        
          ⊗
          
            F
          
        
        Z
        (
        q
        )
        .
      
    
    {\displaystyle \operatorname {Cl} (V,q)\cong \operatorname {Cl} ^{0}(V,q)\otimes _{F}Z(q).}
  

Thus, in odd dimension, the isomorphism class of the full Clifford algebra is determined by the quadratic étale center 
  
    
      
        Z
        (
        q
        )
      
    
    {\displaystyle Z(q)}
  
 together with the Brauer class 
  
    
      
        c
        (
        q
        )
      
    
    {\displaystyle c(q)}
  
.
An explicit computation of 
  
    
      
        c
        (
        q
        )
      
    
    {\displaystyle c(q)}
  
 may be made after diagonalizing

  
    
      
        q
        ≅
        ⟨
        
          a
          
            1
          
        
        ,
        …
        ,
        
          a
          
            n
          
        
        ⟩
        .
      
    
    {\displaystyle q\cong \langle a_{1},\dots ,a_{n}\rangle .}
  

The associated Hasse invariant is the 2-torsion Brauer class

  
    
      
        s
        (
        q
        )
        =
        
          ∏
          
            1
            ≤
            i
            <
            j
            ≤
            n
          
        
        (
        
          a
          
            i
          
        
        ,
        
          a
          
            j
          
        
        )
        ∈
        Br
        ⁡
        (
        F
        )
        [
        2
        ]
        ,
      
    
    {\displaystyle s(q)=\prod _{1\leq i<j\leq n}(a_{i},a_{j})\in \operatorname {Br} (F)[2],}
  

where 
  
    
      
        (
        
          a
          
            i
          
        
        ,
        
          a
          
            j
          
        
        )
      
    
    {\displaystyle (a_{i},a_{j})}
  
 denotes the class of the quaternion algebra generated by 
  
    
      
        i
        ,
        j
      
    
    {\displaystyle i,j}
  
 with 
  
    
      
        
          i
          
            2
          
        
        =
        
          a
          
            i
          
        
      
    
    {\displaystyle i^{2}=a_{i}}
  
, 
  
    
      
        
          j
          
            2
          
        
        =
        
          a
          
            j
          
        
      
    
    {\displaystyle j^{2}=a_{j}}
  
, and 
  
    
      
        i
        j
        =
        −
        j
        i
      
    
    {\displaystyle ij=-ji}
  
. The Clifford invariant is obtained from the Hasse invariant by a universal correction depending only on 
  
    
      
        n
        
          mod
          
            8
          
        
      
    
    {\displaystyle n{\bmod {8}}}
  
:

  
    
      
        c
        (
        q
        )
        =
        s
        (
        q
        )
        ⋅
        
          
            {
            
              
                
                  1
                  ,
                
                
                  n
                  ≡
                  1
                  ,
                  2
                  
                    
                    (
                    mod
                    
                    8
                    )
                  
                  ,
                
              
              
                
                  (
                  −
                  1
                  ,
                  −
                  det
                  q
                  )
                  ,
                
                
                  n
                  ≡
                  3
                  ,
                  4
                  
                    
                    (
                    mod
                    
                    8
                    )
                  
                  ,
                
              
              
                
                  (
                  −
                  1
                  ,
                  −
                  1
                  )
                  ,
                
                
                  n
                  ≡
                  5
                  ,
                  6
                  
                    
                    (
                    mod
                    
                    8
                    )
                  
                  ,
                
              
              
                
                  (
                  −
                  1
                  ,
                  det
                  q
                  )
                  ,
                
                
                  n
                  ≡
                  0
                  ,
                  7
                  
                    
                    (
                    mod
                    
                    8
                    )
                  
                  .
                
              
            
            
          
        
      
    
    {\displaystyle c(q)=s(q)\cdot {\begin{cases}1,&n\equiv 1,2{\pmod {8}},\\(-1,-\det q),&n\equiv 3,4{\pmod {8}},\\(-1,-1),&n\equiv 5,6{\pmod {8}},\\(-1,\det q),&n\equiv 0,7{\pmod {8}}.\end{cases}}}
  

Here 
  
    
      
        det
        q
      
    
    {\displaystyle \det q}
  
 is the determinant of a Gram matrix, viewed in 
  
    
      
        
          F
          
            ×
          
        
        
          /
        
        
          F
          
            ×
            2
          
        
      
    
    {\displaystyle F^{\times }/F^{\times 2}}
  
. In this sense, the Brauer class of the relevant Clifford algebra is the standard Clifford invariant of the quadratic form.
Over 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbf {R} }
  
, this recovers the usual real classification table above. The Brauer group 
  
    
      
        Br
        ⁡
        (
        
          R
        
        )
      
    
    {\displaystyle \operatorname {Br} (\mathbf {R} )}
  
 has two elements, represented by the split class and the class of the quaternion algebra 
  
    
      
        
          H
        
      
    
    {\displaystyle \mathbf {H} }
  
. For a diagonal form of signature 
  
    
      
        (
        p
        ,
        q
        )
      
    
    {\displaystyle (p,q)}
  
, the Hasse invariant is

  
    
      
        s
        (
        q
        )
        =
        [
        
          H
        
        
          ]
          
            
              
                (
              
              
                q
                2
              
              
                )
              
            
          
        
        ,
      
    
    {\displaystyle s(q)=[\mathbf {H} ]^{\binom {q}{2}},}
  

since over 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbf {R} }
  
 the quaternion class 
  
    
      
        (
        a
        ,
        b
        )
      
    
    {\displaystyle (a,b)}
  
 is nontrivial exactly when both 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 are negative. The formula above therefore determines abstractly whether the relevant central simple algebra is split or quaternionic. In even dimension this yields matrix algebras over 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbf {R} }
  
 or 
  
    
      
        
          H
        
      
    
    {\displaystyle \mathbf {H} }
  
; in odd dimension one combines the same Brauer-class computation for 
  
    
      
        
          Cl
          
            0
          
        
        ⁡
        (
        V
        ,
        q
        )
      
    
    {\displaystyle \operatorname {Cl} ^{0}(V,q)}
  
 with the center 
  
    
      
        Z
        (
        q
        )
      
    
    {\displaystyle Z(q)}
  
, which is either 
  
    
      
        
          R
        
        ×
        
          R
        
      
    
    {\displaystyle \mathbf {R} \times \mathbf {R} }
  
 or 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
. When 
  
    
      
        Z
        (
        q
        )
        ≅
        
          C
        
      
    
    {\displaystyle Z(q)\cong \mathbf {C} }
  
, the full Clifford algebra is a complex matrix algebra, because

  
    
      
        
          C
        
        
          ⊗
          
            
              R
            
          
        
        
          H
        
        ≅
        
          M
          
            2
          
        
        (
        
          C
        
        )
        .
      
    
    {\displaystyle \mathbf {C} \otimes _{\mathbf {R} }\mathbf {H} \cong M_{2}(\mathbf {C} ).}