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

It is convenient to write $A \leftrightarrow \{A_1, R(A)\}$. We assume $\dim A_1 < \infty$.