Non-commutative symmetric differences in orthomodular lattices

Cholesky decomposition of positive semidefinite matrices over commutative semirings

We prove that over a commutative semiring every symmetric strongly invertible matrix with nonnegative numerical range has a Cholesky decomposition.

Infinite symmetric products of rational algebras and spaces

We show that the infinite symmetric product of a connected graded-commutative algebra over the rationals is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over the rationals is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold-Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded commutative a…

Unitary and symmetric units of a commutative group algebra

Let F be the field of two elements and G a finite abelian 2-group with an involutory automorphism. The extension of this automorphism to the group algebra FG is called an involutory involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

辛 Leibniz 代数：辛 Lie 代数的一种非交换版本

Symplectic Leibniz algebras as a non-commutative version of symplectic Lie algebras

本文引入辛左/右 Leibniz 代数作为辛 Lie 代数的非交换推广，描述对称辛 Leibniz 代数，并利用双扩张与 $T^*$-扩张构造广泛例子。

Group rings with Lie metabelian set of symmetric elements

Let $R$ be a commutative ring of characteristic zero and $G$ an arbitrary group. In the present paper we classify the groups $G$ for which the set of symmetric elements with respect to the classical involution of the group ring $RG$ is Lie metabelian.

On Rota-Baxter operators of non-zero weight induced by non skew-symmetric solutions of the classical Yang-baxter equation on simple anticommutative algebras

Let $L$ be a simple anti-commutative algebra. In this paper we prove that a non skew-symmetric solution of the classical Yang-Baxter equation on $L$ with $L$-invariant symmetric part induces on $L$ a Rota-Baxter operator of a non-zero weight.

On symmetric units in group algebras

Let $U(KG)$ be the group of units of the group ring $KG$ of the group $G$ over a commutative ring $K$. The anti-automorphism $g\mapsto g\m1$ of $G$ can be extended linearly to an anti-automorphism $a\mapsto a^*$ of $KG$. Let $S_*(KG)=\{x\in U(KG) \mid x^*=x\}$ be the set of all symmetric units of $U(KG)$. We consider the following question: for which groups $G$ and commutative rings $K$ it is true that $S_*(KG)$ is a subgroup in $U(KG)$. We answer this question when either a) $G$ is torsion and $K$ is a commutative $G$-favourable integral domain of characteristic $p\geq 0$ or b) $G$ is non-tor…

Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables

We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various…

Serre duality for non-commutative P^1-bundles

Let E be a locally free, rank n bimodule over a smooth projective scheme X, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of the internal Hom functor. When X is a smooth projective variety, we use the Gorensteinness of A to prove a version of Serre duality on Proj A, the non-commutative P^1 bundle defined by A.

反莱布尼茨代数：模拟李代数的非交换版本

Anti-Leibniz algebras: A non-commutative version of mock-Lie algebra

本文提出反莱布尼茨代数作为模拟李代数的非交换推广，完成低维分类，并利用平均算子与嵌入张量构造出反结合二代数、反结合三代数及反莱布尼茨三代数等新结构。

二元运算的交换结合谱

The associative-commutative spectrum of a binary operation

本文开启二元运算不满足交换律和结合律的定量度量研究，提出交换结合谱概念，给出其一般结果、确定某些二元运算的该度量并提出未来研究问题。

Non-commutative Pieri operators on posets

We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu rela…

量子对称群的有限Gröbner基

Finite Gröbner bases for quantum symmetric groups

针对非交换双边理想Gröbner基未必有限的问题，研究给出了构造Wang量子对称群所用双边理想的有限既约Gröbner基的闭式描述，证明了量子对称群的字问题可判定。

Serre finiteness and Serre vanishing for non-commutative P^1-bundles

Suppose $X$ is a smooth projective scheme of finite type over a field $K$, $\mathcal{E}$ is a locally free ${\mathcal{O}}_{X}$-bimodule of rank 2, $\mathcal{A}$ is the non-commutative symmetric algebra generated by $\mathcal{E}$ and ${\sf Proj}\A$ is the corresponding non-commutative $\mathbb{P}^{1}$-bundle. We use the properties of the internal $\operatorname{Hom}$ functor $\HU(-,-)$ to prove versions of Serre finiteness and Serre vanishing for ${\sf Proj}\A$. As a corollary to Serre finiteness, we prove that ${\sf Proj}\A$ is Ext-finite. This fact is used in \cite{izu} to prove that if $X$ i…

Quantum isometry groups of symmetric groups

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of S_n on the choice of generators in the case n=3. We show that two different choices of generators lead to non-isomorphic quantum isometry groups which …