Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.     

Characteristic modules and tensor products over quasi-hereditary algebras

Let A be a monomial quasi-hereditary algebra with a pure strong exact Borel subalgebra B.It is proved that the category of induced good modules over B is contained in the category of good modules over A;that the characteristic module of A is an induced module of that of B via the exact functor-(?)_B A if and only if the induced A-module of an injective B-module remains injective as a B-module.Moreover,it is shown that an exact Borel subalgebra of a basic quasi-hereditary serial algebra is right serial and that the characteristic module of a basic quasi-hereditary serial algebra is exactly the induced module of that of its exact Borel subalgebra.     

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.     

The localization of 1-cohomology of transitive Lie algebroids

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.     

三角矩阵上的Hochschild 2-循环广义若当导子

In this paper,we prove that every generalized Jordan derivation associate with a Hochschild 2-cocycle from the algebra of upper triangular matrices to its bimodule is the sum of a generalized derivation and an antiderivation.     

Bimodule and twisted representation of vertex operator algebras

In this paper, for a vertex operator algebra V with an automorphism g of order T, an admissible V-module M and a fixed nonnegative rational number n ∈1/T Z_+, we construct an A_(g,n)(V)-bimodule Ag,n(M) and study its properties, discuss the connections between bimodule A_(g,n)(M) and intertwining operators. Especially, bimodule A _(g,n)-1T(M) is a natural quotient of A_(g,n)(M) and there is a linear isomorphism between the space IM~k M Mjof intertwining operators and the space of homomorphisms HomA_(g,n)(V)(A_(g,n)(M)  A_(g,n)(V)M~j(s), M~k(t)) for s, t ≤ n, M~j, M~k are g-twisted V modules, if V is g-rational.     

广义幂级数代数的滤链维数

In this paper, we study the properties of generalized power series modules and the filtration dimensions of generalized power series algebras. We obtain that [[△S,≤]]- gfd([[AS,≤]]) =△-gfd(A) if A is an R-module where R is a perfect and coherent commutative algebra, and(R, ≤) is standardly stratified.     

The lower and upper bounds of the degree distribution of a graded algebra

For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.     

关于Malcev代数的幂零性

We investigate the nilpotence of a Malcev algebra M and of its standard enveloping Lie algebra L(M)=M D(M, M). The main result shows that an ideal A of M is nilpotent in M if and only if the corresponding ideal Ⅰ(A) = A D(A, M)is nilpotent in L(M).     

ON Δ-GOOD MODULE CATEGORIES OF QUASI-HEREDITARY ALGEBRAS

A useful reduction is presented to determine the finiteness of △-good module category F(△)of a quasi-heredltary algebra. As an application of the reduction, the f(△)-finitenetess of quasi-hereditary M-twisted double incidence algebras of posets is discussed. In particular, a complete classification of F(△)-finite M-twisted double incidence algebras is given in case the posets are linearly ordered.     

A Note on Restricted Representations of the Witt Superalgebras

Let F be an algebraically closed field of prime characteristic p 〉 3, and W（n） the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W（n） are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W（n） is proved to have one block.     

Products of Involutions in Steinberg Group over Skew Fields

Consider the stable Steinberg group St(K) over a skew field K. An element x is called an involution if x2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GLn(K) up to conjugation can be represented as BC, where B is lower triangular and C is simultaneously upper triangular. Furthermore, B and C can be chosen so that the elements in the main diagonal of B areβ1,β2, ...,βn, and of C areγ1,γ2,... ,γnCn, where cn∈[K*, K*] and = det A. It is also proved that every element 6 in St(K) is a product of 10 involutions.     

The closures of (U+K)-orbits of essentially normal triangular operator models

The (U + K)-orbit of a bounded linear operator T acting on a Hilbert space H is defined as (U + K)(T)={R-1 T R:R is invertible of the form unitary plus compact on H}.In this paper,we first characterize the closure of the (U + K)-orbit of an essentially normal triangular operator T satisfying H={ker(T-λI):λ∈ρ F (T)} and σ p (T*)=ф.After that,we establish certain essentially normal triangular operator models with the form of the direct sums of triangular operators,adjoint of triangular operators and normal operators,show that such operator models generate the same closed (U + K)-orbit if they have the same spectral picture,and describe the closures of the (U + K)-orbits of these operator models.These generalize some known results on the closures of (U + K)-orbits of essentially normal operators,and provide more positive cases to an open conjecture raised by Marcoux as Question 2 in his article "A survey of (U + K)-orbits".     

Derivations on the algebra of operators in hilbert C*-modules

Let M be a full Hilbert C*-module over a C*-algebra A,and let End*A(M) be the algebra of adjointable operators on M.We show that if A is unital and commutative,then every derivation of End A(M) is an inner derivation,and that if A is σ-unital and commutative,then innerness of derivations on "compact" operators completely decides innerness of derivations on End*A(M).If A is unital(no commutativity is assumed) such that every derivation of A is inner,then it is proved that every derivation of End*A(Ln(A)) is also inner,where Ln(A) denotes the direct sum of n copies of A.In addition,in case A is unital,commutative and there exist x0,y0 ∈ M such that x0,y0 = 1,we characterize the linear A-module homomorphisms on End*A(M) which behave like derivations when acting on zero products.     

群Yetter-Drinfel''d范畴中的Hopf代数

In this note we first show that if H is a finite-dimensional Hopf algebra in a group Yetter-Drinfel＇d category L^LyD（π） over a crossed Hopf group-coalgebra L, then its dual H^＊ is also a Hopf algebra in the category L^LyD（π）. Then we establish the fundamental theorem of Hopf modules for H in the category L^LyD（π）.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Strongly Irreducible Submodules of Modules

Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.     

Selfinjective Koszul algebras of finite complexity

In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G（l, m）, the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL（2, C） over the exterior algebra of a 2-dimensional vector space.     

Inverse limits in representations of a restricted Lie algebra

Let(g,[p]) be a restricted Lie algebra over an algebraically closed field of characteristic p 0.Then the inverse limits of "higher" reduced enveloping algebras {uχs(g)|s∈N} with χ running over g* make representations of g split into different "blocks".In this paper,we study such an infinitedimensional algebra Aχ(g):= ■Uχs(g) for a given χ∈g*.A module category equivalence is built between subcategories of U(g)-mod and Aχ(g)-mod.In the case of reductive Lie algebras,(quasi) generalized baby Verma modules and their properties are described.Furthermore,the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined,and a higher reciprocity in the case of regular nilpotent is obtained,generalizing the ordinary reciprocity.     

全矩阵代数的抛物子代数上具有可导性的非线性映射

Let F be a field of characteristic 0, Mn（F） the full matrix algebra over F, t the subalgebra of Mn（F） consisting of all upper triangular matrices. Any subalgebra of Mn（F） containing t is called a parabolic subalgebra of Mn（F）. Let P be a parabolic subalgebra of Mn（F）. A map φ on P is said to satisfy derivability if φ（x·y） = φ（x）·y＋x·φ（y） for all x,y ∈ P, where φ is not necessarily linear. Note that a map satisfying derivability on P is not necessarily a derivation on P. In this paper, we prove that a map φ on P satisfies derivability if and only if φ is a sum of an inner derivation and an additive quasi-derivation on P. In particular, any derivation of parabolic subalgebras of Mn（F） is an inner derivation.