First Cohomology Group from the Virasoro-Like Algebra to its Larsson Functor Module

Let 𝒲 be the Virasoro-like algebra, and d ₁, d ₂ be the degree derivations of 𝒲. Set ? = W⊕C d ₁⊕C d ₂. Let F ^α(V) be the ?-module defined by Larsson's functor applied to a finite dimensional sl ₂-module V. In this article, the derivations from ? to ?-modules F ^α(V) and the first cohomology group H ¹(?, F ^α(V)) are given.     

On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ?-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.     

q-类似Virasoro-like代数模的导子

记Lq为两个变量的量子环面上的斜导子李代数,当0≠q∈C为非单位时,Lq就是q-类似Virasoro-like代数.本文给出了文中构造的Lq的模上的导子及一上同调群H^1（Lq,M）.     

Induced Projective Modules Over Group von Neumann Algebras

Let G be a group and k a subring of the field of complex numbers. In this paper we study the additive map in reduced K-theory, which is associated with the inclusion of the group algebra kG into the group von Neumann algebra G, and obtain necessary and sufficient conditions for it to be identically zero (or zero modulo torsion). Our results complete the work of Eckmann [Comm. Math. Helvet. 71 (1996), 453–462; Arch. Math. 76 (2001), 241–249] and Schafer [K-theory 19 (2000), 211–217], while reducing to Swan’s theorem on induced representations [Ann. Math. 71 (1960), 552–578], in the case where the group G is finite. (Received: January 2005) Research supported by University of Athens grant Pythagoras 70/3/7298.     

Tilting Modules Over Quasi-Hereditary Nakayama Algebras #

The structure of Ringel duals of quasi-hereditary Nakayama algebras and of quasi-hereditary left serial algebras is described.     

On the Category of Koszul Modules of Complexity One of an Exterior Algebra

In this paper, we prove that there is a natural equivalence between the category F1（x） of Koszul modules of complexity 1 with filtration of given cyclic modules as the factor modules of an exterior algebra A = ∧V of an m-dimensional vector space, and the category of the finite-dimensional locally nilpotent modules of the polynomial algebra of m - 1 variables.     

Geometrical Interpretation of General Complex Algebra and its Application in Relativistic Mechanics

In a generalized formulation of the relativistic dynamics with internal conformation an important role is played by a quadratic polynomial, the coefficients and eigenvalues of which are generated by outer and inner momenta of the relativistic particle. This polynomial induces the general complex algebra, GC. In this paper we explore the geometrical and physical aspects of the evolution generated by the algebraic operations of the GC-algebra. It is shown that the geometrical image of the GC-number is given by a straight line passing through two given points in an euclidean plane. In this representation the straight line is characterized by a norm and an argument. The motions of the straight line are described by hyperbolic trigonometry which brings a correspondence between the Euclidean geometry and the hyperbolic one. It is proved that the evolution equation governed by the generator of the GC-algebra describes the energy conservation law of the relativistic particle. This evolution is depicted on the Euclidean plane as a rotational motion of the straight line, tangent to the circle with radius equal to the mass of the particle. In this way we come to new representation for the momenta in relativistic dynamics.     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

On the Generalized Enveloping Algebra of a Color Lie Algebra

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle , such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L. Supported by the EC project Liegrits MCRTN 505078.     

Bilinear Summation Formulas from Quantum Algebra Representations

The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q^–1, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of ₂₁-series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.     

Kac Determinant Formula for the q-deformed Virasoro Algebra of Hom-typ e

We investigate the highest weight representations of the q-deformed Virasoro algebra of Hom-type. In order to determine its unitarity and irreducible highest weight representations, we present its Kac determinant formula when q is nonzero and non-root of unity.     

Contact代数K(m,n)的不可约模的结构

利用广义限制李代数的概念和性质,研究Contact代数K(m,n)的不可约表示,给出了特征标高度大于等于2且小于P-3的不可约K(m,n)-模的结构.     

On Modular Standard Modules of Association Schemes

We will determine the structure of the modular standard modules of association schemes of class two. In the process, we will give the theoretical interpretation for the p-rank theory for strongly regular graphs, and understand the p-rank as the dimension of a submodule of the modular standard module. Considering the modular standard module, we can obtain the detailed classification more than the p-rank and the parameters.     

The Nilpotency Properties of the Leibniz Algebra Mn(C)D

We study the nilpotency properties of the Leibniz algebras constructed by means of D-mappings on the algebra of complex square matrices M _n (C). In particular, we obtain a criterion for nilpotency of these algebras in terms of the properties of a D-mapping. We prove also that the Leibniz algebras under consideration cannot be simple.     

On product expansions of generalized modular forms

We give examples of generalized modular forms with the property that their divisors are supported at the cusps and the exponents in their q-product expansions take infinitely many values.      

Inverse Spectral Transform for the q-Deformed Volterra Equation

We use the inverse spectral transform to study a q-deformation of the Volterra equation. The q-deformed time dependence of the spectral data is computed, and the one-soliton solution is explicitly constructed.     

Goldie Extending Modules

In this article, we define a module M to be 𝒢-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for 𝒢-extending modules and give a characterization of the Abelian groups which are 𝒢-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are 𝒢-extending. We prove that a minimal cogenerator for 𝒢od-R is 𝒢-extending, but not, in general, extending. It is also shown that if M is (𝒢-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.     

Linear Functional Equations in Banach Modules over a C *-Algebra

The paper is a survey on the Hyers–Ulam–Rassias stability of linear functional equations in Banach modules over a C ^*-algebra. Its contents is divided into the following sections: 1. Introduction; 2. Stability of the Cauchy functional equation in Banach modules; 3. Stability of the Jensen functional equation in Banach modules; 4. Stability of the Trif functional equation in Banach modules; 5. Stability of cyclic functional equations in Banach modules over a C ^*-algebra; 6. Stability of cyclic functional equations in Banach algebras and approximate algebra homomorphisms; 7. Stability of algebra *-homomorphisms between Banach *-algebras and applications.