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) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210 Zbl 05948094 Zbl 1145.17002 Zbl 1145.17001 Zbl 1116.28002 Zbl 1108.26003 Zbl 1106.46005 Zbl 1107.01001 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1107.54001 Zbl 1123.22005 Zbl 1120.17002 Zbl 1120.17001 Zbl 1179.58001 Zbl 1106.46004 Zbl 1105.18001 Zbl 1106.46003 Zbl 1107.13002 Zbl 1107.13001
[2]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056
[3]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001
[4]	E. Cartan, "Leçons sur la théorie des spineurs" , Hermann (1938) Zbl 0022.17101 Zbl 0019.36301 Zbl 64.1382.04

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) MR0060497 Zbl 0057.25901
[a2]	O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) Zbl 0259.10018
[a3]	C. Chevalley, "The construction and study of certain important algebras" , Math. Soc. Japan (1955) pp. Chapt. III MR0072867