Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

Quasinilpotent Operators in Operator Lie Algebras

It is proved that the nilpotent Lie algebra generated by a family of decomposable operators generates an Engel- Banach algebra. We also proved that if a Lie algebra of quasinilpotent operators is essentially nilpotent, then the Banach algebra generated by this Lie algebra consists of quasinilpotent operators.     

平均场倒向重随机微分方程及其应用

研究了平均场倒向随重机微分方程, 得到了平均场倒向重随机微分方程解的存在唯一性.基于平均场倒向重随机微分方程的解, 给出了一类非局部随机偏微分方程解的概率解释.讨论了平均场倒向重随机系统的最优控制问题, 建立了庞特利亚金型的最大值原理.最后讨论了一个平均场倒向重随机线性二次最优控制问题, 展示了上述最大值原理的应用.     

扭Heisenberg-Virasoro代数上Hom-李代数结构

Hom-李代数是一类满足反对称和Hom-Jacobi等式的非结合代数.扭Heisenberg-Virasoro代数是次数不超过1的微分算子代数的中心扩张,它是一类重要的无限维李代数,与一些曲线的模空间有关.文章主要研究扭Heisenberg-Virasoro代数上Hom-李代数结构,确定了扭Heisenberg-Virasoro代数上存在非平凡的Hom-李代数结构.     

Hom-Hopf代数的对极和Drinfel'd偶

本文证明了带有等价Hom- 范畴的Hom- 双代数是Hom-Hopf 代数, 在基本结构定理成立的前提下, 构造了对极使其成为Hom-Hopf 代数. 研究了拟三角Hom-Hopf 代数的对极, 并给出了Hom-Hopf 代数的Radford 公式. 最后, 对有限维Hom-Hopf 代数, 引入了Drinfel''d 偶的概念, 同时证明了Drinfel''d 偶是拟三角Hom-Hopf 代数.     

关于蕴涵BCK—代数的伴随半群

我们证明了蕴涵BCK-代数的伴随半群是一个上半格;具有条件（s)的蕴涵BCK-代数的伴随半群是一个广义布尔代数。更进一步证明了有界蕴涵BCK-代数的伴随半群是一个布尔代数。     

(Quasi-)Poisson enveloping algebras

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.     

L-型模糊格蕴涵代数

This paper introduced the concept of L-fuzzy sub lattice implication algebra and discussed its properties. Proved that the intersection set of a family of L-fuzzy sub lattice implication algebras is a L-fuzzy sub lattice implication algebra, that a L-fuzzy sub set of a lattice implication algebra is a L-fuzzy sub lattice implication algebra if and only if its every cut set is a sub lattice implication algebra, and that the image and original image of a L-fuzzy sub lattice implication algebra under a lattice implication homomorphism are both L-fuzzy sub lattice implication algebras.     

Homotopy G-algebra structure on the cochain complex of hom-type algebras

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. We show that the Hochschild type cochain complex of a hom-associative algebra carries a homotopy G-algebra structure. As a consequence, we get a Gerstenhaber algebra structure on the cohomology of a hom-associative algebra. We also find similar results for hom-dialgebras.     

无限维Heisenberg代数的导子代数和全形

给出了无维Ｈｅｉｓｒｂｅｒｇ代数的全形和全形的导子代数Ｄｅｒ（ｓ）证明（ｓ）不是一个完备李代数，但Ｄｅｒ（ｓ）是一个完备李代数。     

The Lie Algebras in which Every Subspace s Its Subalgebra

In this paper, we study the Lie algebras in which every subspace is its subalgebra (denoted by HB Lie algebras). We get that a nonabelian Lie algebra is an HB Lie algebra if and only if it is isomorphic to g+Cidg, where g is an abelian Lie algebra. Moreover we show that the derivation algebra and the holomorph of a nonabelian HB Lie algebra are complete.     

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.     

单的完备Lie代数

给出了一个Ｈｅｉｓｅｎｂｅｒｇ代数与一个交换Ｌｉｅ代数的直和ｇ０的全形ｈ（ｇ０）和ｈ（ｇ０）的导子代数Ｄｅｒｈ（ｇ０）．证明了ｈ（ｇ０）不是一个完备Ｌｉｅ代数，但Ｄｅｒｈ（ｇ０）是一个单完备Ｌｉｅ代数．     

Modified Ringel-Hall algebras,naive lattice algebras and lattice algebras

For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.     

基于剩余格理论的P-有界分配格的理想集代数

讨论P-有界分配格的理想集代数与剩余格的关系.证明了在适当选取蕴涵算子及相应的剩余算子之后,P-有界分配格的理想集代数就成为剩余格.定义了生成理想,并借助格论上的原子定义了P-有界分配格,然后讨论了它的一些性质,得到了一些好的结论.最后证明了P-有界分配格的理想集代数也是MV代数与R0代数.     

HEYTING代数与FUZZY蕴涵代数

Heyting代数是作为直觉主义命题逻辑的代数模型而引进的Fuzzy蕴涵代数是 [0 ,1]值逻辑的蕴函联结词的一种代数抽象 .本文给出Heyting代数的若干基本性质 ,并证明了Heyting代数是Fuzzy蕴涵代数 ,也是Heyting型Fuzzy蕴涵代数。     

Structure of the Loday–Ronco Hopf algebra of trees

Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto–Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra.We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopf-algebraic structure. In the dual basis for the graded dual Hopf algebra, our formula for the coproduct gives an explicit isomorphism with a free associative algebra. We also obtain a transparent proof of its isomorphism with the non-commutative Connes–Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes–Kreimer Hopf algebra is related to symmetric functions.     

The Lie Algebras in which Every Subspace Is Its Subalgebra

In this paper, we study the Lie algebras in which every subspace is its subalgebra (denoted by HB Lie algebras). We get that a nonabelian Lie algebra is an HB Lie algebra if and only if it is isomorphic to $g\dot{+}\mathbb{C}id_g$, where $g$ is an abelian Lie algebra. Moreover we show that the derivation algebra and the holomorph of a nonabelian HB Lie algebra are complete.     

一类特殊的Koszul Calabi-Yau DG代数

假设一个连通上链DG代数A的基分次代数A~#或者同调分次代数H(A)是由一次元素x,y生成的代数kx,y/(xy+yx).本文证明A是Koszul Calabi-Yau DG代数.