## Clifford algebras and Spin groups

Still lot of editing needed. the pdf file can be obtained here

As a provisional definition, *clifford algebra*over a field can be defined as

where,

Easy to see that has dimension . can also be thought of as

where as vector space, is the tensor algebra, is same as above, except that ‘s are standard basis for . This observation leads to a more general definition of *clifford algebra*, where is a vector space equipped with a symmetric bilinear form

Definition 1Let be a vector space with symmetric biliear form and quadratic form . Then theclifford algebraover can be defined as

Remark 1These are some of the properties that enjoys

- There is a natural inclusion of .
- If then the exterior algebra
- Let denote the
clifford multiplication( induced by the tensor product of ), thenUniversal Property :Let , where is a -algebra, such that , then there exists an unique map such that , that is the following diagram commutes- A map , such that , extends to a -algebra homomorphismThus the orthogonal group has an action on

Proposition 2is a filtered algebra whose associate graded is

Before proving the theorem, recall the following definition

Definition 3If is a -algebra then a filtration of is sequence of subspaces

Theassociate gradedof is defined as

*Proof:* Let be the quotient map

Define, ( fold tensor product). Define filtration on by setting

Define filtration be the filtration on the clifford algebra. Note . Hence, in the associated graded . On the other hand the relation prevails in the associated graded. Hence the associate graded is isomorphic to .

Remark 2is a -graded algebra.

Definition 4Recall,. Define,

and

On we have an involution map, which is induced by the involution on given by,

If then

Let and , then observe

Lemma 5There exist short exact sequences

and

where is the map which sends

Let be a field. Recall, tensor product of -algebras and is a -algebra, denoted by and multiplication is given by

moreover if denotes the set of all matrices. Then we have the following isomorphism

Define

Remark 3If , then

This follows from the fact that the quadratic forms and induces isomorphic innerproduct structure on where the isomorphism sends

Theorem 6If , then we have the following isomorphisms

Corollary 7(Bott Periodicity) As a consequence of (iv) we have

if even, and

if is odd.

To work out the case when the underlying field is . For any field we have the following isomorphisms.

Lemma 8For any field

*Proof:* Let denote the standard basis of and cannonical generatoring set of the *Clifford algebra* .

- To get the first isomorphism we simply produce a map given by sendingandIt is easy to check that the above map is an isomorphism.
- is similar to .

One can explicitly check some of the lower dimension cases( ). Then one can repeatedly use the isomorphisms in previous lemma. One has to work upto dimension when , before one sees the patern, which is called the *Bott periodicity*. TSome of the calculations are as follows calculations are as follows

- In general one gets,

Putting all these observations together we get

Theorem 9TheBott periodicityin case of real number looks like

**Lemma 10** * As -algebras Proof:The isomorphism is given explicitly by the map induced by sending*

*Easy to check that this is an isomorphism of algebras. *

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