Definition:Quadratic Algebra

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A quadratic algebra $A$ is a filtered algebra whose generator consists of degree one 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 $S \subseteq V \times V$.

Thus :

$A = T \left({V}\right) / \left \langle {S}\right \rangle$

and inherits its grading from the tensor algebra $T \left({V}\right)$.

If the subspace of relations may also contain inhomogeneous degree 2 elements, $S \subseteq k \times V \times \left({V \times V}\right)$, 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 $V^* \times V^*$.