Clifford Algebra - Universal Property and Construction

Universal Property and Construction

Let V be a vector space over a field K, and let Q: V → K be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field.

A Clifford algebra Cℓ(V, Q) is a unital associative algebra over K together with a linear map i : V → Cℓ(V, Q) satisfying i(v)2 = Q(v)1 for all v ∈ V, defined by the following universal property: Given any associative algebra A over K and any linear map j : V → A such that

j(v)2 = Q(v)1_A for all v ∈ V

(where 1_A denotes the multiplicative identity of A), there is a unique algebra homomorphism f : Cℓ(V, Q) → A such that the following diagram commutes (i.e. such that f ∘ i = j):

Working with a symmetric bilinear form ⟨·,·⟩ instead of Q (in characteristic not 2), the requirement on j is

j(v)j(w) + j(w)j(v) = 2⟨v, w⟩ for all v, w ∈ V.

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains V, namely the tensor algebra T(V), and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal I_Q in T(V) generated by all elements of the form

for all

and define Cℓ(V, Q) as the quotient algebra

Cℓ(V, Q) = T(V)/I_Q.

The ring product inherited by this quotient is sometimes referred to as the Clifford product to differentiate it from the inner and outer products.

It is then straightforward to show that Cℓ(V, Q) contains V and satisfies the above universal property, so that Cℓ is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra Cℓ(V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of Cℓ(V, Q).

The universal characterization of the Clifford algebra shows that the construction of Cℓ(V, Q) is functorial in nature. Namely, Cℓ can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.