Clifford algebra

A finite-dimensional associative algebra over a commutative ring; it was first investigated by W. Clifford in 1876. Let be a commutative ring with an identity, let be a free -module and let be a quadratic form on . By the Clifford algebra of the quadratic form (or of the pair ) one means the quotient algebra of the tensor algebra of the -module by the two-sided ideal generated by the elements of the form , where . Elements of are identified with their corresponding cosets in . For any one has , where is the symmetric bilinear form associated with .

For the case of the null quadratic form , is the same as the exterior algebra of . If , the field of real numbers, and is a non-degenerate quadratic form on the -dimensional vector space over , then is the algebra of alternions, where is the number of positive squares in the canonical form of (cf. Alternion).

Let be a basis of the -module . Then the elements form a basis of the -module . In particular, is a free -module of rank . If in addition the are orthogonal with respect to , then can be presented as a -algebra with generators and relations and . The submodule of generated by products of an even number of elements of forms a subalgebra of , denoted by .

Suppose that is a field and that the quadratic form is non-degenerate. For even , is a central simple algebra over of dimension , the subalgebra is separable, and its centre has dimension 2 over . If is algebraically closed, then when is even is a matrix algebra and is a product of two matrix algebras. (If, on the other hand, is odd, then is a matrix algebra and is a product of two matrix algebras.)

The invertible elements of (or of ) for which form the Clifford group (or the special Clifford group ) of the quadratic form . The restriction of the transformation

to the subspace defines a homomorphism , where is the orthogonal group of the quadratic form . The kernel consists of the invertible elements of the algebra and . If is even, then and is a subgroup of index 2 in , which in the case when is not of characteristic 2, is the same as the special orthogonal group . If is odd, then

Let be the anti-automorphism of induced by the anti-automorphism

of the tensor algebra . The group

is called the spinor group of the quadratic form (or of the Clifford algebra ).

The homomorphism has kernel . If or and is positive definite, then and coincides with the classical spinor group.

References

[1]	N. Bourbaki, "Elements of mathematics" , Addison-Wesley (1966–1977) (Translated from French)
[2]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955)
[3]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[4]	E. Cartan, "Leçons sur la théorie des spineurs" , Hermann (1938)

Comments

The algebra generated by products of an even number of elements of the free -module is also called the even Clifford algebra of the quadratic form . See also the articles Exterior algebra (or Grassmann algebra), and Cartan method of exterior forms for more details in the case .

References

[a1]	C. Chevalley, "The algebraic theory of spinors" , Columbia Univ. Press (1954)
[a2]	O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973)
[a3]	C. Chevalley, "The construction and study of certain important algebras" , Math. Soc. Japan (1955) pp. Chapt. III