Frobenius Algebras and the Quantum Determinant

Abstract

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) $\dim A_i = \left( {\begin{array}{c}d\\ i\end{array}}\right)$.