Examples of quadratic algebrasExample 1: suppose thatrun 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”:
Example 2:runs over all symmetric matrices. In this case
is the exterior algebra
:
This is the algebra of antisymmetric tensors.