\begin{definition}[Definition:Quadratic Algebra]
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 $V^* \times V^*$.
{{Help}}
{{MissingLinks}}
Category:Definitions/Algebras
\end{definition}