Konrad Schrempf
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.
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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.
We prove that the group of automorphisms of the Lie algebra $\Der_K (Q_n)$ of derivations of the field of rational functions $Q_n=K(x_1,..., x_n)$ over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the $K$-algebra $Q_n$.
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.
A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric $m$-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are des…
Derivations and automorphisms of free nilpotent Lie algebras and their quotiens
本文通过研究自由幂零李代数的导子代数和自同构群，给出了同类型同幂零指数的一般幂零李代数的导子与自同构的自然刻画，并用例子加以说明。
It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.
We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for $2$-step nilpotent Lie algebras determined by graphs, free $2$ and $3$-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand we also exhibit famili…
We investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely generated free group are given as well.
Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable.
Let $\mathcal{R}$ be a commutative ring with identity, $I(X,\mathcal{R})$ be the incidence algebra of a locally finite pre-ordered set $X$. In this note, we characterise the derivations of $I(X,\mathcal{R})$ and prove that every Jordan derivation of $I(X,\mathcal{R})$ is a derivation provided that $\mathcal{R}$ is $2$-torsion free.
On the abelianization of the special derivation Lie algebras of free Lie algebras
本文通过Morita迹证明自由李代数特殊导数李代数的交换化包含无限多个线性无关元素，并展示其中存在被Morita迹杀死的非平凡元素。
In commutative algebra, if $δ$ is a locally nilpotent derivation of the polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ of characteristic 0 and $w$ is a nonzero element of the kernel of $δ$, then $Δ=wδ$ is also a locally nilpotent derivation with the same kernel as $δ$. In this paper we prove that the locally nilpotent derivation $Δ$ of the free associative algebra $K\langle X,Y\rangle$ is determined up to a multiplicative constant by its kernel. We show also that the kernel of $Δ$ is a free associative algebra and give an explicit set of its free generators.
$(σ, τ)$-Derivations of Number Rings with Coding Theory Applications
本文研究数环的$(σ, τ)$-导数，分析其在二次数域和bi-二次数环中的特性，并探讨其在编码理论中的应用。
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…
A nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is called a Weitzenboeck derivation. The classical theorem of Weitzenboeck states that the algebra of constants (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K. Now the algebra of const…
In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra $K[x_1,...,x_m]$ in several variables over a field $K$ of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over $K$ assuming that…
Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra Der A of all K-derivations of A is an A-module in a natural way and if R is the quotient field of A, then RDer A is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDer A of rank k over R (i.e. such that dim_{R}RL=k), then the derived length of L is at most k and L is finite dimensional over its field of constants. In case of solvable Lie algebras over a field of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of …
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…
By the classical theorem of Weitzenboeck the algebra of constants (i.e., the kernel) of a nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants of a locally nilpotent linear derivation of a d-generated relatively free algebra F(V) in a variety V of unitary associative algebras over K. It is known that the algebra of constants of F(V) is finitely generated if and only if V satisfies a polynomial identity which does not hold for t…
Twisted Derivations in Algebraic Number Fields
本文研究了代数数域中扭曲导数的分类，探讨了在特定数域扩展和整数环中，(σ, τ)-导数的性质及内在条件，同时提出关于非零外导数存在的猜想。
This paper solves the rational noncommutative analog of Hilbert's 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of hermitian matrices in its domain, then it is a sum of hermitian squares of noncommutative rational functions. This result is a generalization and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every m…
Derivations of Non-Commutative Group Algebras
本文研究二面体群、双循环群和半二面体群的非交换群代数导子，通过给出维数与显式基，在特征0、奇素数及素域代数扩张情形下完成导子分类。
Realizing free semicircular elements on the full Fock space, we prove an equivalence between rationality of operators obtained from them and finiteness of the rank of their commutators with right annihilation operators. This is an analogue of the result for the reduced $C^*$-algebra of the free group by G. Duchamp and C. Reutenauer which was extended by PA. Linnel to densely defined unbounded operators affiliated with the free group factor. Although their result was motivated from quantized calculus in noncommutative geometry, we state our results in terms of free probability theory.
Rational Normal Curves, Chip Firing and Free Resolutions
本文通过筹码点火博弈与 parcycle 图的组合关联研究有理正规曲线，为每个 Cohen-Macaulay 初始单项式理想显式构造 Gröbner 退化及其极小自由消解，并给出胞腔消解、Hilbert 级数公式和 Eagon-Northcott 复形的组合解释。
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