Quadratic Algebras as Quantum Linear Spaces

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 \(R(A)\subset A_1^{\otimes 2}\).