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PI degree parity in \(q\)-skew polynomial rings. (English) Zbl 1153.16025
Suppose that an associative algebra \(A\) is an iterated skew polynomial extension \(k[x_1;\tau_1,\delta_1]\cdots[x_n;\tau_n,\delta_n]\) such that \(\tau_i(x_j)=\lambda_{ij}x_j\) for all \(j<i\) and \(\delta_i\tau_i=q_i\tau_i\delta_i\) where \(\lambda_{ij},q_i\in k^*\) and \(q_i\neq 1\). It is also assumed that each \(\tau_i\)-derivation \(\delta_i\) extends to a locally nilpotent derivation on \(k\langle x_1,\dots,x_{i-1}\rangle\). Under these assumptions it is shown that \(A\) is a PI-algebra if and only if all \(\lambda_{ij}\) are roots of 1. Clearly in this case the PI-degree of \(A\) is the same as the PI-degree of the related quantum polynomial algebra with multiparameters \(\lambda_{ij}\).

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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