Definition:Quadratic Algebra

Definition
A quadratic algebra $A$ is a filtered algebra whose generator consists of degree $1$ elements, with defining relations of degree $2$.

A quadratic algebra $A$ is determined by a vector space of generators $V = A_1$ and a subspace of homogeneous quadratic relations $S \subseteq V \otimes V$.

Thus:
 * $A = \map T V / \gen S$

and inherits its grading from the tensor algebra $\map T V$.

If the subspace of relations may also contain inhomogeneous degree $2$ elements, $S \subseteq k \oplus V \oplus \paren {V \otimes V}$, this construction results in a filtered quadratic algebra.

A graded quadratic algebra $A$ as above admits a quadratic dual: the quadratic algebra generated by $V^*$ and with quadratic relations forming the orthogonal complement of $S$ in $V^* \otimes V^*$.

Examples
An example of such algebra is the exterior algebra.