Quantization of the external algebra on a Poisson–Lie group

The external algebra $\mathcal M$ on $GL(N)$ is show to be equipped with graded Poisson brackets compatible with the group action. The corresponding Poisson–Hopf superalgebra structures on $\mathcal M$ are classified. They form two families, each of them is parametrized by two continuous parameters $(\alpha,\beta)$. It is shown that the structures from the first family with $\alpha=0=\beta$ appear as the semiclassical limit of the bicovariant differential calculi on the quantum linear group $GL_q(N)$.