Let K,S,D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Vieta formula and decompositions of differential operators. W-polynomials show up naturally, their connections with P-independency, Vandermonde and Wronskian matrices are briefly studied. The different linear factorizations of W-polynomials are analysed. Connections between the existence of LLCM of monic linear polynomials with coefficients in a ring and the left duo property are established at the end of the paper.
Let $S$ be a $3$-dimensional quantum polynomial algebra, and $f \in S_2$ a central regular element. The quotient algebra $A = S/(f)$ is called a noncommutative conic. For a noncommutative conic $A$, there is a finite dimensional algebra $C(A)$ which determines the singularity of $A$. In this paper, we mainly focus on a noncommutative conic such that its quadratic dual is commutative, which is equivalent to say, $S$ is determined by a symmetric regular superpotential. We classify these noncommutative conics up to isomorphism of the pairs $(S,f)$, and calculate the algebras $C(A)$.
Let $A$ be an algebra and let $f$ be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between $[A,A]$, the linear span of commutators in $A$, and span$f(A)$, the linear span of the image of $f$ in $A$. In particular, we show that $[A,A]=A$ implies span$f(A)=A$. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if $C$ is a commutative unital algebra over a field $F$ of characteristic $0$, $A$ is the matrix algebra $M_n(C)$, and the polynomial $f$ is neither an identity nor a central…
On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）
Noncommutative Rational Sums of Squares in Free Algebras
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, University of Chinese Academy of Science（中国科学院数学与系统科学研究院，数学科学学院国家重点实验室）
A representation theoretic study of noncommutative symmetric algebras
利用Minamoto的Fano代数理论，研究了（除环上）Van den Bergh的非交换对称代数$\mathbb{S}^{nc}(M)$.特别地，我们证明了$\mathbb{S}^{nc}(M)$是一致的，并且它的投影范畴$\mathbb{P}^{nc}(M)$等价于相应的双模类.这推广了\cite{minamoto}的主要定理，而\cite{minamoto}又是Beilinson导出等价的推广。作为推论，我们证明了$\mathbb{P}^{nc}(M)$是遗传的，并且有一个类似于$\mathbb{P}^1$的关于$\mathbb{P}^{nc}(M)$上层的结构定理.
The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any $P$-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.
