On the dimension of H-strata in quantum matrices

TL;DR

This paper investigates the structure of prime ideals in quantum matrices, showing that the dimensions of certain strata are bounded and all intermediate values are realized, advancing understanding of quantum algebra topology.

Contribution

It establishes bounds on the dimensions of H-strata in quantum matrices and demonstrates that all intermediate dimensions are attainable, combining stratification and derivation deletion theories.

Findings

Dimensions of H-strata are bounded by min(m, n).

All values between 0 and the bound are achieved.

Provides a detailed topological analysis of prime spectra in quantum matrices.

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 one hand and the theory of deleting derivations of Cauchon on the other. We apply the results obtained to the algebra of $m \times n$ generic quantum matrices to show that the dimensions of the $H$-strata described by Goodearl and Letzter are bounded above by the minimum of $m$ and $n$, and that moreover all the values between 0 and this bound are achieved.