# PI degree parity in q-skew polynomial rings

## TL;DR

This paper investigates the PI degree of iterated skew polynomial rings with q-skew relations, confirming the quantum Gelfand-Kirillov conjecture for certain quantized coordinate rings and extending results to prime factor algebras.

## Contribution

It establishes that the PI degree remains unchanged under specific q-skew extensions and confirms the quantum Gelfand-Kirillov conjecture for various quantized coordinate rings.

## Key findings

- PI degree of iterated skew polynomial rings matches that of simpler rings under q-skew relations.
- Confirmed the quantum Gelfand-Kirillov conjecture for multiple quantized coordinate rings.
- Extended results to completely prime factor algebras.

## Abstract

For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x_1;\tau_1,\delta_1]...b[x_n;\tau_n,\delta_n] agrees with the PI degree of R[x_1;\tau_1]...b[x_n;\tau_n] when each (\tau_i,\delta_i) satisfies a q_i-skew relation for q_i \in k^{\times} and extends to a higher q_i-skew \tau_i-derivation. We confirm the quantum Gel'fand-Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras.