Frobenius Algebras and the Quantum Determinant

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


A quadratic algebra A is said to be a Frobenius algebra (or Frobenius quantum space) of dimension d if (a) \(\dim A_d=1\), \(A_i=0\) for \(i>d\). For all j, the multiplication map \(m:A_j\otimes A_{d-j} \rightarrow A_d\) is a perfect duality. (For example where this pairing is nonsymmetric for \(j=d-j\), see [30]. This asymmetry is the reason why the quantum determinant considered in Example 9.6 might be noncentral.) The algebra A is called a quantum Grassmann algebra if, in addition, (c) \(\dim A_i = \left( {\begin{array}{c}d\\ i\end{array}}\right) \).

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany

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