Examples of quadratic algebras
Example 1: suppose that
run
over
all antisymmetric matrices. In this case the resulting quadratic
algebra
is just the algebra of
polynomials of
variables, where
,
or equivalently the algebra of
symmetric tensors.
Another name for it is “symmetric algebra”:
| (4) |
Example 2: runs over
all symmetric matrices. In this case
is the exterior algebra
:
| (5) |
This is the algebra of antisymmetric tensors.