Quadratic Algebras as Quantum Linear Spaces
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Abstract
A quadratic algebra is an associative graded algebra \(A=\bigoplus _{i=0}^\infty A_i\) with the following properties:

\(A_0=\mathbb {K}\) (the ground field);

A is generated by \(A_1\);

the ideal of relations between elements of \(A_1\) is generated by the subspace of all quadratic relations \(R(A)\subset A_1^{\otimes 2}\).
It is convenient to write \(A \leftrightarrow \{A_1, R(A)\}\). We assume \(\dim A_1 < \infty \).
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