Quadratic Algebras as Quantum Linear Spaces

  • Yuri I. ManinEmail author
Part of the CRM Short Courses book series (CRMSC)


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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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany

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