It is about Clifford Algebra

1. AN INTRODUCTION TO CLIFFORD
ALGEBRA
BY
AMALRAJ.P
ASSISTANT PROFESSOR
DEPARTMENT OF MATHEMATICS
S.D.COLLEGE, ALAPPUZHA

2. INTRODUCTION

3. Clifford algebras extend the real number
system to include vectors and their
product.
They are useful for modelling geometry, and
their vector products .

4. The Clifford Algebra
Anssume n basic vectors of n-
dimensional vector space on the real
space as follows;
1 2 ne ,e ,...,e

5. i j j i
2
i
2
i
2
i
e e e e for i j
e 1 for i 1,2,....p
e 1 for i p 1, p 2,....p q
e 0 for i p q 1, p q 2,....p q r n
 
 
    
        

6. Choose the m basic vectors and consider all
linear combinations of the product of
them . The result is defined as a
multivector of rank m.
If we consider the sum of multi vectors of all
ranks to obtain a vector space of
dimension .
mnC
n
2

7. with which the above product defines a
dimensional vector space and the
associated algebra called clifford algebra
and is denoted by .
n
2
p,q,rCl

8. AXIOMATIC DEVELOPMENT
We postulate that there is a clifford product
that can be used to multiply any element
of the Clifford algebra by one another.
The clifford product of A and B is written AB.
The main axioms of the clifford product is

9. i) The product is associative; that is
(AB)C= A(BC) for any three multi vectors
A, B,and C.
ii) The product is distributive over addition;
that is
A(B+C) = AB+AC

10. SOME SPECIAL CLIFFORD ALGEBRAS

11. CLIFFORD ALGEBRA OF A 1- SPACE
Suppose that the basis for a one space is .
Then the basis for the associated Clifford algegra
is .
So an arbitrary element of Clifford algebra of a
1- space is the form . Here a and b are
scalars.
1e
 11,e
1a be

12. The Clifford product of two general element of the
algebra is:
Different Clifford algebras may be generated
depending on the metric chosen for the space.
     2
0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 1a a e b b e a b a b a b e a b e     

13. In this case the type of Clifford algebras
which we can generate in a 1- space are
dependent only on the choice of a single
scalar value for the product .2
1e

14. It is clear to see that if we choose
we have an algebra isomorphic to the
complex algebra.
The basis 1- element then plays the role of
imaginary unit i .
2
1e 1 

15. Clifford algebra of Complex algebra
1- spce
Or
1
1 1
1 -1
1e
1e
1e
1 i
1 1 i
i i -1

16. CLIFFORD ALGEBRA OF A 2- SPACE
If we choose an orthogonal basis for
the 2- space, then the basis for the
associated clifford algebra is
and the associated clifford product table is
 1 2e ,e
 1 2 1 21, e ,e ,e e

17. 1e 2e 1 2e e
1 2e e
2e
1e 2
1e 1 2e e 2
1 2e e
2 1e e 2
2 1e e
2
2 1e e2
1 2e e 2 2
1 2e e
2
2e

18. The table defines the Clifford algebra on the
2- space.Different clifford algebras may
be generated by choosing different
metrices for the space.
That is, by choosing the two scalar values
for 2 2
1 2e and e

19. A SPECIAL CASE

20. Assume .
In this case, The clifford algebra of 2- space
is isomorphic to the Quaternion.
2 2
1 2e 1, e 1   

21. We see this isomorphism more clearly be
substituting the usual quaternion and
choose the correspondance;
1 2 1 2e i, e j,e e k  

22. A quaternion can be written in terms of these basis
elements as
1 2 1 2Q a be ce de e   
1 1 2(a be ) (c de )e   

23. Hence a quaternion is a complex number with
imaginary unit , whose components are
complex numbers based on as the
imaginary unit.
2e
1e

24. CONCLUSION
---------------------
Clifford algebra is a generalization of vector
products.It helps us to view different algebras in
a single frame.
Using the powerful tool of Clifford algebra, we
can easily establish the relationship between
Complex algebra, Matrix algebra, and the rings
of Quaternions.

25. REFERENCES
---------------------
1) Chris Doran and Anthony Lasenby. Geometric
algebra for Physicists.Cambridge University
Press, 2003.
2) G.Sommer. Geometric Computing With Cliffoed
algebras. Springer- Verlag, 2001.