On the dimension of $H$-strata in quantum algebras

Abstract

We study the topology of the prime spectrum of an algebra supporting a rational torus action. More precisely, we study inclusions between prime ideals that are torus-invariant using the $H$-stratification theory of Goodearl and Letzter on the one hand, and the theory of deleting derivations of Cauchon on the other. We also give a formula for the dimensions of the $H$-strata described by Goodearl and Letzter. We apply the results obtained to the algebra of $m\times n$ generic quantum matrices to show that the dimensions of the $H$-strata are bounded above by the minimum of $m$ and $n$, and that all values between $0$ and this bound are achieved.