On the structure of quantum super groups \(\text{GL}_q(m|n)\) (Q1283937)

The quantum supergroup \(GL_q(m|n)\) is defined as the Hopf envelope of Manin's quantum super semigroup \(M_q(m|n)\). For a quantum super matrix, \(Z=\left(\begin{smallmatrix} A&B\\ C&D\end{smallmatrix}\right)\), the quantum Berezinian of \(Z\) is \(\text{Ber}_q Z= \det_q A\cdot \det_q (D-CA^{-1}B)\). The main result of this paper states that if \(Z\) is invertible then both \(A\) and \(D-CA^{-1}B\) are also invertible. The converse statement is also true under some additional assumptions. Further, the author also obtains an alternative presentation of \(GL_R(m|n)\), replacing the entries of the formal inverse matrix by \(\det_q A^{-1}\) and \(\det_q D^{-1}\) (or \(\det_q A^{-1}\) and \(\det_q((D-CA^{-1}B)^{-1}\)). Finally, in the last section the author discusses some generalizations of the previous results for the Hecke sum of an even and an odd Hecke operator. In particular, it is obtained that the corresponding quantum matrix \(Z\) is invertible if and only if \(A\) and \(D\) are invertible, the quantum Berezinian can be defined as in the classical case and is multiplicative.