Clifford algebra
Let be a vector space over a field , and a symmetric bilinear form. Then the Clifford algebra is the quotient of the tensor algebra by the relations
Since the above relationship is not homogeneous in the usual -grading on , does not inherit a -grading. However, by reducing mod 2, we also have a -grading on , and the relations above are homogeneous with respect to this, so has a natural -grading, which makes it into a superalgebra.
In addition, we do have a filtration on (making it a filtered algebra), and the associated graded algebra is simply , the exterior algebra of . In particular,
The most commonly used Clifford algebra is the case , and is the standard inner product with orthonormal basis . In this case, the algebra is generated by and the identity of the algebra , with the relations
Trivially, , and it can be seen from the relations above that , the complex numbers, and , the quaternions.
On the other ha nd, for we get the particularly answer of
Title | Clifford algebra |
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Canonical name | CliffordAlgebra |
Date of creation | 2013-03-22 13:18:05 |
Last modified on | 2013-03-22 13:18:05 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 9 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 15A66 |
Classification | msc 11E88 |