Recent zbMATH articles in MSC 13Nhttps://www.zbmath.org/atom/cc/13N2021-07-10T17:08:46.445117ZWerkzeugMaximal Lie subalgebras among locally nilpotent derivationshttps://www.zbmath.org/1462.130252021-07-10T17:08:46.445117Z"Skutin, Alexander A."https://www.zbmath.org/authors/?q=ai:skutin.alexander-aThis paper is on investigating two following conjectures that Freudenburg was mentioned in the paper [\textit{G. Freudenburg}, Algebraic theory of locally nilpotent derivations. Berlin: Springer (2006; Zbl 1121.13002)].
Conjecture 1. The lie algebra \(\mathfrak{I}=\mathbb{K}\partial_{x_1}\oplus \cdots \oplus \mathbb{K}[x_1,\cdots,x_{n-1}]\partial_{x_n}\) of triangular derivations is a maximal Lie algebra (with respect to embedding) lying in \(\mathrm{LND}(\mathbb{K}[x_1,\cdots,x_{n}])\).
This is shown that this conjecture is true.
Conjecture 2. Let \(\mathcal{A}\) be a maximal Lie algebra (with respect to embedding) lying in \(\mathrm{LND}(\mathbb{K}[x_1,\cdots,x_{n}])\). Then \(\mathcal{A}\) is conjugate to the Lie algebra \(\mathfrak{I}=\mathbb{K}\partial_{x_1}\oplus \cdots \oplus \mathbb{K}[x_1,\cdots,x_{n-1}]\partial_{x_n}\)
This statement does not hold in general. For example, it is shown that if \(D=-P_z\partial_y+P_y\partial_z\) be the locally nilpotent derivation, with \(P_y=1+y(xz+y^2), P_z=(xz+y^2)/2\), then the conjugacy of a maximal Lie subalgebra makes a contradiction with an example in [Zbl 0874.13021]. But it is proved that with the change of conditions this conjecture can be true. So we get the following theorem.
Theorem [\textit{A. A. Skutin}, Sb. Math. 212, No. 2, 265--271 (2021; Zbl 07349628); translation from Mat. Sb. 212, No. 2, 138--146 (2021)]. Given a maximal Lie algebra \(\mathcal{A}\subset \mathrm{LND}(\mathbb{K}[x_1,\cdots,x_{n}])\), suppose that \(\mathrm{ker}\mathcal{A}:=\bigcap_{D\in \mathcal{A}} \mathrm{ker}D=\mathbb{K}.\) Then \(\mathcal{A}\) is conjugate to the Lie algebra \(\mathbb{K}\partial_{x_1}\oplus \cdots \oplus \mathbb{K}[x_1,\cdots,x_{n-1}]\partial_{x_n}\).Construction of free differential algebras by extending Gröbner-Shirshov baseshttps://www.zbmath.org/1462.130272021-07-10T17:08:46.445117Z"Li, Yunnan"https://www.zbmath.org/authors/?q=ai:li.yunnan"Guo, Li"https://www.zbmath.org/authors/?q=ai:guo.li|guo.li.2|guo.li.1Let \(\mathbf{k}\) be a field of characteristic zero and \(\lambda\in\mathbf{k}\). Then a differential \(\mathbf{k}\)-algebra of weight \(\lambda\) (also called a \(\lambda\)-differential \(\mathbf{k}\)-algebra) is defined as an associative \(\mathbf{k}\)-algebra\(R\) together with a linear operator \(d:R\rightarrow R\) (called a \(\lambda\)-derivation) such that \(d(xy) = d(x)y+xd(y)+\lambda d(x)d(y)\) for all \(x, y\in R\). If \(R\) is unital, it is also required that \(d(1_{R})=0\). If \(X\) is a set, then the free (\(\lambda\)-) differential algebra on \(X\) is a \(\lambda\)-differential \(\mathbf{k}\)-algebra \((\mathcal{D}_{\lambda}(X), d_{X})\) together with the map \(i_{X}:X\rightarrow \mathcal{D}_{\lambda}(X)\) satisfying the following universal property: for any \(\lambda\)-differential algebra \((R, d_{R})\) and map \(f:X\rightarrow R\), there exists a unique homomorphism of \(\lambda\)-differential algebras \(\bar{f}:D_{\lambda}(X)\rightarrow R\) such that \(f=\bar{f}\circ i_{X}\). In this paper the authors prove composition-diamond lemmas on Gröbner-Shirshov bases for associative algebras and (\(\lambda\)-) differential algebras, including their commutative counterparts. Then the composition-diamond lemma for differential algebras is applied to obtain Gröbner-Shirshov bases for free differential algebras on algebras in both the noncommutative and commutative cases. These results, in turn, allowed the authors to obtain canonical bases for the mentioned free objects. It is shown that a Gröbner-Shirshov basis of an algebra can be ``differentially'' extended to a differential Gröbner-Shirshov basis of the free differential algebra on this algebra, in all the cases except for the ``classical'' one, namely for differential commutative algebras with weight zero, when there are obstructions to such an extension. As demonstrations, several examples are given in the cases of free differential algebras on special algebras, including commutative algebras with one generator and some finite group algebras.Tame degree functions in arbitrary characteristichttps://www.zbmath.org/1462.140612021-07-10T17:08:46.445117Z"Gupta, Neena"https://www.zbmath.org/authors/?q=ai:gupta.neena"Sen, Sourav"https://www.zbmath.org/authors/?q=ai:sen.souravSummary: In [Osaka J. Math. 49, No. 1, 53--80 (2012; Zbl 1261.14035)], \textit{D. Daigle} has proved a few results on ``tameness'' of a degree function on an integral domain \(B\) containing \(\mathbb{Q}\), which ensure that, under certain hypotheses, a derivation on \(B\) gives rise to a homogeneous derivation on an associated graded ring of \(B\) induced by the degree function. In this paper, we extend Daigle's results to arbitrary integral domains not necessarily containing \(\mathbb{Q}\).Classification of simple modules of the Ore extension \(K[X][Y; f\frac{d}{dX}]\)https://www.zbmath.org/1462.160272021-07-10T17:08:46.445117Z"Bavula, V. V."https://www.zbmath.org/authors/?q=ai:bavula.vladimir-vSummary: For the algebras \(\Lambda\) in the title of the paper, a classification of simple modules is given, an explicit description of the prime and completely prime spectra is obtained, the global and the Krull dimensions of \(\Lambda\) are computed.