ON CAUCHY-POMPEIU FORMULA FOR FUNCTIONS WITH VALUES IN A UNIVERSAL CLIFFORD ALGEBRA

This paper obtains the Cauchy-Pompeiu formula on certain distinguished boundary for functions with values in a universal Clifford algebra. This formula is just an extension of the Cauchy's integral formula obtained in [11].

Induced Modules of Restricted Lie Superalgebras

In this paper we first prove the PBW theorem for reduced universal enveloping algebras of restricted Lie superalgebras. Then the notion of an induced module is introduced and the dimension formula of induced modules is established. Finally, using the results above, we obtain a property of induced modules pertaining to automorphisms of Lie superalgebras and isomorphisms of modules.

DG Poisson algebra and its universal enveloping algebra

We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We show that A^(ue) has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^(ue). Furthermore, we prove that the notion of universal enveloping algebra A^(ue) is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.

Universal Enveloping Algebras of Simple Symplectic Anti-Jordan Triple Systems

In this work we axe concerned with the universal associative envelope of a finite-dimensional simple symplectic anti-Jordan triple system （AJTS）. We prove that if T is a triple system as above, then there exists an associative algebra U（T） and an injective homomorphism ε ： T→ U（T）, where U（T） is an AJTS under the triple product defined by （a, b, c） = abc- cba. Moreover, U（T） is a universal object with respect to such homomorphisms. We explicitly determine the PBW-basis of U（T）, the center Z（U（T）） and the Gelfand-Kirillov dimension of U（T）.

Realizing Enveloping Algebras via Varieties of Modules

By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L（A） generated by indecomposable constructible sets in the varieties of modules for any finite- dimensional C-algebra A. We obtain a geometric realization of the universal enveloping algebra R（A） of L（A）, this generalizes the main result of Riedtmann. We also obtain Green＇s formula in a geometric form for any finite-dimensional C-algebra A and use it to give the comultiplication formula in R（A）.

Cluster automorphism groups of cluster algebras with coefficients

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra(i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients,cluster algebras with universal geometric coefficients, and cluster algebras from surfaces(except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.

Realization of Poisson enveloping algebra

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.

Actads

In this paper,I introduce a new generalization of the concept of an operad,further generalizing the concept of an opetope introduced by Baez and Dolan(1998),who used this for the definition of their version of non-strict n-categories.Opetopes arise from iterating a certain construction on operads called the+-construction,starting with monoids.The first step gives rise to plain operads,i.e.,operads without symmetries.The permutation axiom in a symmetric operad,however,is an additional structure resulting from permutations of variables,independent of the structure of a monoid.Even though we can apply the+-construction to symmetric operads,there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structures on the previous levels.Defining and investigating these structures is the main purpose of this paper.The structures obtained in this way are what I call n-actads.In n-actads with n>1,the permutations on the different levels give rise to a certain special kind of n-fold category.I also explore the concept of iterated algebras over an n-actad(generalizing an algebra and a module over an operad),and various types of iterated units.I give some examples of algebras over 2-actads,and show how they can be used to construct certain new interesting homotopy types of operads.I also discuss a connection between actads and ordinal notation.

Frobenius bimodules and flat-dominant dimensions

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules.This is motivated by the Nakayama conjecture and an approach of MartinezVilla to the Auslander-Reiten conjecture on stable equivalences.We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions.Furthermore,let A and B be finite-dimensional algebras over a field k,and let domdim(_AX)stand for the dominant dimension of an A-module X.If_BM_A is a Frobenius bimodule,then domdim(A)domdim(_BM)and domdim(B)domdim(_AHom_B(M,B)).In particular,if B■A is a left-split(or right-split)Frobenius extension,then domdim(A)=domdim(B).These results are applied to calculate flat-dominant dimensions of a number of algebras:shew group algebras,stably equivalent algebras,trivial extensions and Markov extensions.We also prove that the universal(quantised)enveloping algebras of semisimple Lie algebras are QF-3 rings in the sense of Morita.

Generalized Integral Representations for Functions with Values in C(V_(3,3))

By using the solution to the Helmholtz equation u-λu = 0(λ≥ 0),the explicit forms of the so-called kernel functions and the higher order kernel functions are given.Then by the generalized Stokes formula,the integral representation formulas related with the Helmholtz operator for functions with values in C(V3,3) are obtained.As application of the integral representations,the maximum modulus theorem for function u which satisfies Hu = 0 is given.

Gelfand–Kirillov Dimensions of the Z^2-graded Oscillator Representations of sl(n)

We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl（n,F）-modules that appeared in the Z2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.Service E-mail this articleAdd to my bookshelfAdd to citation managerE-mail AlertRSSArticles by authors