PI degree parity in q-skew polynomial rings

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 ; τ 1 , δ 1 ] · · · [x n ; τ n , δ n ] agrees with the PI degree of R[x 1 ; τ 1 ] · · · [x n ; τ n ] when each (τ i , δ i ) satisfies a q i -skew relation for q i ∈ k × and extends to a higher q i -skew τ 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.

ON q-SKEW ITERATED ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

“…[x n ; σ n , δ n ] ≃ R n S −1 we can conclude that R n S −1 satisfies polynomial identity and so does R n . Both Theorem 1.2 (1) and Corollary 4.7 of [5] are direct consequences of the above theorem. Moreover we relaxed the assumptions from [5] that δ i 's have to extend to locally nilpotent iterative higher q i -skew σ i -derivations and that R is a noetherian domain.…”

“…The above theorem is a partial generalization of the main result of [5]. Comparing Theorem 6 above with Theorem 4.6 [5], observe that in [5] it is additionally assumed that each δ i , with 1 ≤ i ≤ n, extends to locally nilpotent iterative higher q i -skew σ i -derivation on R i−1 (see [5] for details). Notice also that in [5], R is a noetherian domain which is an algebra over a field k and q i , λ ij ∈ k, where q i ∈ {0, 1}.…”

“…The method of switching indeterminates in the proof of Theorem 6, based on Lemma 5(2), goes back to the paper [5]. The statement (3) from the above lemma explains that, in the process of switching indeterminates in Theorem 6, we need all the λ ij 's to be also invariant under the action of all suitable skew derivations.…”

“…Comparing Theorem 6 above with Theorem 4.6 [5], observe that in [5] it is additionally assumed that each δ i , with 1 ≤ i ≤ n, extends to locally nilpotent iterative higher q i -skew σ i -derivation on R i−1 (see [5] for details). Notice also that in [5], R is a noetherian domain which is an algebra over a field k and q i , λ ij ∈ k, where q i ∈ {0, 1}. This is due to the fact that higher q-skew derivations where used in [5] for erasing derivations from Ore extensions.…”

“…Notice also that in [5], R is a noetherian domain which is an algebra over a field k and q i , λ ij ∈ k, where q i ∈ {0, 1}. This is due to the fact that higher q-skew derivations where used in [5] for erasing derivations from Ore extensions. The assumption that σ i | R 0 is an automorphism of R 0 , for all 1 ≤ i ≤ n, from Theorem 6 was not formally stated in Theorem 4.6 [5] but it was used in its proof.…”