Hopf Algebra Dual to a Polynomial Algebra over a Commutative Ring

For a polynomial algebra in several variables over a commutative ring R with a Hopf algebra structure the existence of the dual Hopf algebra is proved.     

Differential Operators and the Steenrod Algebra

The Novikov-Landweber algebra and the Steenrod algebra are setup in terms of the primitive differential operators acting in the usual way on the integralpolynomial ring Z[x₁,... ,x_n,...]. A commutative wedge productV for differential operators is introduced and it is shown thatthe iterated wedge product is divisible by r! as an integral operator. The divided differentialoperator algebra D is generated over the integers by thedividedoperators under the wedge product. D is additively isomorphic to the abelian group ofsymmetric functions in the variables x_i. Furthermore D is closedunder composition of operators and admits a natural coproductwhich makes it a Hopf algebra in two ways, with respect to thecomposition and wedge products. Under composition D is isomorphicto the Landweber-Novikov algebra. A Hopf sub-algebra is generatedunder composition by the integral Steenrod squares and reduces mod 2 to the Steenrod algebra. An explicitproduct formula for two wedge expressions is developed and usedto derive Milnor's product formula for his basis elements inthe Steenrod algebra. The hit problem in the Steenrod algebrais reformulated in terms of partial differential operators.1991 Mathematics Subject Classification: 55S10.     

The Fusion Algebra of Bimodule Categories

We establish an algebra-isomorphism between the complexified Grothendieck ring of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of . As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.      

A Free Associative Algebra as a Free Module over a Specht Subalgebra

Let k be a field of characteristic 0 and let k<X> be a free associative algebra with finite basis X. Let R=R(k,X) be the universal enveloping algebra of the square of Lie(X), regarded as a subalgebra of k<X> and called the Specht subalgebra of the free algebra. We prove that k<X> is a free (left) R-module, find sufficient conditions for some system of elements in k<X> to be a basis for this module, and obtain an explicit formula that allows us to calculate the R-coefficients of the elements of the free algebra over a special basis of symmetric monomials.     

一类积—微分算子的离散本征值

本文讨论的积-微分算子是一类以众多应用领域为背景的无界、非自伴线性算子.在较一般的假设下,籍助L~2空间的线性算子理论,我们证明了这类算子存在离散本征值的充分条件,并获得了可供实际工作者参考的估计式.     

可除的四元数代数上多项式环的Grobner基

设F是一个特征不等于2的域,A是,上的一个可除代数。本文研究了A上多项式环A[x1,X2,…,xn]中理想是有限生成的,以及它的Grobner基;也表明F[x1,x2,…,xn]中有限子集G是F[x1,x2,…,xn]的Griobner基当且仅当G是A[x1,x2,…,xn]中的Grobner基。     

The Subalgebra of Finite Support in an Incidence Algebra

The automorphisms of the subalgebra of elements of finite support in the incidence algebra are described, and it is shown when this subalgebra is algebraically determined within the incidence algebra.     

Some Polynomial Systems Associated with a Certain Family of Differential Operators

The authors aim at presenting several (presumably new) classes of linear, bilinear, and mixed multilateral generating functions for some general systems of polynomials which are defined by means of a certain family of differential operators. Some of the generating functions considered here are associated with the Stirling numbers of the second kind. Many (known or new) consequences and applications of the results obtained in this paper are also indicated.     

Commuting Differential Operators of Rank 2 with Polynomial Coefficients

Self-adjoint commuting differential operators with polynomial coefficients are considered. These operators form a commutative subalgebra of the first Weyl algebra. New examples of commuting differential operators of rank 2 are found.     

Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra

This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type

\frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      15. Automorphisms Group of an Invariant Subalgebra of a Reductive Lie Algebra      K. N. Ponomarjev 《Acta Appl Math》2005,85(1-3):251-255 We prove that any automorphism of an invariant subalgebra of a reductive Lie algebra over a field of zero characteristic is a standard automorphism. Mathematics Subject Classifications (2000) 17B20, 17B40.      16. 辛代数的极大幂零子代数的保李括积零的线性映射      赵延霞  王登银  贾东芳 《数学研究及应用》2011,31(5):829-839 Let F be a field with char F = 2, l a maximal nilpotent subalgebra of the symplectic algebra sp（2m,F）. In this paper, we characterize linear maps of l which preserve zero Lie brackets in both directions. It is shown that for m ≥ 4, a map φ of l preserves zero Lie brackets in both directions if and only if φ = ψcσT0λαφdηf, where ψc,σT0,λα,φd,ηf are the standard maps preserving zero Lie brackets in both directions.      17. Critical Modules of the Ring of Differential Operators of an Affine Semigroup Algebra      Mutsumi Saito 《代数通讯》2013,41(2):618-631 Let D be the ring of differential operators of an affine semigroup algebra. Regarding the Krull dimension of finitely generated ? d -graded D-modules, we characterize critical ? d -graded D-modules. Moreover, we explicitly describe cyclic ones.      18. 一般扭算子李代数      李立  王焱 《数学的实践与认识》2016,(11):262-266 由扭算子构成的扭算子李代数在李代数理论中占有重要的位置,首先构造了一般形式的扭顶点算子Z~σ(E_(ij),α,β,z),然后给出了一般扭算子李代数g(G,l)[σ],研究了一般扭顶点算子所具有的性质.      19. 关于标准分层代数与它的多项式代数上的滤链维数          下载免费PDF全文 康娜  姚海楼 《数学研究及应用》2006,26(3):440-450 在本文中,我们研究了标准分层代数的Δ-好模滤链维数与它的多项式代数的Δ[x]-好模滤链维数,并得到了一些有趣的结果．      20. Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold      P. B. A. Lecomte  V. Yu. Ovsienko 《Compositio Mathematica》2000,124(1):95-110 Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .

Automorphisms Group of an Invariant Subalgebra of a Reductive Lie Algebra

We prove that any automorphism of an invariant subalgebra of a reductive Lie algebra over a field of zero characteristic is a standard automorphism. Mathematics Subject Classifications (2000) 17B20, 17B40.     

辛代数的极大幂零子代数的保李括积零的线性映射

Let F be a field with char F = 2, l a maximal nilpotent subalgebra of the symplectic algebra sp（2m,F）. In this paper, we characterize linear maps of l which preserve zero Lie brackets in both directions. It is shown that for m ≥ 4, a map φ of l preserves zero Lie brackets in both directions if and only if φ = ψcσT0λαφdηf, where ψc,σT0,λα,φd,ηf are the standard maps preserving zero Lie brackets in both directions.     

Critical Modules of the Ring of Differential Operators of an Affine Semigroup Algebra

Let D be the ring of differential operators of an affine semigroup algebra. Regarding the Krull dimension of finitely generated ? ^d -graded D-modules, we characterize critical ? ^d -graded D-modules. Moreover, we explicitly describe cyclic ones.     

一般扭算子李代数

由扭算子构成的扭算子李代数在李代数理论中占有重要的位置,首先构造了一般形式的扭顶点算子Z~σ(E_(ij),α,β,z),然后给出了一般扭算子李代数g(G,l)[σ],研究了一般扭顶点算子所具有的性质.     

关于标准分层代数与它的多项式代数上的滤链维数

在本文中,我们研究了标准分层代数的Δ-好模滤链维数与它的多项式代数的Δ[x]-好模滤链维数,并得到了一些有趣的结果．     

Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .