Clifford algebra
Let V be a vector space over a field k, and Q:V×V→k a
symmetric bilinear form
. Then the Clifford algebra
Cliff(Q,V) is
the quotient of the tensor algebra 𝒯(V) by the relations
v⊗w+w⊗v=-2Q(v,w) |
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 |
---|---|
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 |