In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford (1845–1879).

The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras.

A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K. The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition $${\displaystyle v^{2}=Q(v)1\ {\text{ for all }}v\in V,}$$ where the product on the left is that of the algebra, and the 1 on the right is the algebra's multiplicative identity (not to be confused with the multiplicative identity of K). The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below.

When V is a finite-dimensional real vector space and Q is nondegenerate, Cl(V, Q) may be identified by the label Cl_p,q(R), indicating that V has an orthogonal basis with p elements with e_i² = +1, q with e_i² = −1, and where R indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. Such a basis may be found by orthogonal diagonalization.

The free algebra generated by V may be written as the tensor algebra ⨁_n≥0 V ⊗ ⋯ ⊗ V, that is, the direct sum of the tensor product of n copies of V over all n. Therefore one obtains a Clifford algebra as the quotient of this tensor algebra by the two-sided ideal generated by elements of the form v ⊗ v − Q(v)1 for all elements v ∈ V. The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. uv). Its associativity follows from the associativity of the tensor product.

The Clifford algebra has a distinguished subspace V, being the image of the embedding map. Such a subspace cannot in general be uniquely determined given only a K-algebra that is isomorphic to the Clifford algebra.

If 2 is invertible in the ground field K, then one can rewrite the fundamental identity above in the form $${\displaystyle uv+vu=2\langle u,v\rangle 1\ {\text{ for all }}u,v\in V,}$$ where $${\displaystyle \langle u,v\rangle ={\frac {1}{2}}\left(Q(u+v)-Q(u)-Q(v)\right)}$$ is the symmetric bilinear form associated with Q, via the polarization identity.

Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case in this respect. In particular, if char(K) = 2 it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfies Q(v) = ⟨v, v⟩, Many of the statements in this article include the condition that the characteristic is not 2, and are false if this condition is removed.