Let k be an algebraically closed field of zero characteristic. The Lie algebra W_2 of all k-derivations of the polynomial ring k[x, y] naturally acts on the polynomial ring k[x, y] and also on the field of rational functions k(x, y). For a fixed non-constant rational function u from k(x,y) we consider the set A_{W_2}(u) of all derivations D from W_2 such that D(u)=0. We prove that A_{W_2}(u) is a free submodule of rank 1 of the k[x,y]-module W_2. A description of the maximal abelian subalgebras as well of the centralizers of elements in the Lie algebra A_{W_2}(u) has been obtained.
In this paper, we investigate the derivations in evolution algebras that are power-associative. This problem is reduced to that of power-associative evolution nilalgebras. We show how to calculate derivations in decomposable algebras. This caculation shows that it is enough to describe derivations in indecomposable evolution algebras. We first determine the derivation algebra of $n$-dimensional indecomposable associative evolution nilalgebras with one-dimensional annihilator. We describe the derivation algebra of indecomposable nilalgebras, up to dimension $6$, that are associative or not. In …
Given a $p$-adic field $K$ and a nilpotent uniform pro-$p$ group $G$, we prove that all primitive ideals in the $K$-rational Iwasawa algebra $KG$ are maximal, and can be reduced to a particular standard form. Setting $\mathcal{L}$ as the associated $\mathbb{Z}_p$-Lie algebra of $G$, our approach is to study the action of $KG$ on a Dixmier module $\widehat{D(λ)}$ over the affinoid envelope $\widehat{U(\mathcal{L})}_K$, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.
For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions $K$, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, $\widehat{U(\mathfrak{g})_{K}},$ is the annihilator of an affinoid highest weight module.
By representing elements in free fields (over a commutative field and a finite alphabet) using Cohn and Reutenauer's linear representations, we provide an algorithmic construction for the (partial) non-commutative (or Hausdorff-) derivative and show how it can be applied to the non-commutative version of the Newton iteration to find roots of matrix-valued rational equations.
Let $\mathbb C_Q$ denote a rational quantum torus with $d$ variables, and $\mathcal Z$ be the centre of $\mathbb C_Q$. In this paper we give a explicit description of the structure of the cuspidal modules for the derivation Lie algebra $\mathcal D$ over $\mathbb C_Q$, with an extra associative $\mathcal Z$-action.
Let $K$ be an algebraically closed field of characteristic zero, $A = K[x_1,\dots,x_n]$ the polynomial ring, $R = K(x_1,\dots,x_n)$ the field of rational functions, and let $W_n(K) = \Der_{K}A$ be the Lie algebra of all $K$-derivations on $A$. If $D \in W_n(K),$ $D\not =0$ is linear (i.e. of the form $D = \sum_{i,j=1}^n a_{ij}x_j \frac{\partial}{\partial x_i}$) we give a description of the centralizer of $D$ in $W_n(K)$ and point out an algorithm for finding generators of $C_{W_n(K)}(D)$ as a module over the ring of constants in case when $D$ is the basic Weitzenboeck derivation. In more gener…
We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computationa…
