Derivations of skew PBW-extensions.
Let \(A\) be a ring with \(1\) and \(R\) a subring. Then \(A\) is called a skew PBW-extension of \(R\) if (1) there exists a finite set of elements \(x_1,\ldots,x_n\in A\) such that \(A\) is a left \(R\)-free module with the basis: \(\{x^\alpha=(x_1^{\alpha_1},\ldots,x_n^{\alpha_n})\mid(\alpha_1,\ldots,\alpha_n)\in(\mathbb N\cup 0)^n\}\), (2) there exist endomorphisms \(\sigma_1,\ldots,\sigma_n\) of \(R\) such that \((ad_{\sigma_i}x_i)r=x_ir-\sigma_i(r)x_i\) (\(=\delta_i(r)\)) for each \(r\in R\), the induced \(\sigma_i\)-derivation of \(A\), and (3) for any \(1\leq i,j\leq n\), there exist elements \(c_{ij},r_{ij},r_{ij}^t\in R\) such that \(c_{ij}c_{ji}=1\) and \(x_ix_j-c_{ij}x_jx_i=\sum_tr_{ij}^tx_t+r_{ij}\). The class of skew PBW-extensions contains Auslander-Gorenstein rings, skew Calabi-Yau algebras and some well known algebras. The author shows some identities of the derivations \(\delta\) of \(A\) mapping \(R\) into itself and \(\delta\sigma_i=\sigma_i\delta\) for each \(i\), and a classification of derivations of \(A\). Suppose that \(R\) is a torsion free \(Z\)-module and \(\sigma_i\) are automorphisms such that \(\delta_i\sigma_i=\sigma_i\delta_i\) for each \(i\), and that for any \(i\neq j\), there exists a central element \(r\in R\) such that \(\sigma_i(r)-\sigma_j(r)\) is not a right zero divisor in \(R\). Let \(\delta\) be a derivation of \(A\) such that \(R\) is stable under \(\delta\) and \(\delta\sigma_i=\sigma_i\delta\) for each \(i\). Then the identities hold: (1) for any index \(m\), there exist elements \(\lambda_m,\lambda_{m0}\in R\) such that \(\delta(x_m)=\lambda_mx_m+\lambda_{m0}\) where \(\lambda_m\) is a central element of \(R\) and \(\lambda_i\delta_i(r)+(ad_{\sigma_i}\lambda_{i0})(r)=0\); (2) \((\lambda_i+\sigma_i(\lambda_j))-(\lambda_j+\sigma_j(\lambda_i))-\delta (c_{ij})=0\); and (3) \(-c_{ij}(\lambda_j+\sigma_j(\lambda_i))r_{ji}^t=\delta(r_{ij}^t)+r_{ij}^t \lambda_t\). Moreover, suppose that under the above assumptions, if there exist derivations \(\delta,\delta',\delta''\) of \(A\) commuting with \(\sigma_i\) for each \(i\) and vanishing on \(R\), then \(\delta[\delta',\delta'']=0\) in \(A\) where \([\delta',\delta'']=\delta'\delta''-\delta''\delta'\). In particular, if \(\delta'''\) is another derivation commuting with \(\sigma_i\) for each \(i\) and vanishing on \(R\), then \([[\delta,\delta'],[\delta'',\delta''']]=0\).
scientific article; zbMATH DE number 6527442